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

Joachim von Below; Gaëlle Pincet Mailly

doi:10.3934/proc.2007.2007.1031

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2007, Volume 2007: 1031-1041. Doi: 10.3934/proc.2007.2007.1031 open access

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

Joachim von Below^1, and
Gaëlle Pincet Mailly²

1.
LMPA Joseph Liouville, FCNRS 2956, Université du Littoral Côte d’Opale, 50, rue F. Buisson, B.P. 699, F–62228 Calais Cedex
2.
LMPA Joseph Liouville, FR 2596 CNRS, Université du Littoral Côte d'Opale, 50, rue F. Buisson, B.P. 699, F-62228 Calais Cedex

Received: August 31, 2006

Revised: March 31, 2007

Published: August 2007

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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.

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Mathematics Subject Classification: 35B05, 35B40,35K55, 35K57, 35K65, 35R45.

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