Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms

Pages: 587 - 600, Volume 8, Issue 2, March 2009

doi:10.3934/cpaa.2009.8.587       Abstract        Full Text (183.5K)       Related Articles

Shaohua Chen - Department of Mathematics, Physics & Geology, Cape Breton University, Sydney, NS, Canada, B1P 6L2, Canada (email)

Abstract: This paper deals with the bounded and blowup solutions of the quasilinear parabolic system $u_t = u^p ( \Delta u + a v) + f(u, v, Du, x)$ and $v_t = v^q ( \Delta v + b u) + g(u, v, Dv, x)$ with homogeneous Dirichlet boundary condition. Under suitable conditions on the lower order terms $f$ and $g$, it is shown that all solutions are bounded if $(1+c_1) \sqrt{ab} < \l_1$ and blow up in a finite time if $(1+c_1) \sqrt{ab} > \lambda_1$, where $\lambda_1$ is the first eigenvalue of $-\Delta $ in $\Omega$ with Dirichlet data and $c_1 > -1$ related to $f$ and $g$.

Keywords:  Bounded solutions, blowup solutions, quasilinear parabolic systems, lower order terms
Mathematics Subject Classification:  Primary: 35K57, 35B40; Secondary: 35J55

Received: March 2008;      Revised: September 2008;      Published: December 2008.