METASTABLE BUBBLE SOLUTIONS FOR THE ALLEN-CAHNEQUATION WITH MASS CONSERVATIONMichael J. Ward yDept. of MathematicsUniv. of British ColumbiaVancouver, British ColumbiaCanada V6T 1Z2AbstractIn a multi-dimensional domain, the slow motion behavior of internal layer solutions with spher-ical interfaces, referred to as bubble solutions, is analyzed for the nonlocal Allen-Cahn equationwith mass conservation. This problem represents the simplest model for the phase separation of abinary mixture in the presence of a mass constraint. The bubble is shown to drift exponentiallyslowly across the domain, without change of shape, towards the closest point on the boundary ofthe domain. An explicit ordinary di�erential equation for the motion of the center of the bubble isderived by extending, to a multi-dimensional setting, the asymptotic projection method developedpreviously by the author to treat metastable problems in one spatial dimension. An asymptoticformula for the time of collapse of the bubble against the boundary of the domain is derived interms of the principal radii of curvature of the boundary at the initial contact point. An analogybetween slow bubble motion and the classical exit problem for di�usion in a potential well is given.Key Words: Internal layers, phase separation, bubble solutions, exponentially small eigenvalues,solvability conditions, dynamic metastability.y This work was supported by the NSERC grant 5-815411

1. IntroductionOne of the simplest models for the phase separation of a binary mixture in the presence of amass constraint is the following constrained Allen-Cahn equation introduced in [24]:ut = �24u+ Q(u)� � ; x 2 D � RN ; (1:1a)@nu = 0 ; x 2 @D ; (1:1b)ZD u(x; t) dx = M : (1:1c)Here � � 1, � = �(t) is a Lagrange mutiplier `parameter' to be determined, D is a convexdomain, u is the concentration of one of the two species, and the mass M is constant. In (1:1a),Q(u) = �V 0(u), where V (u) is a double-well potential with wells of equal depth located at the twopreferred phases s+ > 0 and s� < 0 where V (s�) = 0. The non-local nature of (1:1) is re ected inthe fact that �(t) must satisfy �(t) = V �10 RD Q[u(x; t)] dx in order for the mass to be conserved.Here V0 is the volume of D.Starting from initial data, the solution to (1:1) quickly develops internal layers (or interfaces)of width O(�) that separate regions where u � s+ from regions where u � s� . A similar phaseseparation process, but possibly with very complicated dynamics, leads to the formation of apattern of internal layers for other phase transition models, including the Cahn-Hilliard equation([10]) and the viscous Cahn-Hilliard equation ([20]). In a one-dimensional domain, some recentstudies from various viewpoints (see [1], [5], [7], [11], [12], [13], [15], [17], [19], [21], [22], [23], [27]),have shown that the motion of a pattern of internal layers is exponentially slow (i. e. metastable).This metastable motion results from the exponentially weak interactions that occur between nearbylayers.The motion of internal layers in a multi-dimensional setting is usually very di�erent than inthe one-dimensional case. In a multi-dimensional domain, the interfacial motion is usually drivenby the curvature of the interface and exponentially weak interactions between neighboring layersare asymptotically negligible. A notable exception to this is the work of [25] for the Allen-Cahnequation with a tri-stable Q(u) where it was shown that a steady interfacial pattern can be formedin the neck of a dumb-bell shaped region through a balance between small interfacial curvatureand exponentially weak layer interactions.For (1:1) in a multi-dimensional domain, the asymptotic analysis of [24] for � ! 0 showsthat the normal velocity v of a single closed interface � is given by the volume preserving meancurvature ow v � �2 �� � 1j�j Z� � dS� : (1:2)Here �=2 is the mean curvature of � and j�j denotes the area of �. It was proved in [14], and shownnumerically in [9] for the case N = 2, that a closed convex interface evolving under (1:2) will tendto a spherical interface enclosing the same volume. In the case when several closed interfaces areformed from initial data, a coarsening process typically occurs, which can again ultimately lead toa single spherical interface. This process, studied in [24] for the case of non-overlapping spheresand in [8] for a collection of concentric radially symmetric interfaces, describes the growth of largerregions at the expense of smaller regions until, eventually, only a single closed interface remains.2

This surviving region then tends to a spherical shape under (1:2). A natural question then isto determine the subsequent evolution of a single spherical interface contained in the domain D.Since, from (1:2), v = 0 for such a spherical interface, the asymptotic analysis of [24] gives noindication on the nature of its motion.For the related Cahn-Hilliard equation, it has been proved in [2] and [3] using a dynamicalsystems approach that internal layer solutions with a spherical shape, which are referred to as bub-ble solutions, are metastable. The main qualitative feature is that the bubble drifts exponentiallyslowly across the domain without changing shape, while maintaining a constant radius to conservemass. The analysis in [2] and [3] is based on construcing an approximate invariant manifold, whichcaptures the slow dynamics, and then proving estimates for the attractivity of this manifold. Al-though this analysis proves the existence of slow moving bubble solutions, further work is neededto obtain an explicit ODE that quanti�es the bubble dynamics.The goal of this paper is to give an explicit asymptotic characterization of a similar expo-nentially slow bubble motion that occurs for the analytically more tractable problem (1:1). Ouranalysis for (1:1) is based on an extension to a multi-dimensional setting of the asymptotic projec-tion method developed in [27], [22] and [23] to treat metastable problems in one spatial dimension.The outline of our analysis is as follows. In x2 the method of matched asymptotic expansionsis used to construct a canonical bubble solution to the equilibrium problem for (1:1a) in all ofRN . In x3 the eigenvalue problem associated with the linearization of (1:1a; b) about the canonicalbubble solution is analyzed asymptotically. The analysis of this problem extends the results in[2] by providing explicit asymptotic formulas for the behavior of certain eigenfunctions on theboundary of the domain. These estimates, which are central to an explicit determination of thebubble dynamics, are derived using a boundary layer analysis. In x4 these estimates are usedtogether with the projection method to derive an explicit ODE for the slow motion of the centerx0 = x0(t) of a bubble. The projection method is based on linearizing (1:1) about a canonicalbubble solution where the center x0(t) is to be determined. By imposing solvability conditions forthe linearized problem that must hold in the limit �! 0, an ODE for x0(t) is obtained. In x5 andx6 we study this ODE and determine its unstable equilibrium solution for the cases N = 2 andN � 2, respectively. A plot of the layer structure for a bubble solution is shown in Fig. 1.When N = 3 our main asymptotic result can be summarized roughly as follows (see x6):Suppose that the bubble is inside D at t = 0 and let x0(0) = x00 2 D � RN . Assume that att = 0 there exists a unique point x(�0) 2 @D closest to x00. Then, for exponentially long times, themotion of the center of the bubble is in the direction of x(�0) � x00. Moreover, the distance rm(t)between x(�0) and x0(t) satis�es the asymptotic ODE_rm � ��r�1m �2 (1� rm=R1)�1=2 (1� rm=R2)�1=2 e�2��+��1(rm�rb) : (1:3)Here R1 and R2, with Ri > rm for i = 1; 2, are the principal radii of curvature of @D at x(�0) andrb is the bubble radius. Moreover, � and ��+ are certain constants, which depend weakly on �, thatcan be calculated asymptotically for a given Q(u).The asymptotic analysis leading to (1:3) is valid only when the bubble is strictly containedwithin D (i.e. rm > rb). In x7, we o�er some speculations on the motion that occurs after the3

bubble collapses against @D. In x7 we also make an analogy between the dynamics (1:3) anda similar behavior that occurs in the classical exit time problem (see [18]) for the motion of aBrownian particle con�ned in a domain by a potential well.2. The Canonical Bubble SolutionIn the limit � ! 0, we construct an equilibrium solution to (1:1a) in RN that has radialsymmetry and that has exactly one internal layer centered at some r = rb. Thus, we are todetermine functions �b(�) and Ub(r; �), called the canonical bubble solution, which satisfy�24Ub + Q(Ub) = �b ; 0 < r <1 ; U 0b > 0 ; (2:1a)Ub(rb; �) = 0 ; Ub(r; �)! S�(�) ; as ��1(r� rb)! �1 : (2:1b)In (2:1a), Q(u) has exactly three zeroes on the interval [s�; s+] located at u = s� < 0, u = 0, andu = s+ > 0 withQ0(s�) < 0 ; Q0(0) > 0 ; V (s+) = 0 ; V (u) = � Z us� Q(�) d� : (2:1c)In (2:1b), S�(�) are the asymptotic states, de�ned as the roots ofQ [S�(�)] = �b(�) ; (2:1d)for which S�(�)! s� and �b(�)! 0 as �! 0. The internal layer location rb is de�ned uniquely byUb(rb; �) = 0. Using the method of matched asymptotic expansions, we now construct the solutionto (2:1) in three di�erent regions: in the inner internal layer region where r� rb = O(�) and in thetwo outer regions outside and inside the bubble where r > rb and r < rb, respectively.2.1 The Internal Layer RegionIn this region we introduce � = ��1(r� rb) and ub(�; �) = Ub(rb + ��; �). Then, (2:1) becomesu00b + � (N � 1)rb + �� u0b +Q(ub) = �b ; �1 < � < 1 ; (2:2a)ub(0; �) = 0 ; ub(�; �)! S�(�) ; as �! �1 : (2:2b)To leading order as � ! 0, the solution to (2:2) is simply a planar internal layer pro�le. Speci�cally,for �! 0, we have that �b(�)! 0, S�(�)! s� and ub(�; �)! u0(�), where u0(�) satis�esu000 +Q(u0) = 0 ; �1 < � <1 ; u0(0) = 0 ; u00(�) > 0 ; (2:3a)u0(�) � s+ � a+e��+� ; �!1 ; u0(�) � s� + a�e��� ; �! �1 : (2:3b)The positive constants �� and a� in (2:3b) are de�ned by�� = h�Q0 (s�)i1=2 ; log a� = log (�s�) + Z s�0 ���[2V (�)]1=2 + 1� � s�! d� : (2:3c)Next, we expand the solution to (2:2) asub(�; �) � 1Xj=0 uj(�)�j ; �b(�) � 1Xj=1 �j�j : (2:4a)4

