Global existence and nonexistence for degenerate parabolic systems

Abstract:

The initial-boundary value problems are considered for the strongly coupled degenerate parabolic system \[ \begin {split} u_t=v^p(\Delta u+au), v_t=u^q(\Delta v+bv) \end {split} \] in the cylinder $\Omega \times (0,\infty )$, where $\Omega \subset R^N$ is bounded and $p, q, a, b$ are positive constants. We are concerned with the global existence and nonexistence of the positive solutions. Denote by $\lambda _1$ the first Dirichlet eigenvalue for the Laplacian on $\Omega$. We prove that there exists a global solution iff $\lambda _1\geq \min \{a,b\}$.