Blow up for some nonlinear parabolic problems with convection under dynamical boundary conditions

Pages: 1031 - 1041, Issue Special, September 2007

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Joachim von Below - LMPA Joseph Liouville, FCNRS 2956, Université du Littoral Côte d’Opale, 50, rue F. Buisson, B.P. 699, F–62228 Calais Cedex, France (email)
Gaëlle Pincet Mailly - LMPA Joseph Liouville, FR 2596 CNRS, Université du Littoral Côte d'Opale, 50, rue F. Buisson, B.P. 699, F-62228 Calais Cedex, France (email)

Abstract: Blow up phenomena for solutions of nonlinear parabolic problems including a convection term $\partial_tu = div(a(x)\Deltau) + f(x, t, u,\Deltau)$ in a bounded domain under dissipative dynamical time lateral boundary conditions $\sigma\partial_tu + \partial_vu = 0$ are investigated. It turns out that under natural assumptions in the proper superlinear case no global weak solutions can exist. Moreover, for certain classes of nonlinearities the blow up times can be estimated from above as in the reaction diffusion case [6]. Finally, as a model case including a sign change of the convection term, the occurrence of blow up is investigated for the one–dimensional equation $\partial_tu = \partial^2_xu − u\partial_xu + u^p.

Keywords:  Parabolic problems, dynamical boundary conditions, blow up.
Mathematics Subject Classification:  35B05, 35B40,35K55, 35K57, 35K65, 35R45.

Received: September 2006;      Revised: April 2007;      Published: September 2007.