In addition, the asymptotic states S�(�) are expanded in terms of the unknowns uj(�1) asS�(�) � s� + 1Xj=1 uj(�1)�j : (2:4b)To derive an equation for uj we substitute (2:4) into (2:2) and (2:1d). For some functions Gj =Gj(u0; ::; uj�1) and gj� = gj�(�1; ::; �j�1), we obtain that uj , for j � 1, satis�esLuj � u00j + Q0(u0)uj = �j +Gj(u0; ::; uj�1) ; �1 < � <1 ; (2:5a)uj(�)! ��j��2� + gj�(�1; ::; �j�1) ; as �! �1 ; uj(0) = 0 : (2:5b)The �rst few functions Gj and gj� are given explicitly byG1 = �(N � 1)rb u00 ; G2 = �u212 Q00(u0)� (N � 1)rb �u01 � �rb u00� ; (2:6a)g1� = 0 ; g2� = 12�21��6� Q00(s�) : (2:6b)From (2:3a) and (2:5a), it follows that u00(�) satis�es Lu00 = 0 and u00(�1) = 0. Therefore,the right side of (2:5a) must satisfy a solvability condition. From Green's identity and (2:3b), thiscondition determines �j as �j = �1(s+ � s�) Z 1�1 u00Gj(u0; ::; uj�1) d� : (2:7)With �j given by (2:7), we can evaluate uj(�1) in (2:5b). The solution to (2:5a) can then becalculated up to a multiple of u00(�). The condition uj(0) = 0 �xes the multiple of u00 and thereforedetermines uj uniquely. Such a systematic procedure can be used to calculate all the terms �j anduj for j � 1. Therefore, in principle, we can obtain full asymptotic expansions for ub(�; �), �b(�)and S�(�).The solvability condition for (2:5) with j = 1 yields�1 = �(N � 1)(s+ � s�)rb ; where � � Z 1�1 hu00(�)i2 d� = p2 Z s+s� [V (u)]1=2 du : (2:8)The function u1(�) is given byu1(�) = �1u00(�) Z ��1 (u0(�)� s�)[u00(�)]2 d� � (N � 1)rb u00(�) Z ��1 f(�)[u00(�)]2 d� +D1u00(�) : (2:9)Here f(�) = R ��1[u00(z)]2 dz and D1 is to be chosen so that u1(0) = 0. In terms of u0 and u1, thesolvability condition for (2:5) with j = 2 determines �2 as�2 = (N � 1)(s+ � s�)r2b �rb Z 1�1 u01(�)u00(�) d�� Z 1�1 �[u00(�)]2 d��+ 1(s+ � s�) Z 1�1 12[u1(�)]2Q00 [u0(�)]u00(�) d� : (2:10)5

Further coe�cients are tedious to evaluate explicitly.For some constants Aj�, it can be shown from (2:5) that ub(�; �) has the far �eld behaviorub(�; �) � S�(�) + � 1Xj=0 �j�jAj�� e���� ; as �! �1 : (2:11)Here A0� = �a� and A1� = �(N � 1)a�=(2rb). A three-term asymptotic expansion for S�(�) isS�(�) = s� � ��1��2� + �2 �12�21��6� Q00(s�)� �2��2� � + O(�3) : (2:12)Since (2:11) is not uniformly valid for � = O(��1), we must construct outer solutions bothinside and outside the bubble. This non-uniform behavior arises from the fact that the curvatureterm in (2:2a) is O(1) when � = O(��1) and that the far �eld linearization of (2:2a) is done abouts� rather than S�(�).2.2 The Outer SolutionsIn the region outside the bubble for r > rb, we write Ub(r; �) = S+(�) + u+(r; �), whereu+ � S+. Then, by linearizing (2:1a) about S+(�), we obtain that u+(r; �) satis�esu00+ + (N � 1)r u0+ � ���1��+�2 u+ = 0 ; r > rb ; u+ ! 0 as r !1 : (2:13)Here ��+ = ��Q0 [S+(�)]�1=2. From (2:12) it follows that��+ = �+ �1 + ��12�4+Q00(s+) + O(�2)� ; as �! 0 : (2:14)The exact solution to (2:13) isu+(r; �) = C+ (r=rb)1�N=2 Km ���+��1r� ; m = (N � 2)=2 : (2:15)Here Km(z), with Km(z) � (�=2z)1=2 e�z as z !1, is the modi�ed Bessel function of the secondkind of order m. The constant C+ in (2:15) is to be found by matching (2:15) with (2:11). Tomatch these terms we introduce an intermediate variable r� de�ned by r = rb + �(�)r�, where�� �(�)� 1 as �! 0. Then, in terms of r�, (2:15) and (2:11) become to leading orderu+ � C+�1=2 �2��+��1rb��1=2 e���+��1rb e���+��1�(�)r� ; (2:16a)Ub � S+(�) � �a+e��+��1�(�)r� : (2:16b)Since ��+ � �+ = O(�), then �(�)��1r�(��+ � �+)! 0 as �! 0. Therefore, we can match the rightsides of (2:16a) and (2:16b) to obtainC+ � �a+��1=2 �2��+��1rb�1=2 e��+��1rb : (2:17)For r > rb, the outer solution is Ub(r; �) = S+(�) + u+(r; �), where u+ is given in (2:15).6

We can determine the outer solution inside the bubble, where 0 � r < rb, in a similar way.Substituting Ub(r; �) = S�(�) + u�(r; �) into (2:1a), where u� � S�, we obtain an equation foru�(r; �). The solution to this equation that is regular at r = 0, and which matches to (2:11) isu�(r; �) = C� (r=rb)1�N=2 Im ������1r� ; C� � a� �2������1rb�1=2 e������1rb : (2:18)Here Im(z) is the modi�ed Bessel function of the �rst kind of order m = (N � 2)=2. In (2:18), thedecay constant ��� is de�ned by ��� = ��Q0 [S�(�)]�1=2, and has the expansion��� = �� �1 + ��12�4�Q00(s�) + O(�2)� ; as �! 0 : (2:19)We summarize our results for the canonical bubble solution by using the large argumentexpansions of Km(z) and Im(z) to simplify (2:15) and (2:18). This yields,Ub(r; �) � S+(�)� a+ (r=rb)(1�N)=2 e���+��1(r�rb) ; r > rb ; (2:20a)Ub(r; �) = ub(�; �) � 1Xj=0 uj(�)�j ; � = ��1(r � rb) = O(1) ; (2:20b)Ub(r; �) � S�(�) + a� (r=rb)(1�N)=2 e������1(rb�r) ; 0 � r < rb : (2:20c)The functions uj(�) for j � 1 and the coe�cients in the asymptotic expansions for S�(�) and �b(�),written in (2:4), can be obtained from (2:5). A few of these terms are given explicitly in (2:8),(2:9), (2:10) and (2:12).2.3 Comparison of Asymptotic and Numerical ResultsWe now compare our asymptotic results for �b and ��+ with corresponding numerical resultscomputed from the full problem (2:2) using the boundary value solver COLSYS [4]. Since COLSYSallows for nonlinear boundary conditions and for interior point constraints, the problem (2:2) canbe readily solved using this package by re-writing it as a �rst order system for the 5 unknowns ub,u0b, S� and �b. We then truncate (2:2) to a �nite domain j�j < L by imposing ub(�L) = S�. Wechose L = 14 in the computations below. To obtain numerical solutions for increasing values of �,we used a continuation strategy starting from the planar interface solution.The computations were done for the following two forms of Q(u):Q = Qo(u) � 2(u� u3) ; (2:21a)Q = Qa(u) � u(u+ 1)(c0 � u)(c1 � u) ; c1 = c0 + (3 + c0 � 2c20)5(c0 � 1) ; c0 = 1:2 : (2:21b)The heteroclinic orbit constants ��, a�, s� and � for u0(�), obtained from (2:3) and (2:8), areQ = Qo =) �� = 2 ; a� = 2 ; s� = �1 ; � = 4=3 ; (2:22a)Q = Qa =) �+ = 1:8668 ; �� = 2:7828 ; a+ = 1:8000 ; a� = 2:2101 ;s+ = 1:2 ; s� = �1:0 ; � = 1:9044 : (2:22b)7

Using the values in (2:22), we can evaluate the asymptotic result for ��+, given in (2:14), and theleading order asymptotic result �b � ��1, where �1 is given in (2:8).When Q = Qo(u), rb = 1 and N = 2, in Fig. 2 we plot the numerical solution ub(�; �) to (2:2)at two values of �. For this form of Q(u) and with rb = 1, in Fig. 3 and Fig. 4 we compare thenumerically computed values of ��+ and �b with the corresponding asymptotic results (2:14) and�b � ��1. For N = 2 and N = 3, these �gures show that the asymptotic results provide a closedetermination of the corresponding numerical results for the range 0 < � < :15. In Tables 1a and1b we show a similar agreement between the asymptotic and numerical values of ��+ and �b for theform Q = Qa(u) with rb = 1 and c0 = 1:2.3. Spectral Estimates for the Linearized ProblemWe now study the spectral properties associated with linearizing the �nite domain problem(1:1a; b) about the canonical bubble solution Ub(r; �), which is de�ned in all of RN . The eigenvalueproblem associated with this linearization is�24�+Q0 [Ub(r; �)]� = �� ; x 2 D ; (3:1a)@n� = 0 ; x 2 @D ; ��; �� = 1 : (3:1b)Here �u; v� � RD uv dx and r = jx� x0j, where x0 is the location of the center of the bubble. Theeigenvalues and eigenfunctions of (3:1) are labeled by �j and �j , for j = 0; 1; ::, with �j ! �1 asj !1.In the analysis below we assume that the bubble is strictly inside D so that the distance from@D to the internal layer region is O(1) (see Fig. 1). With this assumption, the method of matchedasymptotic expansions is used to asymptotically calculate those eigenpairs �j , �j for which �j ! 0as � ! 0. The analysis is rather similar to that in x2, with the exception that we must insert aboundary layer term for �j near @D in order to satisfy the boundary condition (3:1b) exactly. Thisboundary layer analysis, which gives explicit asymptotic estimates for certain �j on @D, is validonly when D is convex. These spectral results are then used in x4.3.1 The Principal Eigenpair �0 and �0The principal eigenfunction �0 is radially symmetric in r except in an O(�) region near @D.In the internal layer region we set � = ��1(r � rb) and �0(�; �) = �0(rb + ��). Since, from(2:20b), Ub(r; �) �P1j=0 uj(�)�j in this region, the coe�cient Q0(Ub) in (3:1a) has the expansionQ0(Ub) = Q00 + (�u1 + �2u2)Q000 + �22 u21Q0000 + � � � ; as �! 0 : (3:2)Here we have de�ned Q00 � Q0(u0), Q000 � Q00(u0), etc. We then expand �0(�; �) and �0(�) as�0(�; �) � 1Xj=0�0j(�)�j ; �0(�) � 1Xj=0 �0j�j : (3:3)Substituting (3:2) and (3:3) into (3:1a), and collecting powers of �, we obtain, on �1 < � < 1,that L�00 = �00�00 ; (3:4a)8

