A doubly critical degenerate parabolic problem

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2004; 27:1619–1627 (DOI: 10.1002/mma.487)MOS subject classi�cation: 35K 55; 35K 65; 35B 33

A doubly critical degenerate parabolic problem

Michael Winkler∗;†

Lehrstuhl I f�ur Mathematik der RWTH Aachen; W�ullnerstr. 5-7; D-52056 Aachen; Germany

Communicated by M. Fila

SUMMARY

It is shown that the Dirichlet problem for

ut = up(�u+ u) in �× (0; T ); p¿0

where �⊂Rn is critical in that it has �rst eigenvalue one, is globally solvable for any continuouspositive initial datum vanishing at @�. Moreover, for p¡3 all solutions are bounded and tend to somenonnegative eigenfunction of the Laplacian as t → ∞, while if p¿3 then there are both bounded andunbounded solutions. Finally, it is shown that unlike the case p∈ [0; 1), all steady states are unstableif p¿1. Copyright ? 2004 John Wiley & Sons, Ltd.

KEY WORDS: degenerate di�usion; global existence

INTRODUCTION

Let us consider nonnegative solutions of the problem

ut = up�u+ up+1 in �× (0; T )u|@� = 0

u|t=0 = u0

(0.1)

where p¿0; �⊂Rn is a smooth bounded domain and u0 ∈C 0( ��) is positive in � andvanishes at @�.The PDE in (0.1) arises, particularly for p=2, in astrophysics as well as in di�erential

geometry; with di�erent source terms, it is frequently used in the modelling of �ows throughporous media (for p¡1) and in population dynamics (cf. References [1–4] and the refer-ences therein). Problem (0.1) marks the borderline case of the corresponding problem forut = up�u+ uq: For q¡p+ 1, this problem has a global bonded solution for each u0, while

∗Correspondence to: Michael Winkler, Lehrstuhl I f�ur Mathematik der RWTH Aachen, W�ullnerstr. 5-7, D-52056Aachen, Germany.

†E-mail: [email protected]

Published online 3 August 2004Copyright ? 2004 John Wiley & Sons, Ltd. Received 7 March 2003

1620 M. WINKLER

for q¿p + 1 there are both global solutions and solutions that blow up (i.e. become un-bounded) in �nite time [5,6]. As to the case of the critical source exponent q=p + 1 tobe considered here, it is fairly well-known (cf. References [1,5,7,8]) that concerning globalsolvability, the size of �, measured in terms of the principal eigenvalue �1 of −� in �, playsa crucial role; to be more precise,

• if � is large such that �1¡1 then all solutions blow up in �nite time, whereas• if � is small such that �1¿1 then each solution exists globally and tends to zero as

t → ∞.The exceptional case �1 = 1 thus marks another borderline between two regimes characterizedby totally di�erent behaviour of solutions, so that the competition ‘blow-up vs convergenceto zero’ seems to deserve a closer look for this critical domain size. To the best of ourknowledge, the only result concerning this question is formulated in Reference [5], where itwas shown that

• if �1 = 1 and u0=dist(x; @�) is bounded above and below by positive constants then sois u(t)=dist(x; @�) uniformly in t¿0 and hence the solution is global in time.

Moreover, denoting by � the principal eigenfunction with max� �=1, it is proved there thatif p¡3 then u approaches one of the steady states �� with �¿0 as t → ∞. However, thequestion what happens if u0 is either small or large near the boundary seems to be open sofar. Can one construct initial data steep enough near @� such that the solution will becomeunbounded in in�nite or even in �nite time? Are there data decaying rapidly enough near @�such that u will tend to zero as t → ∞?We shall see in Section 2 that �nite time blow-up does not occur, and that indeed p=3

appears as a critical exponent:

• If �1 = 1 then all solutions are global in time (Theorem 2.1).• If �1 = 1 and

◦ p¡3 then all solutions are bounded and moreover, u(t)→ �� as t → ∞ for some�¿0. The case �=0 indeed occurs (Theorem 2.4);

◦ p¿3 then (0.1) has both bounded and unbounded solutions, depending on the bound-ary behaviour of u0 (Theorem 2.5).

