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

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March 2009, 8(2): 587-600. doi: 10.3934/cpaa.2009.8.587

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

1.	Department of Mathematics, Physics & Geology, Cape Breton University, Sydney, NS, Canada, B1P 6L2

Received March 2008 Revised September 2008 Published December 2008

Abstract
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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, lower order terms, quasilinear parabolic systems.

Mathematics Subject Classification: Primary: 35K57, 35B40; Secondary: 35J55.

Citation: Shaohua Chen. Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms. Communications on Pure & Applied Analysis, 2009, 8 (2) : 587-600. doi: 10.3934/cpaa.2009.8.587

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