Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 10, pp. 1-14.
Title: Nonlinear convection in reaction-diffusion equations under
dynamical boundary conditions
Authors: Gaelle Pincet Mailly (Univ. Lille Nord de France, Calais Cedex, France)
Jean-Francois Rault (Univ. Lille Nord de France, Calais Cedex, France)
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 09, 2013.
Math Subject Classifications: 35K55, 35B44.
Key Words: Nonlinear parabolic problem; dynamical boundary conditions;
lower and upper-solution; blow-up; global solution.