The occurrence of ‘many’ global unbounded solutions in presence of suciently strongdegeneracies is in sharp contrast to both the linear case (p=0) as well as some semilin-ear and weakly degenerate equations related to (0.1) with certain superlinear source terms(cf. References [9–11], for example). In such problems, global unbounded solutions wereproved to exist, if at all, only for some non-generic initial data.Apart from the questions mentioned so far, the present problem shows another interest-

ing feature which underlines the fundamental di�erence between degenerate equations of thegeneral type ut =f(u)(�u + g(u)) with f(0)=0 and f(s)¿0 for s¿0 on the one handand semi-linear equations where f≡ 1 on the other. Namely, problem (0.1) will reveal inSection 3 that, roughly speaking, steady states which are stable for ut =�u+ g(u) may beunstable for ut =f(u)(�u + g(u)). Even more drastically, we shall see that all equilibriaof (0.1) are unstable if p¿1, while each of them is stable for ut =�u + u. The fact thatthese steady states are also stable for (0.1) in the case p¡1 may furthermore be interpretedas a feature distinguishing weakly degenerate from strongly degenerate equations, where the

Copyright ? 2004 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2004; 27:1619–1627

CRITICAL PARABOLIC PROBLEM 1621

latter ones are characterized by the property that f(s) be small enough near s=0 such thatI :=

∫ 10 ds=f(s)=∞. Weakly degenerate equations (where I¡∞) can be transformed into

porous medium-type equations vt =��(v) + (v) (see Reference [8]).

1. PRELIMINARIES AND BASIC ESTIMATES

As usual in the context of degenerate parabolic problems, we attempt to obtain a solutionof (0.1) by suitably regularizing the equation. To be precise, let �= �i ↘ 0 as i→ ∞ andu0� ∈C1( ��) ful�ll u0�|@� = � and u0 + �=26u0�6u0 + 2� as well as u0� ↘ u0 as �↘ 0. Due tothe classical parabolic comparison principle (see Reference [5, p. 1987], for an appropriateversion), the problem

u�t = up� (�u� + u�) in �× (0; T�)

u�|@� = �

u�|t=0 = u0�

(1)

is actually non-degenerate (since �=2 is a subsolution) and thus is solvable in a maximumtime interval (0; T�); T� ∈ (0;∞]. Letting �→ 0, we invoke the results in Reference [8] tosuccessfully construct a solution of the original problem:

Lemma 1.1There is Tmax ∈ (0;∞] such that (0.1) has a unique positive classical solution u on �× (0; Tmax).u can be obtained as the monotonic and G0

loc( ��× [0; Tmax))∩C 2;1loc (�× (0; Tmax))-limit of the

solutions u� to (1).

ProofFor p¿1, the existence and approximation assertions are proved in Theorem 1.2.2 in Refer-ence [8], and the proof is easily adapted to the case p¡1. For uniqueness, see Theorem 1.2.4in Reference [8] or Theorem 4.2 in Reference [5].

The following semi-convexity estimate is well-known. For a more detailed proof see Ref-erence [8], Lemma 3.1.3 (cf. also Reference [3, p. 9]).

Lemma 1.2We have

ut

u¿− 1

ptin �× (0; Tmax) (2)

ProofTo prove (2) for u� instead of u, observe that z := u�t=u� satis�es z|@� =0 and zt =pz2 +up−1� (2u�xzx + u�zxx). Parabolic comparison of z with large-data solutions y(t) of y′=py2 on�× (�; T�) and taking � and then � to zero yields the claim.

The next observation is easily made but nevertheless of great importance in all that follows.