L�01 = �u1�00Q000 � (N � 1)rb �000 + �01�00 + �00�01 ; (3:4b)L�02 = �(N � 1)rb �001 + (N � 1)�r2b �000 � u1�01Q000 � u2�00Q000 � 12u21�00Q0000+ �00�02 + �01�01 + �02�00 : (3:4c)Here L is de�ned by Lv � v00 + Q00v and we require that �0i(�)! 0 as �! �1.Since Lu00 = 0, with u00(�1) = 0, the system (3:4) must satisfy the solvability condition thatL�0i, for i = 0; 1; 2, is orthogonal to u00 with respect to the inner product u; v� � R1�1 uv d�.For (3:4a) this condition yields �00 = 0. Thus �00 = R0u00, where R0 is a normalization constant.Substituting �00 = R0u00 and �00 = 0 into (3:4b), we then re-write the right side of (3:4b) byexplicitly using the equation (2:5a) (with j = 1) satis�ed by u1. This yieldsL�01 = R0Lu01 + �01R0u00 : (3:5)Since Lu01; u00� = u01; Lu00� = 0, the solvability condition for (3:5) gives �01 = 0. Therefore,�01 = R0u01. The arbitrary multiple of u00, which can be added to �01, can be chosen to be zero.Substituting �0i = R0u0i and �0i = 0 for i = 1; 2 into (3:4c), and using (2:5a) with j = 2, we canre-write (3:4c) as L�02 = R0 �Lu02 � (N � 1)r2b u00� + �02R0u00 : (3:6)The solvability condition for (3:6) gives �02 = (N � 1)r�2b and, thus, �02 = R0u02. Therefore, theinner expansion for �0 is�0(�; �) = R0 hu00(�) + �u01(�) + �2u02(�) +O(�3)i ; (3:7)and the expansion of the eigenvalue is�0(�) = �2 (N � 1)r2b +O(�3) : (3:8)Although �0 is positive, and hence would normally lead to exponential growth, this mode is sup-pressed in x4 by the presence of the mass constraint.In the outer region for r < rb we let �0 � R0��(r; �). In this region Ub � S�(�), so that (3:1a)becomes�00� + (N � 1)r �0� � ���1~����2 �� = 0 ; 0 < r < rb ; ~��� = ��� �1 + �0=(���)2�1=2 : (3:9)Since �0 = O(�2) as � ! 0, it follows that ��� � ~��� = O(�2). The exact solution to (3:9), which isregular at r = 0, and which matches to (3:7) is��(r; �) = B� (r=rb)1�N=2 Im �~�����1r� ; B� � a��� �2������1rb�1=2 e�~�����1rb ; (3:10)where m = (N � 2)=2. 9

In the outer region for r > rb, but at anO(1) distance away from @D, we write �0 � R0�+(r; �).In this region Ub � S+(�), so that (3:1a) becomes�00+ + (N � 1)r �0+ � ���1~��+�2 �+ = 0 ; r > rb ; ~��+ = ��+ �1 + �0=(��+)2�1=2 : (3:11)Note that ~��+ � ��+ = O(�2). The exact solution to (3:11), which matches to (3:7), is�+(r; �) = B+ (r=rb)1�N=2 Km �~��+��1r� ; B+ � a+�+��1=2 �2��+��1rb�1=2 e~��+��1rb ; (3:12)where m = (N � 2)=2.Notice that �+ is exponentially small for r > rb, and has the following asymptotic estimateon @D:@n�+ = ~��+��1B+ (r=rb)1�N=2K 0m(~��+��1r) [r�n +O(�)] ; where r = jx� x0j ; x 2 @D : (3:13)Here n is the unit outward normal to @D, r = (x� x0)r�1 and r�n denotes the dot product. Theanalysis leading to (3:13) requires that D is convex.In x4 we need an asymptotic formula for �0 on @D. However, since �+ fails to exactly satisfythe boundary condition (3:1b), we cannot use �+ directly to obtain this formula. Instead, toestimate �0 on @D, we must �rst add a boundary layer term to �+ which is localized near @D. Torepresent �0 near @D, we introduce a local coordinate system de�ned near @D. We set � = n=�,where �n is the distance from x 2 D to @D, and we let � denote N � 1 coordinates orthogonal ton. In the region � = O(1), we write �0 = R0(�++ �L), where �L = �L(�; �) is the boundary layerfunction. Then, to leading order, we obtain from (3:1) that@���L � (~��+)2�L = 0 ; � < 0 ; �L ! 0 ; as � ! �1 ;@��L = �h(�) ; � = 0 ; h(�) � �[@n�+] ����=0 : (3:14)The solution to (3:14), which decays exponentially as � ! �1, is�L = �� h(�)=~��+� e~��+� : (3:15)From (3:13), (3:15) and �0 = R0(�+ + �L), we obtain in an O(�) neighborhood of @D that�0 � R0B+ (r=rb)1�N=2 �Km(~��+��1r)� e~��+n��1 �r�n K 0m(~��+��1r)� ����=0� : (3:16)Finally, using (3:12), the result ~��+ = ��+ + O(�2), and the large argument asymptotics of Km andK0m, we calculate on @D that�0 � R0a+�+ (r=rb)(1�N)=2 e���+��1(r�rb) [1 + r � n] ; r = jx� x0j ; x 2 @D : (3:17)We can summarize our results for �0 by using ~��� � ��� and the large argument expansions ofKm(z), Im(z) and K 0m(z) to simplify (3:10) and (3:12). This yields�0 � R0a��� (r=rb)(1�N)=2 e������1(rb�r) ; 0 � r < rb ; (3:18a)�0 � R0 hu00(�) + �u01(�) + �2u02(�) + � � �i ; � = ��1(r� rb) = O(1) ; (3:18b)�0 � R0a+�+ (r=rb)(1�N)=2 e���+��1(r�rb) ; r > rb ; n = O(1) : (3:18c)10

To normalize �0 by ��0; �0� = 1, we note that R�10 �0 is O(1) for r � rb = O(�), but isexponentially small elsewhere. Therefore, a Laplace-type evaluation of this inner product yields��0; �0� � � R20 N rN�1b Z 1�1[u0(�)0 ]2 d� ; =) R0 � �N �rN�1b ���1=2 : (3:19)Here N is the surface area of the unit ball in RN and � is de�ned in (2:8). A similar asymptoticevaluation provided ��0; 1� � �R0NrN�1b (s+ � s�) = O(�1=2) : (3:20)Finally, since �0 has no nodal lines it is clear that �0 must be the �rst eigenfunction. The principaleigenvalue �0(�) has the estimate (3:8).3.2 The Translation EigenfunctionsLet Ub(r; �) satisfy (2:1) with r = jx� x0j, and let xj and x0j denote the jth coordinate of xand x0, respectively. Then, by di�erentiating (2:1a) with respect to xj , we obtain for j = 1; ::; Nthat �24 �@xjUb(r; �)�+ Q0(Ub) �@xjUb(r; �)� = 0 : (3:21)By comparing (3:21) and (3:1a), it is clear that the eigenvalue problem (3:1a) in RN has N zeroeigenvalues with eigenfunctions �j = @xjUb(r; �) for j = 1; ::; N . In RN , these zero eigenvaluesare a consequence of the translation invariance. For the �nite domain problem, the bubble isassumed to be strictly inside D, and thus it follows from (2:20a) that @xjUb(r; �) fails to satisfy theboundary condition (3:1b) by only exponentially small terms. Therefore, it is natural to expectthat the translation eigenvalues and eigenfunctions associated with the in�nite domain problem getperturbed only very slightly by the presence of the �nite domain. Speci�cally, we show that (3:1)hasN exponentially small eigenvalues �j with corresponding eigenfunctions �j � Rj �@xjUb + �Lj�,for j = 1; ::; N . Here, Rj is a normalization constant and �Lj is a boundary layer function localizednear @D, which allows (3:1b) to be satis�ed. To estimate �j , we can use Green's identity appliedto (3:1a) and @xjUb to derive�j�@xjUb; �j� = ��2 Z@D �j @n �@xjUb� dS ; (3:22)where dS is the surface area element on @D. Substituting �j � Rj �@xjUb + �Lj� and (2:20a) into(3:22) we �nd, for j = 1; ::; N , that �j = O(�pe�C��1 ) for some p and C > 0. A more preciseestimate for �j is given below once we have determined �Lj .To calculate �Lj we set Ub � S+(�) and � = 0 in (3:1a). Then, near @D, we write (3:1) interms of the local (�; �) coordinate system introduced in x3.1. To leading order, �Lj satis�es@���Lj � (��+)2�Lj = 0 ; � < 0 ; �Lj ! 0 ; as � ! �1 ;@��Lj = �h(�) ; � = 0 ; h(�) � �@n[@xjUb] ����=0 : (3:23)By solving (3:23) we obtain �Lj and thus�j = Rj h@xjUb + ��h(�)=��+� e��+n��1i ; (3:24)11

where �n is the distance from x 2 D to @D.In x4 we require an asymptotic formula for �j on @D. To derive such a formula, we use (2:20a)to obtain on @D that@xjUb = ��+a+r�1��1 (r=rb)(1�N)=2 e���+��1(r�rb)[(xj � x0j) + O(�)] ; (3:25a)@n �@xjUb� = �(��+)2a+r�1��2 (r=rb)(1�N)=2 e���+��1(r�rb)(xj � x0j) [r�n+ O(�)] : (3:25b)Therefore, by substituting (3:25) and (2:20a) into (3:24) and by replacing ��+ by �+ in the pre-exponential factors, we obtain the following asymptotic formula for �j on @D for j = 1; ::; N :�j � Rj�+a+��1r�1 (r=rb)(1�N)=2 e���+��1(r�rb) (xj � x0j) [1 + r�n] ; x 2 @D : (3:26)In (3:25) and (3:26), r = jx� x0j, r = (x� x0)r�1, and n is the unit outward normal to @D.To estimate �j and Rj we need to evaluate �@xjUb; @xjUb� for � ! 0. From (2:20) it followsthat the dominant contribution to this integral arises from the region near r = rb. Therefore, using(2:20b) for Ub, we calculate that�@xjUb; @xjUb� � (�N)�1 rN�1b � N : (3:27)Here � is de�ned in (2:8) and N is the surface area of the unit ball in RN . Therefore, for �! 0,the normalization constant Rj satis�esRj � ��N= �rN�1b � N��1=2 : (3:28)Next, by substituting (3:25b), (3:26), (3:27) and �j � Rj@xjUb into (3:22), we obtain the followingasymptotic formula for the exponentially small eigenvalues �j for j = 1; ::; N :�j � a2+�3+N�N Z@D r1�Ne�2��+��1(r�rb) �xj � x0jr �2 r�n [1 + r�n] dS : (3:29)In x5 and x6 we use Laplace's method to asymptotically evaluate the surface integral in (3:29).However, from (3:29), it is immediately clear that, for some p, �j = O ��pe�2��+��1(rmax�rb)�. Herermax is the radius of the largest ball in RN , centered at x0, that can be inscribed within D.Finally, in x4, we need a rough asymptotic estimate for the inner product ��j ; 1�. Since, bysymmetry, the integral of �j vanishes identically on radially symmetric domains centered at x0, itfollows from the exponential decay of �j for r > rb that��j ; 1� = O ��pe���+��1(rmax�rb)� ; for j = 1; ::; N : (3:30)3.3 The Other Eigenpairs with �j ! 0 as �! 0For simplicity we consider only the two-dimensional case N = 2. In x3.1 and x3.2 we calculated�j and �j for j = 0; 1; 2. We now calculate the other eigenpairs for which �j ! 0 as � ! 0. Sincethese other eigenvalues are negative and since mass is conserved we are lead to the conclusion that12

