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66 9th AIMS CONFERENCE – ABSTRACTS
Special Session 15: Nonlinear Evolution Equations, Inclusions andRelated Topics
Mitsuharu Otani, Waseda University, JapanTohru Ozawa, Waseda University, Japan
N. U. Ahmed, University of Ottawa, CanadaS. Migorski, Jagiellonian University, PolandI. I. Vrabie, Al. I. Cuza University, Romania
This session will focus on the recent developments in the theory of Nonlinear Evolution Equations andRelated Topics including the theory of abstract evolution equations in Banach spaces as well as the studies(the existence , regularity and asymptotic behaviour of solutions) of various types of Nonlinear PartialDi↵erential Equations. Some new trends in both abstract linear and nonlinear evolution inclusions as wellas concrete applications in Control Theory, Partial Di↵erential Equations, Variational Inequalities, andmathematical models described by evolution inclusions will be disscussed. Open problems and new resultsin the above mentioned areas based on a interplay between various disciplines as Multi-Valued Analysis,Measure Theory, Fourier Analysis and Spectral Theory, will find their place within this Special Session.
Optimal feedback control for di↵erential in-clusions on Banach spaces
Nasir Uddin AhmedUniversity of Ottawa, [email protected]
In this paper we consider a class of partially observedsemilinear dynamic systems on infinite dimensionalBanach spaces governed by di↵erential inclusion rep-resenting measurement uncertainty. In the paper wepresent a result on existence of an optimal operatorvalued function as the feedback control law minimiz-ing the maximum risk. We also present necessaryconditions of optimality for the feedback control lawfrom a class of operator valued functions defined onfinite intervals and taking values from the space ofcompact operators. This is then extended to coversystems with uncertainty in the dynamics itself. Thenecessary conditions remain valid also for a class ofstochastic systems driven by Wiener martingale.
[1] N.U.Ahmed, Optimal Relaxed Controls for Sys-tems Governed by Impulsive Di↵erential Inclu-sions,Nonlinear Funct. Anal. & Appl., 10(3), (2005),427-460.
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Doubly nonlinear parabolic equations withvariable exponents
Goro AkagiKobe University, [email protected]
This talk is concerned with doubly nonlinearparabolic equations involving variable exponents.Let ⌦ be a bounded domain in Rd with smoothboundary @⌦. We discuss the following doubly non-
linear parabolic problem (P):
@t
⇣|u|m(x)�2u
⌘��p(x)u = 0 in ⌦⇥ (0, T ),
|u|m(·)�2u = v0 on ⌦⇥ {0},where @t = @/@t, T > 0, v0 = v0(x) is a given initialdata and �p(x) is the so-called p(x)-Laplacian, a typ-ical example of nonlinearity with variable exponents,together with either the Dirichlet condition
u = 0 on @⌦⇥ (0, T ),
or the Neumann condition
|ru|p(x)�2@nu = 0 on @⌦⇥ (0, T ),
where @nu denotes the outward normal derivative ofu on @⌦. The existence of solutions is proved bydeveloping an abstract theory on doubly nonlinearevolution equations governed by gradient operators.In contrast to constant exponent cases, two nonlin-ear terms have inhomogeneous growth and some dif-ficulty may occur in establishing energy estimates.Our method of proof relies on an e�cient use ofLegendre-Fenchel transforms of convex functionalsand a modified energy method.
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Existence for a class of nonlinear delayreaction-di↵usion systems
Monica-Dana BurlicaAlexandru Ioan Cuza University Iasi & Gh. AsachiTechnical University Iasi, [email protected] Rosu
We consider an abstract nonlinear multi-valuedreaction-di↵usion system with delay and, using somecompactness arguments coupled with metric fixedpoint techniques, we prove some su�cient conditionsfor the existence of at least one C0-solution. A global
SPECIAL SESSION 15 67
asymptotic stability result is also obtained and somespecific examples are analyzed.
Acknowledgements. Supported by a grant ofthe Romanian National Authority for Scientific Re-search, CNCS–UEFISCDI, project number PN-II-ID-PCE-2011-3-0052.