1622 M. WINKLER

Lemma 1.3If p =1 then ∫

�u1−p(t)�≡ const:∈ (0;∞]

If p=1 then ∫�ln u(t)�≡ const:∈ [−∞;∞)

ProofMultiplying (1) by u−p

� � and integrating over �× (t1; t2); 0¡t1¡t2¡T�, yields in the casep =1 ∫

�u1−p� (t2)�−

∫�u1−p� (t1)�= (1− p)�1�

∫��

Letting �↘ 0, the claim follows from the monotone convergence theorem. The proof for p=1is similar.

2. GLOBAL EXISTENCE OF SOLUTIONS AND LARGE TIME BEHAVIOUR

Let us return to the question whether our borderline case �1 = 1 inherits one of the featurescharacterizing the neighbouring cases �1¡1 or �1¿1, that is, the tendency towards eitherblow-up or convergence to zero. A �rst partial answer excludes �nite time blow-up.

Theorem 2.1The solution of (0.1) is global in time.

ProofIn view of the parabolic comparison principle, we may without loss of generality assume thatu0 is so large that

u0¿c0�� (3)

for some c0¿0, with � :=

{1 for p622p for p¿2

According to Reference [5], the problem

−�W =W 1−p in �W|@� = 0 (4)

has a (unique) positive solution, and it is shown there that

W¿c1�� in � (5)

for some c1¿0.



Let y(t) solve y′=−yp+1 on (0;∞); y(0)=c0=c1, and v(x; t):=y(t)W (x); (x; t)∈�× (0;∞).Then v¡u� on the parabolic boundary of �× (0; T�) and since vt−vp(�v+v)= (y′+yp+1)W−vp+160, we have v¡u� in �× (0; T�) and thus, letting �↘ 0,

u¿c(T )W in �× (0; T ) (6)

for any given T¿0.Now suppose u blew up at some �nite time T , i.e. ‖u(tk)‖L∞(�) → ∞ for some sequence

of times tk ↗T . We claim that then for a subsequence, leaving indices unchanged,

u(tk)→ ∞ a:e: in � (7)

Indeed, let q(x; t) := u(x; t)=m(t) with m(t) := ‖u(t)‖L∞(�). By (2),

−�q6q+1

ptm(t)u1−p (8)

Multiplying by (q − �)+; �¿0, and integrating over �, we obtain

∫�

|∇(q − �)+|26∫�(q − �)2+ + �

∫�(q − �)+ +

1ptmp(t)

∫�q1−p(q − �)+

6∫�(q − �)2+ + �|�|+ c(�)

tmp(t)(9)

As tk ↗T; m(tk)→ ∞, so that for all j ∈N there is kj ∈N such that∫�

|∇(q − �j)+|26∫�(q − �j)2+ +

1j

at any t= tk with k¿kj (10)

where e.g. �j :=1=j. Assuming without loss of generality that tkj ↗T , we write qj(x):=q(x; tkj)and obtain from (10) that for a subsequence, (qj − �j)+*Q in W 1;2

0 (�) as well as

qj →Q in L2(�) and a:e: in � (11)

where ∫�

|∇Q|26∫�Q2

As �1 = 1, however, this means

Q= �� (12)

for some �∈ [0; 1], so that (7) will follow as soon as we have excluded the case �=0.To achieve this, we once again employ (8) which combined with (6) can for t= tkj be

read as

−�qj6qj + �jW 1−p


1624 M. WINKLER

where �j → 0 as j → ∞. Letting vj ∈C0( ��)∩C2(�) be the solution of −�vj= qj in �;vj|@� =0, we obtain from this −�qj6−�(vj + �jW ) and thus

qj6vj + �jW (13)

by elliptic comparison. Due to the de�nition of q, there is (xj)j∈N ⊂� such that qj(xj)=1 forall j; extracting a further subsequence, we may assume xj → x0 ∈ ��. As qj →Q in L2(�) and‖qj‖L∞(�) = 1, Theorem 8.33 in Reference [12] shows that vj → v in C1( ��), where −�v=Q in� and v|@� =0. In particular, vj(xj)→ v(x0), so that (13) yields v(x0)¿1, whereby Q= −�vcannot vanish identically and � must be positive. Hence, (7) has been proved.Now if p¿1 then (7) together with the dominated convergence theorem shows that