the radially symmetric bubble solution is stable with respect to in�nitesmal perturbations havingzero mean.In the internal layer region, these eigenfunctions for m = 2; 3:: have the following form�2m�1 = �m[��1(r � rb); �] cos(m�) ; �2m = �m[��1(r � rb); �] sin(m�) : (3:31)Here (r; �) is a polar coordinate system centered at x0. The corresponding eigenvalues �2m�1(�)and �2m(�) can each be expanded in powers of �. The coe�cients in these expansions, which aredetermined from the internal layer region, are found to agree to all orders in �. Therefore, to withinall algebraic terms in �, we have �2m�1(�) � ��m(�) and �2m(�) � ��m(�). For an arbitrary domainthat is not concentric with r = rb, we expect that the gap width �2m(�)��2m�1(�) is exponentiallysmall as �! 0. However, we do not estimate the gap width here.Substituting � = ��m and (3:31) into (3:1a), we obtain on �1 < � <1, that �m(�; �) satis�es�00m + �rb + ���0m � �2m2(rb + ��)2�m +Q0(Ub)�m = ��m�m ; �m(�1) = 0 : (3:32)Here � = ��1(r � rb). Next, we expand ��m(�) and �m(�; �) as�m(�; �) � 1Xj=0 �mj(�)�j ; ��m(�) � 1Xj=0 ��mj�j : (3:33)Substituting (3:2) and (3:33) into (3:32), and collecting powers of �, we obtain that �mj(�), forj � 0, satis�es a system similar to (3:4). The system for �mj for j = 0; 1; 2 is obtained bysetting N = 2 in (3:4), replacing �0j and �0j in (3:4) by �mj and ��mj , and adding the new termm2r�2b �m0 to the right side of (3:4c). Therefore, by repeating the calculations in (3:5)� (3:8), wereadily obtain for m = 2; 3; :: that�m(�; �) � Rm hu00(�) + �u01(�) + �2u02(�) + O(�3)i ; (3:34a)�2m(�) � �2m�1(�) � ��m(�) = �2 (1�m2)r2b + O(�3) : (3:34b)The eigenfunctions are then given by combining (3:34a) and (3:31). The normalization constantRm, chosen so that ��2m; �2m� = 1, is given asymptotically byRm � (��rb�)�1=2 : (3:35)In a similar way as was done in x3.1, we can calculate the behavior of the eigenfunctions �2m�1and �2m in the two outer regions inside and outside the bubble. Since the results from such ananalysis are not needed later we do not carry out the details here. In x4 we only need an order ofmagnitude estimate for the inner products ��2m; 1� and ��2m�1 ; 1�. This estimate can be obtainedfrom the observations that these inner products vanish identically on radially symmetric domainscentered at x0, and the eigenfunctions are exponentially small in the outer regions where r > rb.Therefore, the estimate (3:30) also holds for these inner products. Such an estimate also holds13

for any eigenfunction �j of (3:1) in RN , with N � 2, that is not radially symmetric but that islocalized near r = rb and is exponentially small for r > rb.3.4 Comparison of Asymptotic and Numerical ResultsFor each of the two forms of Q(u) given in (2:21) we use COLSYS ([4]) to numerically computethe principal eigenvalue �0 as a function of �. When Q = Qo and rb = 1, in Fig. 5 we compare,for N = 2 and N = 3, the numerically computed �0 with the asymptotic result (3:8). This �gureshows the close agreement between the asymptotic and numerical results for �0 over the range0 < � < :20. For the case Q = Qa, rb = 1 and c0 = 1:2 in (2:21b), in Table 2a we show thefavorable agreement between the asymptotic and numerical values for �0 over the same range in �.For the form Q = Qo with rb = 1 and N = 2, in Table 2b we compare the asymptoticformula (3:34b) for �3 and �4 with corresponding full numerical results computed from (3:32)using COLSYS. The asymptotic results for these higher eigenvalues are again found to be in closeagreement with the numerical results.4. The Analysis of Slow Bubble Dynamics using the Projection MethodWe now consider the time-dependent problem (1:1) for u = u(x; t) and � = �(t), where themass M in (1:1c) is constant. We assume that the initial data for (1:1) is a canonical bubblesolution u(x; 0) = Ub[jx� x00j; �] with � = �b(�). Alternatively, we can view this initial data asrepresenting the end result of a coarsening process that has taken place between many bubbles.The bubble is assumed to lie entirely within D and, as shown in Appendix A, its radius rb can becalculated for �! 0 in terms of the mass M asrb = r0b + �r1b +O(�2) ; where r0b = �� NN ��V s+ �Ms+ � s� ��1=N : (4:1)Here V is the volume of D and the correction term r1b is given explicitly in (A:5a).We now look for a solution to (1:1) where the bubble is translated without change of shape.The projection method is used to determine the trajectory x0 = x0(t), with x0(0) = x00, of thecenter of the bubble. The analysis below is valid up until the edge of the bubble is at an O(�)distance from @D. touches @D.To use the projection method we �rst set u(x; t) = Ub[jx� x0j; �]+w(x; t) and �(t) = �b+�(t),where x0 = x0(t). Assuming that w � Ub and � � �b uniformly in time, we obtain the followinglinearized problem from (1:1):L�w � �24w +Q0(Ub)w = @tUb + �+ wt ; x 2 D ; (4:2a)@nw = �@nUb ; x 2 @D ; (4:2b)ZD wdx = 0 ; w(x; 0) = 0 : (4:2c)We then expand w =P1j=0 cj(t)�j , where �j for j � 0 are the normalized eigenfunctions of (3:1).Since r = jx� x0(t)j in (3:1a), these eigenfunctions now depend parametrically on time. ApplyingGreen's identity to (3:1) and (4:2a; b) we derive�wt; �j�� �w;L��j� = ��2 Z@D �j@nUb dS � �@tUb; �j�� ��1; �j� ; (4:3)14

where �u; v� � RD uv dx. Now since cj(t) = ��j ; w� and ��j ; �j� = 1, we calculate thatc0j � ��j ; wt� = �w; �jt� = 1Xk=0 ck��k ; �jt� = 1Xk=0k 6=j ck��k ; �jt� : (4:4)Next, substituting L��j = �j�j and (4:4) into (4:3), we �nd that cj(t) for j = 0; 1; ::, satis�esc0j � �jcj � 1Xk=0k 6=j ck��k ; �jt� = Fj ; cj(0) = 0 : (4:5a)Here Fj is de�ned by Fj = ��@tUb; �j�� ��1; �j�� Bj : (4:5b)In (4:5a), �j is an eigenvalue of (3:1) and we have de�ned the boundary term Bj byBj = �2 Z@D �j@nUb dS : (4:6)Using (2:20a) to calculate @nUb, we �ndBj � � a+��+ Z@D �j (r=rb)(1�N)=2 e���+��1(r�rb) r�n dS : (4:7)In addition, to satisfy the mass constraint in (4:2c) we require that1Xj=0 cj��j ; 1� = 0 : (4:8)Using the asymptotic estimates for �j and �j obtained in x3, we now derive N +1 consistencyconditions that determine the function � = �(t) and an evolution equation for the N -vectorx0(t). These conditions are that (4:8) is asymptotically satis�ed and that each cj(t) in (4:5a) isexponentially small over the exponentially long time interval induced by the exponentially smalleigenvalues �j for j = 1; ::; N . These two conditions are su�cient to ensure that w� Ub uniformlyin time. It is easy to show from (3:1) that ��k ; �jt� = ���j ; �kt� and that ��k ; �jt� is exponentiallysmall. Thus, the last term on the left side of (4:5a) corresponds to a skew-symmetric matrix. Sincesuch a matrix does not lead to exponential growth of cj(t) in (4:5a), this term is insigni�cant inthe derivation of the consistency conditions below.When the bubble is strictly inside D, we recall from x3 that �0 > 0 with �0 = O(�2), that�1,..,�N are exponentially small and that �j < 0 with �j = O(�2) for j � N + 1. Thus, in (4:5a),the only terms that can lead to growth in cj(t) are those corresponding j = 0; 1; ::N . In addition,by comparing (3:20) with (3:30) it follows that ��j ; 1� for j � 1 is exponentially smaller than��0; 1�. Therefore, to asymptotically satisfy the constraint (4:8) we require that c0(t) = 0 for alltime. Thus, to eliminate exponential growth for c0(t) on the time interval t = O(��10 ), we requirethat the right side of (4:5a) vanish when j = 0. Next, we observe from (3:26), (3:28) (3:29) and(4:7) that, although Bj is exponentially small, it has the same exponential estimate as that for the15