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Dynamic analysis of a nonlinear Timoshenkoequation
Jorge Esquivel-AvilaUniversidad Autonoma Metropolitana, [email protected]
An initial boundary value problem in a bounded do-main for the Timoshenko equation with a nonlineardamping and a nonlinear source term is considered.Exploiting an idea of a potential well under suitableassumptions on the damping and the source terms,we prove blow-up as well as convergence to the zeroand nonzero equilibria and we give rates of decay tothe zero equilibrium.
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Leadership in crowd dynamics: modelling viatwo-scale interactions
Joep EversEindhoven University of Technology, [email protected] Muntean
We describe a two-scale model for the dynamics of acrowd by distinguishing between a number of lead-ers considered at the micro-scale and a large crowdviewed macroscopically. Leaders are of particularinterest in view of crowd control. We encounter thefollowing fundamental questions: - How to describea leader in mathematical terms? - Does the presenceof a leader essentially change the macro-scale be-haviour? - Which specific macroscopic patterns canbe caused by the leader(s)? - How can we validateour two-scale model with experimental data? Thecrowd in our model consists of two subpopulationsin a corridor, assumed to be two-dimensional. Tocapture the evolution of the crowd, we use a time-dependent mass measure for each subpopulation.Each of the two mass measures satisfies the continu-ity equation. Hence, their evolution is governed bythe partial velocity field, which we have to provide.The actual modelling consists therefore in findingthe right velocity field, which we expect to dependfunctionally on both mass measures. In this talk, Iexplain the model we use, and the particular formwe choose for the velocity field. Furthermore I givea flavour of the crowd-dynamical phenomena thatarise from this model.
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Nodal and multiple constant sign solutions forequations with the p-Laplacian
Michael FilippakisUniversity of Piraeus, [email protected] P. Agarwal, Donal O’Regan, Nikolaos S.Papageorgiou
We consider nonlinear elliptic equations driven bythe p-Laplacian with a nonsmooth potential (hemi-variational inequalities). We obtain the existence ofmultiple nontrivial solutions and we determine theirsign (one positive,one negative and the third nodal).Our approach uses nonsmooth critical point theorycoupled with the method of upper-lower solutions.
�!1 ⇧1 �Asymptotic behavior of blow-up solutions forthe heat equations with nonlinear boundaryconditions
Junichi HaradaWaseda University, [email protected]
We consider the heat equation equation on the halfspace with nonlinear boundary conditions (ut = �u,@⌫u = uq). Here we discuss the asymptotic behaviorof blow-up solutions for the Sobolev subcritical case.In particular, we provide a su�cient condition on theinitial data for the single point blow-up at the origin.Furthermore we study the spacial singularities of theblow-up profile.
�!1 ⇧1 �Pohozaev-Otani type inequalities for weak so-lutions of some quasilinear elliptic equationsin unbounded domains
Takahiro HashimotoAichi Medical University, [email protected]
In this talk, we are concerned with the followingquasilinear elliptic equations:
(E)
⇢�div �a(x)|ru|p�2ru = b(x)|u|q�2u in ⌦,
u = 0 on @⌦,
where ⌦ is a domain in RN (N � 2) with smoothboundary, 1 < p, q <1, a(x), b(x) � 0.
When a(x) = b(x) ⌘ 1, a complementary ex-istence result was founded for the problem of inte-rior, exterior and whole space. The main purposeof this talk is to obtain the nonexistence results viaPohozaev-Otani type inequalities for some class ofweak solutions of (E). We also discuss the existenceresult for exterior problem and the regularity of so-lutions of (E).
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68 9th AIMS CONFERENCE – ABSTRACTS
Fractional Sobolev spaces via Riemann - Li-ouville derivatives and some imbeddings
Dariusz IdczakUniversity of Lodz, [email protected] Walczak
Using the Riemann - Liouville derivatives of testfunctions we introduce fractional Sobolev spaces offunctions of one variable, defined on a bounded inter-val. Next, we characterize elements of these spaceswith the aid of the Riemann - Liouville derivatives aswell as with the aid of some integral representations.We also study some imbeddings of the introducedspaces and prove their compactness.
Acknowledgements.The project was financed withfunds of National Science Centre, granted on the ba-sis of decision DEC-2011/01/B/ST7/03426.
