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

Abstract

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