exponentially small eigenvalues. Therefore, unless the right side of (4:5a) vanishes for j = 1; ::; N ,we would obtain an O(1) response for cj(t) over the exponentially long time intervals induced by�1; ::; �N. This would then violate the assumption that w � Ub. Therefore, we must require thatthe right side of (4:5a) vanish for j = 1; ::; N . In summary, the N + 1 consistency conditions arethat �@tUb; �j� = ���1; �j�� Bj ; j = 0; ::; N : (4:9)Finally, notice that when j � N +1, the right side of (4:5a) is exponentially small and �j = O(�2)with �j < 0. Since cj(t) for j � N + 1 is decreasing in time, the conditions (4:9) are su�cient toensure that w is exponentially small uniformly in time.There are three main observations, which follows from the results in x3, that enable us toasymptotically decouple (4:9) into two separate sub-systems: one for j = 0 and the other forj = 1; ::; N . Firstly, from (3:17), (3:19), (3:26) and (3:28), we observe that Bj in (4:7) has the sameasymptotic order as � ! 0 for each j = 0; ::; N . Secondly, it follows from (3:20) and (3:30) that��0; 1� is exponentially larger than ��j ; 1� for j = 1; ::; N . Thirdly, if we choose the time scale sothat �@tUb; �j� for j = 1; ::; N has the same asymptotic order as Bj for j = 1; ::; N , then since �0is radially symmetric (except in a O(�) region near @D), it follows that �@tUb; �0� is exponentiallysmaller than B0. These three observations show that we can neglect the left side of (4:9) whenj = 0 and we can neglect the term �1; �j� for j = 1; ::; N . Therefore, using (4:7), we obtain thefollowing asymptotically decoupled problem for x0(t) and �(t):��1; �0� � �� a+��+ Z@D �0 (r=rb)(1�N)=2 e���+��1(r�rb) r�n dS ; (4:10a)�@tUb; �j� � �� a+��+ Z@D �j (r=rb)(1�N)=2 e���+��1(r�rb) r�n dS ; j = 1; ::; N : (4:10b)Next, we evaluate the various terms in (4:10). To evaluate �j on @D, we use the estimate(3:17) for j = 0 and the estimate (3:26) for j = 1; ::; N . The inner product ��0; 1� is estimated in(3:20). Finally, to calculate �@tUb; �j� for j = 1; ::; N we note that the dominant contribution tothis integral arises from the region near r = rb. Substituting @tUb = �U 0b r� _x0 and �j � Rj@xjUbinto this inner product, we obtain, for j = 1; ::; N , that�@tUb; �j� � �Rj _x0j ZRN [U 0b ]2r�2(xj � x0j)2 dx = �RjN _x0j ZRN [U 0b(r; �)]2rN�1 dr dN : (4:11)Here _x0j � dx0j=dt. Then, using (2:20b), (4:11) becomes�@tUb; �j� � �Rj _x0j (�N)�1 NrN�1b � ; j = 1; ::; N : (4:12)By substituting (3:17), (3:26), (3:20) and (4:12) into (4:10) we obtain the following slow motionresult:Proposition (Slow Motion Result): Assume that the bubble solution for (1:1) is strictly in-side of D and that D is convex. Then, the slow motion of the bubble is described by u(x; t) �Ub[jx� x0(t)j; �] and � � �b + �(t), where�(t) � � a2+�2+N(s+ � s�) Z@D r1�Ne�2��+��1(r�rb) [1 + r�n] r�n dS ; (4:13a)_x0 � �Na2+�2+N� Z@D r1�N e�2��+��1(r�rb) r [1 + r�n] r�n dS : (4:13b)16

In (4:13), r = r(�; t) � jx(�) � x0(t)j, r = r(�; t) � [x(�)� x0(t)]=r(�), n = n(�) is the unitoutward normal to @D, N is the surface area of the unit N -ball, and � = (�1; ::; �N�1) are thesurface coordinates that parametrize @D. In addition, rb, a+, �+ and ��+ are given in (4:1), (2:3c),(2:3c) and (2:14), respectively. The following result for the equilibrium problem is an immediateconsequence of (4:13):Corollary (Equilibrium Bubble Location): When D is convex, the equilibrium location x0 =xe0 for the center of the bubble satis�esZ@D r�Ne�2��+��1(r�rb) (xj � x0j) [1 + r�n] r�n dS = 0 ; j = 1; ::; N : (4:14)The equilibrium value of � is obtained by setting x0 = xe0 in (4:13a).The geometrical implications of (4:13) and (4:14) for the dynamics and the equilibria of thebubble solution are examined in x5 for the case N = 2 and in x6 for the case N � 2.5. Slow Bubble Motion in N = 2 DimensionsWe now examine the dynamics under (4:13b) when N = 2. In this case � is arclength along@D and thus dS = d� in (4:13). Suppose that at time t = 0, the distance r(�; 0) = jx(�)� x0(0)jis minimized at a unique point x(�0) 2 @D with arclength coordinate � = �0. In the limit � ! 0,we now show from (4:13b) that x(�0) remains the closest point on @D to x0(t) as t increases.To show this we need to introduce some notation. Let � = �m(t), with �m(0) = �0, denotethe arclength coordinate at time t where the minimum value of r(�; t) = jx(�) � x0(t)j over allx(�) 2 @D is attained. Assume that �m(t) is the unique such minimum point. We want to showthat _�m = 0. Let rm = rm(t) � r[�m(t); t] denote the minimum distance from x0(t) to @D at time tand let rm(t) denote the unit vector rm(t) = (x[�m(t)]� x0(t)) r�1m . For �m(t) to be the minimizer,the following necessary condition must holdx0 [�m(t)] � rm(t) = 0 : (5:1)Now, for �! 0, the dominant contribution to the integral in (4:13b) arises from the region � � �m(t)where r(�; t) is minimized. Since � is arclength, the Taylor expansion of r(�; t) near �m(t) isr(�; t) = rm + 12 �r�1m + �m� (� � �m)2 + � � � ; as � ! �m : (5:2)Here �m is the curvature of @D at � = �m(t). Since D is assumed to be convex, then �m � 0. Next,using (5:2) and the condition rm � n = 1 that holds at the minimum point, we can use Laplace'smethod on (4:13b) to obtain_x0 � �3=2� r�1=2m(1 + �mrm)1=2 e�2��+��1(rm�rb) rm ; as �! 0 : (5:3a)Here we have de�ned � by � = 2a2+�2+��1 ����+��1=2 : (5:3b)In deriving (5:3a) we have assumed that the strict inequality �mrm > �1 holds so that r�� > 0 at�m(t). 17

Since, from (5:3a), _x0 is in the direction of rm(t), it follows that _rm(t) < 0. Since this distanceis decreasing, the surface coordinate where r(�; t) is minimized cannot be a discontinuous functionof t. Therefore, we can assume that � = �m(t) is di�erentiable. Finally, to establish that _�m = 0,we �rst di�erentiate (5:1) with respect to t to get_�m (1 + �mrm) = x0(�m) � _x0 : (5:4)Then, from (5:1), (5:3a) and (5:4), it is clear that _�m = 0. Hence, �m(t) = �0 for t > 0. Finally,we can use Laplace's method on (4:13a) to calculate �(t). We summarize our result as follows:Proposition (Slow Bubble Motion for N = 2): Assume that at t = 0, x(�0) is the unique pointon @D which is closest to x0(0) = x00. Then, for � ! 0, the motion of the center of the bubble isin the direction of x(�0)� x00 for exponentially long times, and the distance rm(t) = jx(�0)� x0(t)jsatis�es the asymptotic ODE _rm � � �3=2� r�1=2m(1 + �mrm)1=2 e�2��+��1(rm�rb) : (5:5)In addition, �(t) satis�es� � � ���1=22(s+ � s�) r�1=2m(1 + �mrm)1=2 e�2��+��1(rm�rb) : (5:6)Here �m is the curvature of @D at �0 and � is de�ned in (5:3b).Using Laplace's method on (3:29), we can also calculate the exponentially small eigenvalues�j for j = 1; 2. We �nd that�j � ��1=2�+r�1=2m(1 + �mrm)1=2 hrm � iji2 e�2��+ ��1(rm�rb) ; j = 1; 2 : (5:7)Here ij is the unit basis vector in the xthj direction.The dynamics (5:5) is asymptotically valid provided that rm(t) > rb. When rm(t)�rb = O(�),the bubble begins to collapse against @D on a faster time-scale. We de�ne an approximate collapsetime tc by the condition rm(tc) = rb. Suppose that rm(0) = r0 > rb. From (5:5), tc is given bytc = ��3=2� Z r0rb z1=2(1 + �mz)1=2 e2��+��1(z�rb) dz : (5:8)We can then integrate (5:8) by parts twice to obtain, for �! 0, thattc � ��1=22���+ f(r0) �1� �4��+ � 1r0 + �m1 + �mr0�� e2��+��1(r0�rb) : (5:9)Here f(r0) � r1=20 (1+�mr0)1=2. To evaluate the exponentiated terms in (5:5) and (5:9) we use thetwo term expansions for ��+ and rb given in (2:14) and (A:4). These terms are signi�cantly moreaccurate than if evaluated them by using only the leading order behaviors ��+ � �+ and rb � r0b .18

5.1 Explicit Examples of Slow DynamicsTo generate convex domains we follow [16]. Let the origin be contained in D and let (x1; x2)be a point on @D. Let p denote the perpendicular distance from the origin to the tangent line to@D that passes through (x1; x2). Let � denote the angle between this perpendicular line and thepositive x1 axis. Then, when � ranges over 0 � � � 2�, we sweep out a closed domain D whoseboundary is given parametrically byx1(�) = p(�) cos(�) � p0(�) sin(�) ; x2(�) = p(�) sin(�) + p0(�) cos(�) : (5:10)The curvature �(�) of @D, the length L of @D, and the area V of D are given in terms of p(�) by�(�) = � hp(�) + p00(�)i�1 ; L = Z 2�0 p(�) d� ; V = 12 Z 2�0 �[p(�)]2 � [p0(�)]2� d� :(5:11)Thus given any p(�) with p(�) = p(�+2�), p(�) > 0 and p(�)+ p00 (�) > 0, we obtain the boundaryof a strictly convex domain (� < 0).We now illustrate our results for two choices of p(�) and for the two choices of Q(u) given in(2:21) (see (2:22) for some heteroclinic orbit constants for these Q(u)). For convenience, in theexamples below we give the radius rb and then calculate the mass M = M(�) in terms of rb using(A:2). To calculate M and the terms in (5:5) for each choice of Q(u), rb and �, we need to evaluate�1, ��+, � and �� given in (2:8), (2:14), (5:3b) and (A:3), respectively. When Q = Qo(u), rb = :75and � = :15, we use (2:21a) and (2:22a) to obtain�1 = 8=9 ; ��+ = 1:9 ; �� = log 2 ; � = 9:8233 : (5:12a)Alternatively, when Q = Qa(u), rb = :75 and � = :15, we �nd from (2:21b) and (2:22b) that�1 = 1:1542 ; ��+ = 1:8176 ; �+ = 0:7980 ; �� = 0:5173 ; � = 4:9625 : (5:12b)In the examples below, solutions to (5:5) were computed using the Sandia ODE solver [26]. Thedynamical re-scaling method of [27] was used to generate appropriate time steps.Example 5.1: Let p(�) = 3 + 0:4 sin3(�) � 0:5 cos2(�), Q = Qo(u), rb = :75 and � = :15. From(5:11) we obtain that V = 23:34 and from (5:12a), (2:22a) and (A:2) we calculate that M = 19:03.In Fig. 6 we show the motion of the center of the bubble starting from the three di�erent initialconditions labeled by Oi, for i = 1; 2; 3. The closest point on @D to Oi (labeled by ? in Fig. 6) iscomputed numerically and the curvature �m of @D at this closest point, which is needed in (5:5),is calculated from (5:11). The bubble then moves in the direction of the arrows and begins tocollapse against @D when its center is at Ci. In Fig. 7 we plot the function log10(1 + t) versusrm�rb for each of the three initial conditions. Notice that the bubble is essentially stationary overa very long time interval. For each i, in Table 3 we give the coordinates of the initial location Oiand we give the initial distance rm(0) to @D. We also show the favorable comparison between theasymptotic collapse time (5:9) and the corresponding numerical value computed from (5:5).Example 5.2: Let p(�) = 3 + 1:4 sin3(�), Q = Qa(u), rb = :75 and � = :15. From (5:11), (5:12b),(2:22b) and (A:2) we calculate that V = 26:74 and M = 26:72. In Fig. 8 we plot the motion19