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Variational problems associated withTrudinger-Moser inequalities in unboundeddomains
Michinori IshiwataFukushima University, [email protected]
In this talk, we are concerned with the existenceand the nonexistence of maximizers for variationalproblems associated with the Trudinger-Moser in-equality in unbounded domains. Particularly, we areinterested in the variational problems for a version ofTM-typeinequality with singular weight functions.
�!1 ⇧1 �Well-posedness for anisotropic degenerateparabolic equations with non-homogeneousboundary conditions
Kazuo KobayasiWaseda University, [email protected]
We study the well-posedness for anisotropic degen-erate parabolic-hyperbolic equations with initial andnon-homogeneous boundary conditions. We provea comparison theorem for any entropy sub- andsuper-solution, which immediately deduces the L1
contractivity and therefore, uniqueness of entropysolutions. The method used here is based upon thekinetic formulation and the kinetic techniques devel-oped by Lions, Perthame and Tadmor. By adaptingand modifying those methods to the case of Dirichletboundary problems for degenerate parabolic equa-tions we can establish a comparison property. More-over, in the quasi-isotropic case the existence ofentropy solutions is proved.
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A new class of nonlinear evolution equations
Masahiro KuboNagoya Institute of Technology, [email protected]
Codifying applications of subdi↵erential evolutionequations to variational inequalites, we introduce anew class of evolution equations associate with sub-di↵erentials depending on unknonwn functions. Weconsider three kinds of evolution equations: parabolic(I), parabolic (II) and hyperbolic evolution equa-tions; and two types of dependence of the subdi↵er-entials on the unknown functions: local and non-localtypes of dependence. We prove the existence of a so-lution to a hyperbolic problem with a non-local typedependence on unknown functions and study its sin-gular limit to a solution to a parabolic (II) problem.
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Existence and non existence of solutions toinitial boundary value problems for nonlinearevolution equations with strong dissipation
Akisato KuboFujita Health University, [email protected]
The main purpose is to investigate existence and non-existence of global solutions of the initial Dirichlet-boundary value problem for evolution equations withthe strong dissipation. Many authors studied classesfor which initial boundary value problems possessglobal solutions. We consider a related problem andseek global solutions and blow-up solutions of it de-pending on whether it belongs to such classes ornot.
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Existence of sliding motions for nonlinear evo-lution equations in Banach space
Laura LevaggiFree University of Bolzano, [email protected]
In this work the existence problem for solutions ofa class of controlled nonlinear evolution equationsin a Banach space is presented, where the consid-ered feedback laws can depend nonlinearly on thestate variable. The proof is based on the usualFaedo-Galerkin approximation method and the useof techniques related to the monotonicity and coer-civity assumptions on the evolution operator. Theresult is then applied to the existence of the so-calledsliding motions for infinite dimensional systems, ei-ther under distributed or boundary control. Slidingmode methods are used to control finite-dimensionalsystems and are based on the idea of constraining theevolution on a manifold, designed in order to attain
SPECIAL SESSION 15 69
the control aim. The idea here is to exploit thesetechniques to define a feedback control that is ableto make each Galerkin approximation fulfill approxi-mately the sliding constraint defining the prescribedmanifold and show that the limit evolution is viable.
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Fractional du Bois-Reymond lemma and itsapplications.
Marek MajewskiUniversity of Lodz, [email protected] Kamocki, Stanislaw Walczak
In the paper a fractional du Bois-Reymond lemmafor functions of one variable with Riemann-Liouvillederivatives of the order ↵ 2 (n � 1
2, n) with n � 1
is presented. We use this lemma to show that anycritical point of a fractional Lagrange functional is asolution to it’s Euler-Lagrange equation. The abovetechnique can be apply to the problem of the ex-istence of solutions to the fractional counterpart ofclassical Dirichlet problem.
Acknowledgements.The project was financed withfunds of National Science Centre, granted on the ba-sis of decision DEC-2011/01/B/ST7/03426.
