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Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

José M. Arrieta; Anibal Rodriguez-Bernal; Philippe Souplet — 2004

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions $u$ : either the space derivative $u_{x}$ blows up in finite time (with $u$ itself remaining bounded), or $u$ is global and converges in $C^{1}$ norm to the unique steady state. The main difficulty is to prove $C^{1}$ boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out the method of...

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Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

Blow up versus global boundedness of solutions of reaction diffusion equations with nonlinear boundary conditions

Robin type conditions arising from concentrated potentials

Dissipative dynamics of reaction diffusion equations in $ℝ^{n}$

Advanced Search

Formula preview

Currently displaying 1 – 4 of 4

Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

Blow up versus global boundedness of solutions of reaction diffusion equations with nonlinear boundary conditions

Robin type conditions arising from concentrated potentials

Dissipative dynamics of reaction diffusion equations in ℝ n

Dissipative dynamics of reaction diffusion equations in $ℝ^{n}$