∫�u1−p(tk)�→ 0 as tk ↗T (14)

since by (6) and (5), some multiple of �1−(p−1)� serves as a uniform L1-majorant for theintegrand. Lemma 1.3 however says that this is impossible.In the case p=1 a similar argument leads to the absurd conclusion

∫�ln u(tk)�→ ∞ as tk ↗T (15)

Finally, if p¡1, we use (11) to see that∫� q1−p

j �→ ∫�Q1−p�¿0 and thus

∫�u1−p(tkj)�=m1−p(tkj)

∫�q1−pj �→ ∞ as tkj ↗T (16)

again contradicting Lemma 1.3.

For small p we do not even have blow-up in in�nite time.

Corollary 2.2If p¡3 then u is bounded in �× (0;∞).ProofWe again may assume u0 to be large such that u0¿c0�. As c0� solves (0.1), a comparisonargument yields u¿c0� in �× (0;∞). Using this bound from below instead of (6), we nowassume u(0; tk)→ ∞ for a sequence tk ↗ ∞ and then proceed in a way similar to the one inthe proof of (14)–(16) to derive a contradiction.

The latter result encourages us, at least for p¡3, to search for a relationship between the!-limit set of a solution and the set {�� | �¿0} of nonnegative equilibria of (0.1). Notethat the following assertion does not require any assumption on time regularity (such as e.g.∫∞1

∫� u2t ¡∞ which, by the way, does not seem to be available here); instead, we once again

use the one-sided estimate from Lemma 1.2 in the proof of



Lemma 2.3Suppose p¿0 and (u(tk)=mk)k∈N is bounded in L2(�) for some sequence of times tk ↗ ∞and given numbers mk¿1. Then there is �¿0 and a subsequence kj ↗ ∞ such that

u(tkj)mkj

→ �� in L2(�) as t → ∞

ProofFor qk := u(tk)=mk , Lemma 1.2 yields −�qk6qk +1=ptkm

pk q

1−pk . As 1=ptkm

pk → 0 by assump-

tion, we may perform the same testing and extraction procedure as in the proof of Theorem 2.1to obtain sequences kj ↗ ∞ and �j ↘ 0 such that (qkj − �j)+*Q in W 1;2

0 (�) and qkj →Q inL2(�), where

∫� |∇Q|26 ∫�Q2, so that Q coincides with some nonnegative multiple of �.

Now we are able to give a precise description of the asymptotic behaviour in the casep¡3.

Theorem 2.4Suppose p¡3. Then

u(t)→ �� in L2(�) as t → ∞

where �= �(p; u0)∈ [0;∞) is given by

�=

(∫� u1−p

0 �∫� �

2−p

)−1=(p−1)if p =1

exp(∫

� ln(u0=�)�∫� �

)if p=1

ProofBy Corollary 2.2 and Lemma 2.3, each sequence tk ↗ ∞ contains a subsequence tkj ↗ ∞ suchthat u(tkj)→ �� for some �¿0, so that it suces to identify �.First, assume p¿1 and u0¿c0� for some c0¿0. Then I0 :=

∫� u1−p

0 �¡∞ and Lemma 1.3states that

∫� u1−p(t)�≡ I0. As u¿c0� in �× (0;∞) by comparison, the dominated conver-

gence theorem yields∫� u1−p(tkj)�→ �1−p

∫� �

2−p and thus indeed �= �0 := (I0=∫�

�2−p)−1=(p−1). For general u0, we �rst apply Fatou’s lemma to infer �1−p∫� �

2−p6I0 andtherefore �¿�0. To obtain the opposite inequality, we use an approximating sequence ofinitial data u0�, �= �i ↘ 0, such that u0�|@� =0, u0�¿c0�� with c0�¿0 and u0� ↘ u0 in �as �= �j ↘ 0. By the previous step, the corresponding solutions u� satisfy u�(t)→ �� ast → ∞, where �� := (I�=

