IJPAM: Volume 58, No. 3 (2010)

GLOBAL EXISTENCE AND LARGE TIME BEHAVIOR OF
SOLUTIONS FOR PARABOLIC SYSTEMS WITH
NONLINEAR GRADIENTS TERMS

Abeer AL-Elaiw

, Slim Tayachi

Department of Mathematics
Faculty of Girls Education
King Faisal University
P.O. Box 789, Al-Hofuf, 31982, KINGDOM OF SAUDI ARABIA
e-mail: a.alelaiw@gmail.com

Department of Mathematics
Faculty of Science
University ``El-Manar'' of Tunis
Tunis, 1060, TUNISIA
e-mail: slim.tayachi@fst.rnu.tn

Abstract.In this paper we study the global existence of mild solutions for the nonlinear parabolic system: $\ds \pa_tu=\D u+a\vert\nb v\vert^p$ , $\ds \pa_tv=\D v+b\vert\nb u\vert^q$ , , $x \in {\rb}^N$ , where $a,\; b\in \rb$ , $N\geq 1$ and $1<p\leq q<2$ . Under the condition $pq>\frac{q}{N+1}+\frac{N+2}{N+1}$ and suitable smallness conditions on the initial values we prove the existence of global solutions. We study also the large time behavior for some of these global solutions. We prove that if the initial values $\textstyle u(0,x)\sim \o_1\left(x/\vert x\vert\right)\vert x\vert^{-{2-p(q-1)\over pq-1}}$ , $v(0,x)\sim \o_2\left(x/\vert x\vert\right)\vert x\vert^{-{2-q(p-1)\over pq-1}}$ as $\vert x\vert\rightarrow \f$ and under suitable conditions on $\o_1$ , $\o_2$ the resulting solutions are asymptotically self-similar. The asymptotic behavior is established in the $W^{1,\f}$ -norm and is stable under some small perturbations.