of the center of the bubble for three di�erent initial conditions. The corresponding trajectorieslog10(1 + t) versus rm � rb are shown in Fig. 9. The coordinates of the initial conditions and acomparison of the asymptotic and numerical values for the collapse times are given in Table 4.5.2 The Equilibrium Bubble Location N = 2For � ! 0, we now use (4:14) to determine the center x0 = xe0 of the (unstable) equilibriumbubble solution. Suppose that the center xin of the largest inscribed circle B for D is uniquelyde�ned. In particular, this occurs when D is strictly convex (i.e. � < 0 on @D). In this case, weshow that xe0 is located at an O(�) distance from xin. In other words, xe0 is asymptotically close tothat point in D which is furthest from the boundary.Suppose that B is uniquely determined and that B is tangent to @D at exactly two pointsx(�1) 2 @D and x(�2) 2 @D with �1 6= �2. Let rin be the radius of B and let C be the chordjoining x(�1) and x(�2). We assume that the order of contact of B with @D is such that the strictinequality �irin > �1 holds for i = 1; 2. Here �i is the curvature of @D at � = �i. Let ni be theunit outward normal to @D at �i. Then, the following local conditions must hold at �1 and �2:n1 = �n2 ; (x(�i)� xin)jx(�i)� xinj � ni = 1 ; i = 1; 2 : (5:13a)Moreover, since B makes exactly two-point contact with @D, the following global condition mustbe satis�ed: jx(�)� xinj � rin > 0 ; 8 x(�) 2 @D ; with � 6= �1 ; � 6= �2 : (5:13b)From (5:13b) it is clear that if we set xe0 = xin in (4:14), the dominant contribution to theintegral in (4:14) would arise from the regions near �1 and �2. Therefore, by using Laplace's methodon (4:14) together with (5:13a), we �nd that (4:14) is asymptotically satis�ed when �1 = �2. Thus,in this case, the center of the equilibrium bubble solution coincides with the center xin of B. Anexample of this case is illustrated in Fig. 10 where we show the location of the equilibrium bubblesolution corresponding to the domain given in Fig. 6.When the curvatures at the two contact points are unequal, the bubble center xe0 still lies onC but is now at an O(�) distance from xin. Let ri = jx(�i) � xe0j and ri = (x(�i) � xe0)r�1i fori = 1; 2. Since xe0 2 C and jxe0 � xinj = O(�), we have ri � ni = 1, r1 = �r2 and r1 � r2 = O(�).These relations are used when applying Laplace's method to (4:14) to obtainr�1=21(1 + �1r1)1=2 e�2��+��1(r1�rb) � r�1=22(1 + �2r2)1=2 e�2��+��1(r2�rb) : (5:14)Since r1 = 2rin � r2, we conclude thatr1 � rin + �8��+ log�1 + �2rin1 + �1rin� ; r2 � rin � �8��+ log� 1 + �2rin1 + �1rin� : (5:15)Note that when j�1j > j�2j, then r1 > rin and r2 < rin. Thus, the center of the equilibrium bubblesolution is located on C at an O(�) distance from xin in the direction of the contact point wherethe magnitude of the curvature is smaller. 20

We now brie y consider the case when B makes exactly three-point contact with the boundaryof a strictly convex domain. Let xin be the unique location of the center of B. Then, by usingLaplace's method on (4:14), it is clear that (4:14) is asymptotically satis�ed when xe0 = xin providedthat the curvatures at the three contact points are equal and that the angles between adjacentline segments joining xin to the contact points are 120� apart. An example of this situation isillustrated in Fig. 11 where we show the location of the equilibrium bubble solution correspondingto the triangular shaped domain given in Fig. 6. In the more typical case when the angles are not120� apart or when the curvatures at the contact points are unequal, the center xe0 is shifted byan O(�) amount away from xin. A similar analysis as was given above for the two-point contactcase can be done to calculate this shift precisely. The result is as follows:Corollary (Three-Point Contact): Assume that B is uniquely de�ned and that B makes exactlythree-point contact with @D at x(�i) 2 @D for i = 1; 2; 3. Suppose also that �irin > �1 fori = 1; 2; 3, where �i � 0 is the curvaure of @D at x(�i). Then, x0 = x0(�) satis�esx0(�) = xin + �x10 + O(�2) ; (5:16)where x10 is the solution to the linear system(n3 � n1) � x10 = (2��+)�1 �log (A1=A3) + log ��n1 � t2=n3 � t2� ;(n1 � n2) � x10 = (2��+)�1 �log (A2=A1) + log ��n2 � t3=n1 � t3� : (5:17)Here Ai = (1 + �irin)�1=2 and ni and ti are the unit outward normal vector and the unit tangentvector at x(�i) 2 @D, respectively. A similar result is obtained in [28] for equilibrium spike-typesolutions to a class of reaction-di�usion equations in a multi-dimensional domain.Finally, when the domain is not strictly convex the center xin of B may not be uniquelydetermined. For instance, consider the rectangular domain jx1j < b1, jx2j < b2 with b1 > b2. Then,rin = b2 and xin is any point on the line segment x2 = 0 with jx1j < b1 � b2. In this degeneratecase, a simple application of Laplace's method on (4:14) shows only that the bubble must lie on thisline segment. However, by examining the subdominant contributions to the integral in (4:14) awayfrom the contact points it is clear that (4:14) will be asymptotically satis�ed only when xe0 = (0; 0).6. Slow Bubble Motion in N � 2 DimensionsWe now analyze (4:13) when N � 2. Let � = (�1; ::; �N�1) be a parameterization of theN � 1 dimensional surface @D. Suppose that at time t = 0, the distance r(�; 0) = jx(�) � x0(0)jis minimized at a unique point x(�0) 2 @D where � = �0. Then, as in the case N = 2, it can beshown that the motion of the center of the bubble, in the limit � ! 0, is in the direction of thevector x(�0)� x0(0) for t > 0.To derive an explicit ODE for the distance rm(t) between x(�0) and x0(t) we need to evaluatethe following surface integral asymptotically:IN�1 = Z@D r1�Ne�2��+��1(r�rb) [1 + r�n] r�n dS : (6:1)The dominant contribution for IN�1 arises from the region near � = �0. The Hessian of r(�; t) at �0can be diagonalized by choosing the parametrization of @D such that each �j , for j = 1; ::; N � 1,21

corresponds to arclength along one of the principal directions passing through �0. An applicationof the multi-dimensional Laplace's method (see [6]) then yieldsIN�1 � 2� ����+rm�(N�1)=2 H(rm) e�2��+��1(rm�rb) ; as �! 0 : (6:2)The function H(rm) is de�ned to be the product of N � 1 factors of the formH(rm) � (1� rm=R1)�1=2 (1� rm=R2)�1=2 : : : (1� rm=RN�1)�1=2 : (6:3)Here Rj � 0 for j = 1; ::; N � 1, are the principal radii of curvature of @D at x(�0). In obtaining(6:2) we have assumed that the non-degeneracy condition Rj > rm for j = 1; ::; N � 1 holds.Substituting (6:2) into (4:13) leads to the following explicit result.Proposition (Slow Bubble Motion N � 2:) Assume that at t = 0, x(�0) is the unique point on@D which is closest to x0(0) = x00. Then, for exponentially long times and for � ! 0, the motionof the center of the bubble is in the direction of x(�0) � x00 and the distance rm(t) satis�es theasymptotic ODE _rm � ��rm � �rm�(N+1)=2 H(rm) e�2��+��1(rm�rb) : (6:4)In addition, �(t) satis�es� � � ��N(s+ � s�) � �rm�(N�1)=2 H(rm) e�2��+��1(rm�rb) : (6:5)Here � is de�ned by � = 2Na2+�2+N� � ���+�(N�1)=2 : (6:6)When N = 2, the results agree with those in x5.Using Laplace's method on (3:29), we can estimate the exponentially small eigenvalues for�! 0 as �j � ��+ � �rm�(N�1)=2 H(rm) hrm � iji2 e�2��+��1(rm�rb) ; j = 1; ::; N : (6:7)Here ij is the unit basis vector in the xthj direction.For the ODE (6:4), suppose that rm(0) = r0 > rb. The collapse time tc is again de�ned byrm(tc) = rb. For �! 0, it is easy to show from (6:4) that tc satis�estc � �(1�N)=22���+ f(r0)"1� �4��+ N � 1r0 � N�1Xi=1 1(Ri � r0)!# e2��+��1(r0�rb) : (6:8)Here f(r0) � r(N�1)=20 [H(r0)]�1 and H(s) is de�ned in (6:3).Example 6.1 Set N = 3 and let D be an ellipsoid with boundary x21=32 + x22=42 + x23=22 = 1.Suppose that the center of a bubble of radius rb = :75 is initially located at the point (x1; x2; x3) =22