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Boundary control and hidden trace regularityof a semigroup associated with a beam equa-tion and non-dissipative boundary conditions
Richard MarchandSlippery Rock University, [email protected] Lasiecka, Timothy J. McDevitt
New control-theoretic results will be presented forsecond order (in time) PDE scalar equations withnon-monotone feedback boundary conditions. Whilethe analysis of monotone structures is based on asuitable version of monotone semigroup theory, thenon-monotone case seems to require detailed mi-crolocal analysis on the boundary, showing that theunderlying semigroup is of Gevrey’s class. A par-ticular type of “hidden regularity” exhibited by theboundary traces is instrumental for showing well-posedness of the associated control system within astandard finite energy space even when the controlsare not necessarily collocated. Although the analysisis applicable to multidimensional problems, the talkwill focus on a one-dimensional Euler-Bernoulli beamequation. Theoretical results will be complementedby numerical simulations that illustrate the spectralproperties of the system operators.
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Existence of time-periodic solutions for themicropolar fluid equations with the spin-vortex interaction boundary condition
Kei MatsuuraWaseda University, [email protected] Otani
It will be presented that the time-periodic prob-lem for the micropolar fluid equation in a regularbounded domain ⌦ of R3 is solvable if the norms ofexternal forces are su�ciently small. As the bound-ary conditions on the velocity field u and the mi-crorotation field ! we impose the no-slip conditionu|@⌦ = 0 on the velocity and assume that the spinand the vorticity are proportional on the boundary,i.e., ! = ✓
2curlu on @⌦ with some constant ✓ 2 [0, 1].
For the existence of time-periodic solutions the small-ness of the parameter ✓ is required as well. The sta-bility and the uniqueness of time-periodic solutionswill also be discussed.
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Fuzzy stochastic di↵erential equations-di↵erent approaches and recent results.
Mariusz MichtaUniversity of Zielona Gora and Opole University,Poland, [email protected]
In the talk we present di↵erent approaches to the no-tion of fuzzy stochastic di↵erential equations drivenby semimartingales and also to the notion of theirsolutions. We also present existence and uniquenesstheorems to such equations. Presented results extendto stochastic case possible approches known from thetheory of deterministic fuzzy di↵erential equations.
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Nonlinear subdi↵erential inclusions with ap-plications to contact mechanics
Stanislaw MigorskiJagiellonian University, Krakow, [email protected]
We consider two classes of nonlinear subdi↵eren-tial inclusions in a framework of evolution triple ofspaces. The first class involves inclusions with ahistory-dependent term for which we provide an ex-istence and uniqueness result. In the second class,we consider time-dependent possibly nonconvex non-smooth functions and their Clarke subdi↵erentialsoperating on the unknown function. We prove theexistence of a weak solution and study the asymp-totic behavior of a sequence of solutions when asmall parameter in the inertial term tends to zero.We prove that the limit function is a solution of a
70 9th AIMS CONFERENCE – ABSTRACTS
first order inclusion. Finally, we give applications toquasi-static viscoelastic frictional contact problems.
Acknowledgement. Supported by the MarieCurie International Research Sta↵ Exchange SchemeFellowship within the 7th European CommunityFramework Programme under Grant Agreement No.295118.
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Stochastic delay inclusions with noncontinu-ous multifunctions
Jerzy MotylUniversity of Zielona Gora, [email protected]
The talk deals with the Ito’s type delay stochasticdi↵erential inclusion
dX(t) 2 F (Xt)dt+G(Xt)dW (t), X0 = ⇠,
where W denotes an m-dimensional Wiener process,Xt(s) = X(t + s) and set-valued functions F,G de-fined on C([�r, 0], Rd) take on closed and convex val-ues. The existence of local and global solutions of theinclusion with so-called ”upper separated” multifunc-tions will be discussed. Some examples of noncontin-uous ”upper separated” multifunctions will be alsopresented.
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Regularity and singularities of optimal convexshapes in the plane
Arian NovruziUniversity of Ottawa, [email protected] Lamboley, Michel Pierre
We focus here on the analysis of the regularity orsingularity of solutions ⌦0 to shape optimizationproblems among convex planar sets, namely:
J(⌦0) = min{J(⌦), ⌦ convex, ⌦ 2 Sad},where Sad is a set of 2-dimensional admissible shapesand J : Sad 7! R is a shape functional.
