Nonlinear Convection in Reaction-diffusion Equations under dynamical boundary conditions

Gaëlle Pincet Mailly (1) , Jean-François Rault (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

We investigate 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 determinated.

Mots clés: Nonlinear parabolic problem; Dynamical boundary conditions; Lower and upper-solution; Blow-up; Global solution

Domaines: Equations aux dérivées partielles [math.AP]

Gaëlle Pincet Mailly, Jean-François Rault. Nonlinear Convection in Reaction-diffusion Equations under dynamical boundary conditions. 2012. ⟨hal-00668330v2⟩