(0; 0; 0:25). Then, until the bubble collapses against @D, the motion of the center of the bubble istowards the closest point on @D, which is (0; 0; 2). The principal radii of curvature at (0; 0; 2) areR1 = 4:5 and R2 = 8:0. With rm(0) = 1:75 and Q = Qo(u), in Fig. 12 we plot the trajectorieslog10(1 + t) versus rm(t) � rb for three di�erent values of �. These trajectories were computednumerically from (6:4) using the dynamical re-scaling method of [27]. Numerical values for theparameters ��+ and � used in (6:4), which were calculated from (2:8), (2:14), (2:22a) and (6:6),are given in Table 5. In Table 5 we also show the favorable comparison between the asymptoticcollapse time (6:8) and the corresponding numerical result computed from (6:4).Finally, we remark that the determination of the equilibrium bubble center xe0 proceeds as inthe N = 2 case considered in x5.2. Suppose that the center xin of the largest sphere that can beinscribed within D is uniquely determined. Then, by using Laplace's method on (4:14) it followsthat xe0 is located at an O(�) distance away from xin. For instance, consider the case of two-pointcontact and let N = 3. Then, xe0 is located along the chord C joining the contact points. Moreover,in analogy with (5:15), we obtain from (4:14) and (6:2) thatr1 � rin + �8��+ log" 1� rin=R(2)11� rin=R(1)1 ! 1� rin=R(2)21� rin=R(1)2 !# ; r2 = 2rin � r1 : (6:9)Here R(i)j is the jth principal radius of curvature of @D at the contact point �i and r1 is the distancealong C from the contact point at �1 to the center xe0 of the equilibrium bubble solution.7. DiscussionAlthough the metastability results in x4-6 pertain only to the Allen-Cahn equation, we an-ticipate that a similar formal asymptotic analysis can be done to explicitly characterize the slowbubble motion for the Cahn-Hilliard equation and for the viscous Cahn-Hilliard equation, intro-duced in [20]. Such an analysis would complement the rigorous results of [2] and [3], which provedthe existence of slow bubble motion for the Cahn-Hilliard equation.Another topic for further investigation is to analyze the motion of the bubble once it hasbecome attached to the boundary. Recall that the results in x4-6 are valid only until the bubblebegins to collapse against @D. For the two-dimensional case, we now speculate on the qualitativebehavior of the subsequent motion. To satisfy the Neumann boundary condition and to minimizethe surface energy at a given time, the bubble interface must intersect @D at right angles andshould be asymptotically close to the arc of a circle that encloses the required mass. The bubbleshould then move on a fast time scale in the direction where the magnitude of the curvature isincreasing the most. This process decreases the surface energy and should continue until a localminimum of the surface energy is attained. Such a minimum presumably occurs near a region of@D where the magnitude of the curvature has a local maximum. If @D contains some segmentswhere the curvature is constant, the bubble motion along @D may become stuck and an asymptoticmetastability analysis may be required. It would be worthwhile to examine these issues in detail.Finally, we remark on an interesting comparison between the eigenvalue problem (3:1) and theeigenvalue problem that is associated with the exit time behavior for a Brownian particle con�nedin a domain D by a potential well V (see [18]). To determine the expected time for this particle to23

leave D, we must calculate for � ! 0, the principal eigenpair �0, �0 of the Fokker-Plank equation�4p+r � [prV ] = �p ; x 2 D � RN ; p = 0 ; x 2 @D : (7:1)Assume that V has the form V = V [jx� x0j] for some x0 2 D and that V (0) = 0 and V 0(r) > 0for r > 0. By setting p = e�V=(2�)� in (7:1), we obtain a problem very similar in form to (3:1)L�� � �4� + F (r; �)� = �� ; x 2 D ; � = 0 ; x 2 @D ; (7:2a)F (r; �) = ���14 [V 0(r)]2 + 12 hV 00(r) + (N � 1)r�1V 0(r)i ; r = jx� x0j : (7:2b)Now since ~�0 � e�V=(2�) satis�es L� ~�0 = 0, is of one sign, and fails to satisfy the boundarycondition by only exponentially small terms, it follows that �0 = ~�0 +�L, where �L is a boundarylayer function. Since V is radially symmetric about x0, the error made in satisfying the boundarycondition is maximized at that point xm 2 @D which is closest to x0. A boundary layer analysis for�0 then leads to the estimate �0 = O(�qe���1V (rm)) where rm = jxm�x0j (see [18]). Therefore, theparticle is most likely to exit @D at xm and the expected time for exit is O(��10 ). This estimate for�0 is qualitatively similar to the estimate (6:7) that was derived for the bubble solution. Therefore,although the number of exponentially small eigenvalues for (3:1) and (7:2) di�er, our conclusionthat the bubble will move in the direction of the closest point on @D over an exponentially longtime scale has a natural correspondence with similar behavior in the exit-time problem.AcknowledgmentsI am grateful to Prof. D. Austin, Prof. J. B. Keller and Prof. R. Miura for some valuablediscussions on the material in x5.2. I would also like to thank Dr. L. Reyna for his comments andProf. N. Alikakos for sending me his preprints.Appendix A: Calculating the Bubble RadiusThe radius rb of the canonical bubble solution Ub[jx� x0j; �] is determined asymptotically interms of the mass M by M � RD Ub dx. Let D+ and D� denote the subregions of D that areoutside (i. e. r > rb) and inside (i. e. r < rb) the bubble, respectively. To asymptotically evaluatethe mass constraint it is convenient to decompose it asM � (V � V�)S+(�) + V�S�(�) + ZD+ [Ub � S+(�)] dx+ ZD� [Ub � S�(�)] dx : (A:1)Here V is the volume of D, V� = rNb N=N is the volume of D�, N is the surface area of the unitN -ball and S�(�) is given in (2:12). Since the dominant contributions to the integrals in (A:1) arisefrom the region near r = rb, the integrals can be evaluated for � ! 0 using (2:20b). Substituting(2:20b) and (2:12) into (A:1), and retaining terms of order �, we obtain thatM � V s+ + V� (s� � s+) + � ��1V� ���2+ � ��2� �� �1��2+ V + NrN�1b (�� � �+)� : (A:2)Here �1 is de�ned in (2:8) and �� > 0 is de�ned by�+ = Z 10 [s+ � u0(�)] d� ; �� = Z 0�1 [u0(�)� s� ] d� ; (A:3)24

where u0(�) is given in (2:3). To obtain rb(�) in terms of M we expandrb = r0b + �r1b + O(�2) : (A:4)Substituting (A:4) into (A:2) and collecting powers of �, we �nd thatr0b = �� NN ��V s+ �Ms+ � s� ��1=N ; r1b = � r0b(s+ � s�)N ��01��2+ � VV 0�� + � ; (A:5a)where V 0� and are de�ned byV 0� = N�1N(r0b)N ; = �01(��2� � ��2+ ) + Nr0b (�+ � ��) : (A:5b)Here �01 is given by the right side of (2:8) upon replacing rb by r0b .REFERENCES[1] N. Alikakos, P. W. Bates, G. Fusco, Slow Motion for the Cahn-Hilliard Equation in One SpaceDimension, J. Di�er. Equations 90, (1991), pp. 81-135.[2] N. Alikakos, G. Fusco, Slow Dynamics for the Cahn-Hilliard Equation in Higher Spatial Di-mensions, Part 1: Spectral Estimates, Carr Reports in Mathematical Physics, University ofRome, (1993).[3] N. Alikakos, G. Fusco, Slow Dynamics for the Cahn-Hilliard Equation in Higher Spatial Di-mensions, Part 2: The Motion of Bubbles, preprint, (1993).[4] U. Ascher, R. Christiansen, R. Russell, Collocation Software for Boundary value ODE's,Math. Comp. 33, (1979), pp. 659-679.[5] P. W. Bates, J. P. Xun, Metastable Patterns for the Cahn-Hilliard Equation, J. Di�. Equations111, (1994), pp. 421-457.[6] N. Bleistein, Mathematical Methods For Wave Phenomena, Computer Science and AppliedMathematics Series, Academic Press, Orlando, (1984).[7] L. Bronsard, D. Hilhorst, On the Slow Dynamics for the Cahn-Hilliard Equation in One SpaceDimension, Proc. Roy. Soc. London A, Vol. 439, (1992), pp. 669-682.[8] L. Bronsard, B. Stoth, Volume Preserving Mean Curvature Flow as a Limit of a NonLo-cal Ginzburg Landau Equation, Carnegie Mellon University Research Report No. 94-NA-008,(1994).[9] L. Bronsard, B. Wetton, A Numerical Method for Tracking Curve Networks Moving withCurvature Motion, to appear, (1995), J. Comp. Phys.[10] J. Cahn, J. Hilliard, Free Energy of a Non-Uniform System. I. Interfacial Free Energy, J.Chem. Phys. 28, (1958), pp. 258-267.[11] J. Carr, R. Pego, Metastable Patterns in Solutions of ut = �2uxx � f(u), Comm. Pure Appl.Math. 42, (1989), pp. 523-576.[12] C. M. Elliot, D. A. French, Numerical Studies of the Cahn-Hilliard Equation for Phase Sepa-ration, IMA J. Appl. Math. Vol. 38, (1987), pp. 97-128.[13] G. Fusco, J. Hale, Slow Motion Manifold, Dormant Instability and Singular Perturbation, J.Dynamics and Di�. Equations 1, (1989), pp. 75-94.25

[14] M. Gage, On an Area-Preserving Evolution for Plane Curves, Contemp. Math. 51, (1986),pp. 51-62.[15] C. Grant, Slow Motion in One-Dimensional Cahn-Morrel Systems, SIAM J. Math. Anal.Vol. 26 No. 1, (1995), pp. 21-34.[16] C. C. Hsiung, A First Course in Di�erential Geometry, Wiley Interscience Series in Pure andApplied Mathematics, New York (1981).[17] M. Kuwamura, S. I. Ei, M. Mimura, Very Slow Dynamics for Some Reaction-Di�usion Systemsof the Activator-Inhibitor Type, Japan J. Indust. Appl. Math. 9, (1992), pp. 35-77.[18] B. J. Matkowsky, Z. Schuss, The Exit Problem for Randomly Perturbed Dynamical Systems,SIAM J. Appl. Math. Vol. 33 No. 2, (1977), pp. 365-382.[19] J. Neu, Unpublished Notes.[20] A. Novick-Cohen, On the Viscous Cahn-Hilliard Equation, in `Material Instabilities in Con-tinuum Mechanics and Related Mathematical Problems', (J. Ball ed.), Oxford Science Publi-cations, Clarendon Press, (1988), pp. 329-342.[21] T. Ohta, M. Mimura, Pattern Dynamics in Excitable Reaction-Di�usion Media, in `Formation,Dynamics and Statistics of Patterns', Vol. 1 (K. Kawasaki et al., eds.), World Scienti�c,(1990), pp. 55-112.[22] L. G. Reyna, M. J. Ward, Metastable Internal Layer Dynamics for the Viscous Cahn-HilliardEquation, to appear, Methods and Applications of Analysis (1995).[23] L. G. Reyna, M. J. Ward, Resolving Weak Internal Layer Interactions for the Ginzburg-LandauEquation, European J. Appl. Math. Vol. 5, Part 4, (1994), pp. 495-523.[24] J. Rubinstein, P. Sternberg, Nonlocal Reaction-Di�usion Equations and Nucleation, IMA J.Appl. Math. Vol. 48, (1992), pp. 249-264.[25] J. Rubinstein, P. Sternberg, J. B. Keller, Front Interaction and Nonhomogeneous Equilibriafor Tristable Reaction-Di�usion Equations, SIAM J. Appl. Math. Vol. 53 No. 6, (1993),pp. 1669-1685.[26] L. F. Shampine, M. K. Gordon, Computer Solution of Ordinary Di�erential Equations, theInitial Value Problem, W. H. Freeman publishers, San Fransisco (1975).[27] M. J. Ward, Metastable Patterns, Layer Collapses, and Coarsening for a One-DimensionalGinzburg-Landau Equation, Studies in Appl. Math. Vol. 91 No.1, (1994) pp. 51-93.[28] M. J. Ward, An Asymptotic Analysis of Localized Solutions for some Reaction-Di�usion Modelsin Multi-Dimensional Domains, submitted, Studies in Appl. Math 4/95 (20 pages).26