Our main goal is to obtain qualitative proper-ties of these optimal shapes by using first and secondorder optimality conditions, including the infinite di-mensional Lagrange multiplier due to the convexityconstraint. We prove two types of results:
i) under a suitable convexity property of the func-tional J , we prove that ⌦0 is a W 2,p-set, p 2 [1,1].This result applies, for instance, with p = 1 whenthe shape functional can be written as J(⌦) = R(⌦)+P (⌦), where R(⌦) = F (|⌦|, Ef (⌦),�1(⌦)) involvesthe area |⌦|, the Dirichlet energy Ef (⌦) or the firsteigenvalue of the Laplace-Dirichlet operator �1(⌦),and P (⌦) is the perimeter of ⌦,
ii) under a suitable concavity assumption on thefunctional J , we prove that ⌦0 is a polygon. Thisresult applies, for instance, when the functional isnow written as J(⌦) = R(⌦)� P (⌦), with the samenotations as above.
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Splitting methods for semilinear evolutionequations with applications to nonlinearSchrodinger equations
Masahito OhtaTokyo University of Science, [email protected]
We study splitting methods for seminear evolutionequation u0 = Au+F (u), where A is an m-dissipativeoperator in a Hilbert spaceX, and F : D(A)! D(A)is a Lipschitz continuous function on bounded sub-sets of D(A). Under appropriate assumptions on F ,we prove the second order convergence of the Strangsplitting method in X for initial data in D(A2).This abstract result unifies several known results fornonlinear Schrodinger equations.
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Viability of a time dependent closed set withrespect to a semilinear delay evolution inclu-sion
Marius PopescuAl. I. Cuza University of Iasi, Romania, [email protected] Necula
We prove a su�cient condition for a time-dependentclosed set to be viable with respect to a delay evo-lution inclusion governed by a strongly-weakly u.s.c.perturbation of an infinitesimal generatorof a C0-semigroup. This condition is expressed in terms of anatural concept involving tangent sets, generalizingtangent vectors in the sense of Bouligand and Severi.
Acknowledgements. Supported by a grant of theRomanian National Authority for ScientificResearch,CNCS–UEFISCDI, project number PN-II-ID-PCE-2011-3-0052.
�!1 ⇧1 �Asymptotic behavior of solutions to a coupledsystem of Maxwell’s equations and a con-trolled di↵erential inclusion
Volker ReitmannSt. Petersburg State University, [email protected] Kalinichenko, Sergey Skopinov
We consider the parameter-dependent coupled sys-tem consisting of the one-dimensional Maxwell’s
SPECIAL SESSION 15 71
equations and the heat equation which is under-stood as di↵erential inclusion. This system is calledMaxwell’s equations with thermal e↵ect introducedby H.M. Yin. The use of a boundary control in theinclusion guarantees the boundedness and conver-gence of solutions and excludes the presence of ablow-up. The techniques which are used by us areLyapunov functionals and multiscaling methods.
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Global existence and exponential stability fora nonlinear delay evolution equation with non-local initial condition
Daniela RosuAlexandru Ioan Cuza University Iasi & Gh. AsachiTechnical University Iasi, [email protected] Burlica
We consider a delay evolution equation subjectedto a nonlocal initial condition and governed by anonlinear Lipschitz perturbation of an m-dissipativeoperator. We prove some existence and asymptoticstability results for global C0-solutions and we ex-emplify the abstract results by a delayed porousmedia equation subjected either to periodic or toanti-periodic conditions.
Acknowledgements. Supported by a grant ofthe Romanian National Authority for Scientific Re-search, CNCS–UEFISCDI, project number PN-II-ID-PCE-2011-3-0052.
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Multiple positive solutions for periodic prob-lems with concave terms
Vasile StaicuUniversity of Aveiro, Portugal, [email protected]. Aizicovici, N. Papageorgiou
We consider a nonlinear periodic problem (P�),driven by the scalar p-Laplacian, with a paramet-ric concave term � |x|q�2 x and a Caratheodory per-turbation f (t, x) such that x ! f (t, x) exhibits a(p� 1)�superlinear growth near +1 (the convexterm). Using variational techniques, based on thecritical point theory, and suitable truncation tech-niques we prove a bifurcation-type theorem describ-ing the nonexistence, existence and multiplicity ofpositive solutions as the parameter varies. Namely,we show that a critical parameter value �⇤ > 0 existssuch that: for all � 2 (0,�⇤) , the problem (P�) has atleast two positive solutions; for � = �⇤, the problem(P�) has at least one positive solution, and, for all� > �⇤, the problem (P�) has no positive solutions.
