Electron. J. Diff. Equ., Vol. 2013 (2013), No. 10, pp. 1-14.

Nonlinear convection in reaction-diffusion equations under dynamical boundary conditions

Gaelle Pincet Mailly, Jean-Francois Rault

Abstract:
We study the blow-up phenomena for positive solutions of nonlinear reaction-diffusion equations including a nonlinear convection term $\partial_t u = \Delta u - g(u) \cdot \nabla u + f(u)$ in a bounded domain of $\mathbb{R}^N$ under the dissipative dynamical boundary conditions $\sigma \partial_t u + \partial_\nu u =0$. Some conditions on g and f are discussed to state if the positive solutions blow up in finite time or not. Moreover, for certain classes of nonlinearities, an upper-bound for the blow-up time can be derived and the blow-up rate can be determined.

Submitted July 10, 2012. Published January 9, 2013.
Math Subject Classifications: 35K55, 35B44.
Key Words: Nonlinear parabolic problem; dynamical boundary conditions; lower and upper-solution; blow-up; global solution.

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Gaëlle Pincet Mailly
LMPA Joseph Liouville FR 2956 CNRS
Université Lille Nord de France
50 rue F. Buisson, B. P. 699, F-62228 Calais Cedex, France
email: mailly@lmpa.univ-littoral.fr
Jean-Fran&ccdil;ois Rault
LMPA Joseph Liouville FR 2956 CNRS
Université Lille Nord de France
50 rue F. Buisson, B. P. 699, F-62228 Calais Cedex, France
email: jfrault@lmpa.univ-littoral.fr

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