∫� �

2−p)−1=(p−1) and I� :=∫� u1−p

0� �. A comparison argument showsthat u6u� and thus �6�� for all �= �i. As I� → I as �↘ 0 by the monotone convergencetheorem, this implies �6�0, as claimed.The proof in the case p=1 is very similar, with obvious changes in the terms involving

logarithms. If p¡1 then the argument becomes even simpler since no negative integrationpowers are involved and therefore it is not necessary to assume u0¿c0� �rst.


1626 M. WINKLER

The fact that the above restriction to small p was not of purely technical nature is shownin the following theorem which states that for p¿3 both bounded and unbounded solutionsexist, depending only on the behaviour near @� of u0.

Theorem 2.5Let p¿3.

(i) If∫� u1−p

0 �¡∞ then u(t)→ ∞ a.e. in � as t → ∞.In particular, this conclusion holds if u0¿c0�� for some �¡ 2

p−1 and c0¿0.(ii) If u06c1� for some c1¿0 then u6c1� in �× (0;∞).

Proof

(i) If the assertion were false, Lemma 2.3 would provide a sequence tk ↗ ∞ such thatu(tk)→ �� in L2(�) for some �¿0, so that in the case �¿0∫

�(��)1−p�6 lim inf

tk→∞

∫�u1−p(tk)�

by Fatou’s lemma. In view of Lemma 1.3 and the hypothesis, this would imply∫� �

2−p¡∞, a contradiction.(ii) This is obvious, since c1� is a (super-)solution of (0.1).

3. STABILITY AND UNSTABILITY OF EQUILIBRIA

Assume for a moment that p=0 in (0.1), i.e. let us consider the linear equation ut =�u+u.Multiplication of the equation satis�ed by the di�erence v= u1 − u2 of two solutions by vyields @t

∫� v260 which implies that each equilibrium is stable with respect to the norm in

L2(�). The loss of this nice property for p¿1 is a simple consequence of Theorems 2.4and 2.5.

Corollary 3.1

(i) If p¡1 then each positive steady state w= ��, �¿0, of (0.1) is stable with respectto the topology of C0( ��).

(ii) If 16p¡3 then each w= ��, �¿0, is unstable from below in the topology ofC0( ��)∩W 1;2(�). More precisely, there is a sequence of positive initial data u0k6wsuch that u0k→w in C0( ��)∩W 1;2(�) as k → ∞, but each of the corresponding solu-tions uk converges to zero in L2(�) as t → ∞.

(iii) If p¿3 then w= ��, �¿0, is unstable from above with respect to the topology ofC0( ��)∩W 1; q(�), 16q¡(p−1)=(p−3). There exist u0k¿w, u0k¿0 in �, u0k |@� =0,such that u0k →w in C0( ��)∩W 1; q(�), but uk(t)→ ∞ a.e. in � as t →∞.

Proof

(i) This easily follows from Lemma 1.3 and the fact that the functional u0 → ∫� u1−p

0 �is continuous on C0( ��).



(ii) If p¿1, let e.g. u0k := (��−1=k)++1=k�2=(p−1). Then u0k → �� in C0( ��)∩W 1;2(�).As

∫� u1−p

0k �=∞, however, Theorem 2.4 states that uk(t)→ 0 as t → ∞.In the case p=1 we may e.g. employ u0k := (��− 1=k)+ + 1=k�e−�−2

.(iii) Fix �∈ (1− 1=q; 2=(p − 1)) and take u0k := ��+ 1=k�� this time. Then u0k → �� in

C0( ��)∩W 1; q(�), but uk(t)→ ∞ as t → ∞ by Theorem 2.5.

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A doubly critical degenerate parabolic problem