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On a nonstandard nonlinear parabolic prob-lem for the coupling surface – deep oceantemperatures with latent heat and coalbedoterms.
Lourdes TelloUniversidad Politecnica de Madrid, [email protected]. Diaz, A. Hidalgo
We study a global climate model for the coupling ofthe mean surface temperature with the deep oceantemperature. The nonlinear model presents somenonstandard facts: the boundary condition, repre-senting the mean surface temperature, is not onlyof dynamic type (involving the time derivative ofthe trace of the solution) but also a surface di↵usiveterm. The model includes also some delicate nonlin-ear terms such as the coalbedo e↵ect and the latentheat, which here are formulated in terms of suitable(multivalued) maximal monotone graphs of R2. Weprove the existence of bounded weak solutions andshow some numerical experiences. Other qualitativeproperties of the solutions will be also presented.
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Robust feedback stabilization of solutions ofstochastic evolution equations with delay
Govindan TrivelloreInstituto Politecnico Nacional, [email protected]. Ahmed
In this talk, we consider the problem of state feed-back robust stabilization against uncertainty in theclass of relatively A- bounded operators of mild so-lutions of stochastic evolution equations with delay.An example is included to illustrate the theory.
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Periodic solutions of some double-di↵usiveconvection systems based on Brinkman-Forchheimer equation
Shun UchidaWaseda Universty, [email protected] Otani
We consider the following system:
8>>>>><
>>>>>:
@t~u = ⌫�~u� a~u�rp+ ~gT + ~hC + ~f1 in ⌦⇥[0, S],@tT + ~u·rT = �T + f2 in ⌦⇥[0, S],@tC + ~u·rC = �C + ⇢�T + f3 in ⌦⇥[0, S],r·~u = 0 in ⌦⇥[0, S],~u|@⌦ = 0; T |@⌦ = 0; C|@⌦ = 0,
72 9th AIMS CONFERENCE – ABSTRACTS
where N = 2, 3 and ⌦ is a bounded domain withsmooth boundary.This system describes the double-di↵usive convection between the temperature T andthe concentration of solute C. The first equation ofthe system comes from the Brinkman-Forchheimerequation, which describes the behavior of the fluidvelocity ~u and the pressure p in some porous medium.In this talk, we discuss the existence of a solution forthis system under the time periodic condition withperiod S.
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Nonlinear delay evolution inclusions with non-local conditions on the initial history
Ioan VrabieAl. I. Cuza University, [email protected]
We consider a nonlinear delay evolution di↵erentialinclusion, governed by a multi-valued perturbationof an m-dissipative operator, subjected to a non-local condition on the initial history. Under somenatural assumptions, allowing to handle periodic,anti-periodic and mean-type problems with delay,we prove the existence in the large of at least oneC0-solution. Two applications concerning nonlineardelay parabolic problems are presented.
Acknowledgements. Supported by a grant ofthe Romanian National Authority for Scientific Re-search, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0052.
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Uniqueness and non-degeneracy of groundstates of quasilinear Schrodinger equations
Tatsuya WatanabeKyoto Sangyo University, [email protected]
We consider the following quasilinear elliptic prob-lem:
��u� �(|u|↵)|u|↵�2u = g(u) in RN
where N � 3, > 0 and ↵ > 1. This equationcan be obtained as a stationary problem of modifiedSchrodinger equations which appear in the study ofplasma physics.
In this talk, we discuss the existence and thevariational characterization of the ground state andpresent our recent results on the uniqueness and non-degeneracy.
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Life span of positive solutions for a semilinearheat equation with non-decaying initial data
Yusuke YamauchiWaseda University, [email protected]
We consider the following semilinear heat equation:
ut = �u+ up, (x, t) 2 RN ⇥ (0,1),
where p > 1, N � 1. In this talk, we discuss theupper bound of the life span of positive solutions ofthe equation for initial data having positive limit in-ferior at space infinity. The proof is based on a slightmodification of Kaplan’s method.
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