Electron. J. Diff. Eqns., Vol. 2007(2007), No. 158, pp. 1-7.

Solution to nonlinear gradient dependent systems with a balance law

Zoubir Dahmani, Sebti Kerbal

Abstract:
In this paper, we are concerned with the initial boundary value problem (IBVP) and with the Cauchy problem to the reaction-diffusion system
$$ \displaylines{ u_t-\Delta u = -u^n |\nabla v |^p,\cr v_t-d \Delta v = u^n |\nabla v|^p, }$$
where $1\leq p\leq 2$, d and n are positive real numbers. Results on the existence and large-time behavior of the solutions are presented.

Submitted April 15, 2007. Published November 21, 2007.
Math Subject Classifications: 35B40, 35B50, 35K57.
Key Words: Reaction-diffusion systems; global existence; asymptotic behavior; maximum principle.

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Zoubir Dahmani
Department of Mathematics, Faculty of Sciences
University of Mostaganem, Mostaganem, Algeria
email: zzdahmani@yahoo.fr
Sebti Kerbal
Department of Mathematics and Statistics
Sultan Qaboos Uiverstiy, Alkhod
Muscat, Sultanate of Oman
email: skerbal@squ.edu.om

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