Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
George Smyrlis, Nikolaos Papageorgiou, Dimitrios Kravvaritis
Technological Educational Institute of Athens, Department of Mathematics (Smyrlis), National
Technical University, Department of Mathematics (Papageorgiou, Kravvaritis)
Multiple solutions for semilinear Neumann problems
Let Ω ⊆ RN (N ≥ 1) be a bounded domain with a C2-boundary ∂Ω. We consider the following nonlinear
Neumann problem
∂u
−∆u(z) = f (z,u(z)) a.e. in Ω, = 0, on ∂Ω, (2)
∂ν
where ∆ stands for the Laplace differential operator, f (z,x), z ∈ Ω, x ∈ R is of Carathéodory type with
linear growth in x, ν is the exterior normal to ∂Ω and ∂u
is the normal derivative in the direction ν.
∂ν
Under certain hypotheses on the reaction term f (z,x), we derive multiple nontrivial smooth (weak) solutions
for the problem (1), possibly of constant sign. Our approach combines variational methods based on
the critical point theory, together with techniques from Morse theory.
Christian Stinner
Universität Zürich
Large time behavior in a quasilinear viscous Hamilton-Jacobi equation with degenerate
diffusion
The large time behavior of nonnegative solutions to the quasilinear degenerate diffusion equation
∂tu − ∆pu = |∇u| q is investigated for p > 2 and q > 0 in a bounded domain. Qualitative properties of the
solutions vary greatly according to the relative strength of the diffusion and the source term. In particular,
we show how the relation between the parameters influences the existence of nontrivial steady states as
well as the existence of solutions which are global in time. Moreover, we study the convergence of global
solutions towards steady states and characterize the stationary solutions. Part of the presented results
were obtained in joint works with G. Barles and Ph. Laurençot.
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