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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

Shaohua Chen 1,

1. 

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

Received  March 2008 Revised  September 2008 Published  December 2008

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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