� ��+ (num.) ��+ (asy.) ��+ (num.) ��+ (asy.)0.025 1.8605 1.8606 1.8541 1.85450.075 1.8476 1.8483 1.8268 1.82990.125 1.8340 1.8360 1.7966 1.80530.151 1.8266 1.8296 1.7798 1.79250.175 1.8196 1.8237 1.7636 1.78070.199 1.8124 1.8178 1.7466 1.7689Table 1a: Comparison of asymptotic and numerical values for ��+ whenrb = 1 and Q = Qa(u) with c0 = 1:2 : The second and third columnsare for N = 2 and the fourth and �fth columns are for N = 3 :� �b(�) (num.) �b(�) (asy.) �b(�) (num.) �b(�) (asy.)0.025 0.02159 0.02164 0.04313 0.043280.075 0.0645 0.0649 0.1283 0.12980.125 0.1070 0.1082 0.2119 0.21640.151 0.1290 0.1307 0.2546 0.26140.175 0.1492 0.1515 0.2935 0.30300.199 0.1693 0.1723 0.3319 0.3445Table 1b: Comparison of asymptotic and numerical values for �b(�) whenrb = 1 and Q = Qa(u) with c0 = 1:2 : The second and third columnsare for N = 2 and the fourth and �fth columns are for N = 3 :� �0(�) (num.) �0(�) (asy.) �0(�) (num.) �0(�) (asy.)0.025 0.6234 �10�3 0.6250 �10�3 0.1247 �10�2 0.1250 �10�20.075 0.5591 �10�2 0.5625 �10�2 0.01117 0.011250.125 0.01552 0.01563 0.03094 0.031250.151 0.02265 0.02280 0.04511 0.045600.175 0.03046 0.03063 0.06057 0.061250.199 0.03945 0.03960 0.07832 0.07920Table 2a: Comparison of asymptotic and numerical values for �0(�) whenrb = 1 and Q = Qa(u) with c0 = 1:2 : The second and third columnsare for N = 2 and the fourth and �fth columns are for N = 3 :27

� �3(�) (num.) �3(�) (asy.) �4(�) (num.) �4(�) (asy.)0.025 -0.0018758 -0.0018750 -0.0050020 -0.0050000.075 -0.01694 -0.01688 -0.04516 -0.045000.125 -0.47351 -0.04688 -0.12620 -0.125000.151 -0.06941 -0.06840 -0.18487 -0.182410.175 -0.09368 -0.09187 -0.24925 -0.245000.199 -0.12179 -0.11880 -0.32351 -0.31681Table 2b: Comparison of asymptotic and numerical values for �3(�) and�4(�) when rb = 1 ; N = 2 ; and Q = Qo(u) :label i x01 x02 rm(0) tc (asy.) (5.9) tc (num.)1 0.5 0.5 1.92789 0.627320 �1012 0.627072 �10122 -0.5 -0.5 1.89187 0.151560 �1012 0.151096 �10123 -0.5 1.25 1.61271 0.206595 �109 0.206501 �109Table 3: For the data in Ex:5:1 we give the coordinates x01 ; x02 ; and the initial distancerm(0) for each initial condition in Fig. 6 and Fig. 7. The last two columns comparecompare the asymptotic collapse time (5.9) with the corresponding numerical result.label i x01 x02 rm(0) tc (asy.) (5.9) tc (num.)1 0.4 1.4 2.12714 0.516034 �1014 0.515910 �10142 -0.5 1.6 1.93534 0.482419 �1012 0.482292 �10123 0.5 0.5 2.06601 0.115260 �1014 0.115230 �1014Table 4: For the data in Ex:5:2 we give the coordinates x01 ; x02 ; and the initial distancerm(0) for each initial condition in Fig. 8 and Fig. 9. The last two columns comparethe asymptotic collapse time (5.9) with the corresponding numerical result.28

� ��+ � tc (asy.) (6.8) tc (num.)0.10 1.8667 9.6429 0.544797 �1016 0.544675 �10160.14 1.8133 9.9265 0.421828 �1011 0.421632 �10110.18 1.7600 10.227 0.571291 �108 0.570823 �108Table 5: For the data in Ex:6:1 we give the values of ��+ and � used in the numericalsolution to (6.4). The last two columns compare the asymptotic collapse time(6.8) with the corresponding numerical result.

29

O(�) x0u � s+ �D nr = rbu � s�@D- � ] *

Figure 1: A bubble of radius r = rb in a two-dimensional convex domain D.30

�1:0�0:50:00:51:0�8:0 �4:0 0:0 4:0 8:0

ub(�)�

Q = Qo(u)rb = 1N = 2Figure 2: Plot of the computed canonical bubble solution at two di�erent values of � when Q = Qo,rb = 1 and N = 2. The solid line is for � = :04 and the dotted line is for � = :16.

31

1:71:81:92:00:0 0:05 0:10 0:15 0:20

��+�Q = Qo(u)rb = 1

N = 2N = 3 �����Figure 3: Comparison of asymptotic and numerical values for ��+ for N = 2 and N = 3 whenQ = Qo and rb = 1. The solid (dotted) lines are the numerical (asymptotic) results.

32

0:00:050:100:150:200:250:0 0:05 0:10 0:15 0:20

�b(�)�

Q = Qo(u)rb = 1N = 2

N = 3oo

Figure 4: Comparison of asymptotic and numerical values for �b(�) for N = 2 and N = 3 whenQ = Qo and rb = 1. The solid (dotted) lines are the numerical (asymptotic) results.33

0:00:020:040:060:080:0 0:05 0:10 0:15 0:20

�0(�)�

Q = Qo(u)rb = 1N = 2N = 3BBMO

Figure 5: Comparison of asymptotic and numerical values for �0(�) for N = 2 and N = 3 whenQ = Qo and rb = 1. The solid (dotted) lines are the numerical (asymptotic) results.34

�2:0�1:00:01:02:03:0

�3:0 �2:0 �1:0 0:0 1:0 2:0 3:0y

x?

?? �o1 � �c1�o2��c2

�o3��c3 : :++kkQQ

Figure 6: For Q = Qo(u), � = :15 and rb = :75, we plot the motion of the center of a bubble solutionfor three di�erent initial conditions. Here @D is parametrized by p(�) = 3+0:4 sin3(�)�0:5 cos2(�).The center of the bubble is initially at Oi for i = 1; 2; 3. The center then moves in the directionof the arrows until the point Ci where the bubble begins to collapse against the wall. The closestpoint on @D to Oi is labeled by ?.35

0:02:04:06:08:010:012:014:0

0:0 0:2 0:4 0:6 0:8 1:0 1:2 1:4log10(t+ 1)

rm � rb o1

c1

o2c2

o3c3

��=

�

so

sI

Figure 7: Plots of log10(1+ t) versus rm(t)�rb are given for each of the trajectories shown in Fig. 6.The initial and end points for these trajectories are labeled in accordance with Fig. 6.36

�1:00:01:02:03:04:0

�3:0 �2:0 �1:0 0:0 1:0 2:0 3:0y

x??

?�o1 � �c1�o2��c2 �o3 ��c33 3kk W W

Figure 8: Same caption as for Fig. 6 except that now Q = Qa(u), � = :15, rb = :75, and @D isparametrized by p(�) = 3 + 1:4 sin3(�).37

0:02:04:06:08:010:012:014:016:0

0:0 0:2 0:4 0:6 0:8 1:0 1:2 1:4 1:6log10(t+ 1)

rm � rb o1

c1

o2c2

o3c3

��

=

~@@I

wk

Figure 9: Plots of log10(1+ t) versus rm(t)�rb are given for each of the trajectories shown in Fig. 8.The initial and end points for these trajectories are labeled in accordance with Fig. 8.38

�2:0�1:00:01:02:03:0

�3:0 �2:0 �1:0 0:0 1:0 2:0 3:0y

x� ??

Figure 10: For the data in Fig. 6 we show the largest inscribed circle for D (labeled by the dashedcircle), which is centered at the label �. The chord between the two tangent points of this circleto @D is shown. Since the curvature at the two tangent points is the same, the equilibrium bubblesolution (labeled by the solid circle) is also centered at the point �.39

�1:00:01:02:03:04:0

�3:0 �2:0 �1:0 0:0 1:0 2:0 3:0y

x� ???Figure 11: For the data in Fig. 8 we show the largest inscribed circle for D (labeled by the dashedcircle), which is centered at the label �. The equilibrium bubble solution (labeled by the solidcircle) is also centered at �.

40

0:02:04:06:08:010:012:014:016:018:0

0:0 0:2 0:4 0:6 0:8 1:0 1:2 1:4log10(t+ 1)

rm � rb� = 0:18� = 0:14� = 0:10

���

Figure 12: Plots of log10(1+t) versus rm(t)�rb are given for three di�erent values of � correspondingto a bubble of radius rb = 0:75 with Q = Qo(u) which is initially centered at the point (x1; x2; x3) =(0; 0; 0:25) inside the ellipsoid x21=32 + x22=42 + x23=22 = 1. The motion of the center of the bubbleis towards the point (0; 0; 2) and it begins to collapse against the boundary of the ellipsoid whenrm = rb.41