EUDML

We consider three types of semilinear second order PDEs on a cylindrical domain $Ω \times (0, \infty)$ , where $Ω$ is a bounded domain in $ℝ^{N}$ , $N \geq 2$ . Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of $Ω \times (0, \infty)$ is reserved for time $t$ , the third type is an elliptic equation with a singled out unbounded variable $t$ . We discuss the asymptotic behavior, as $t \to \infty$ , of solutions which are defined and bounded on $Ω \times (0, \infty)$ .

Group Actions on Strongly Monotone Dynamical Systems.

Janusz Miercynski; Peter Polácik — 1989

Mathematische Annalen

Imbedding vector fields in scalar parabolic Dirichlet $B V P_{s}$

Peter Poláčik; Krzysztof Rybakowski — 1995

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball.

Peter Polácik; Xu-Yan Chen — 1996

Journal für die reine und angewandte Mathematik

Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations

Thomas Bartsch; Peter Poláčik; Pavol Quittner — 2011

Journal of the European Mathematical Society

We prove a Liouville type theorem for sign-changing radial solutions of a subcritical semilinear heat equation $u_{t} = Δ u + {|u|}^{p - 1} u$ . We use this theorem to derive a priori bounds, decay estimates, and initial and final blow-up rates for radial solutions of rather general semilinear parabolic equations whose nonlinearities have a subcritical polynomial growth. Further consequences on the existence of steady states and time-periodic solutions are also shown.

Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations

Juraj Húska; Peter Poláčik; Mikhail V. Safonov — 2007

Annales de l'I.H.P. Analyse non linéaire

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Realization of the dynamics of ODE's in scalar parabolic PDE's

Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations

Domains of attraction of equilibria and monotonicity properties of convergent trajectories in parabolic systems admitting strong comparison principle.