Global existence for semilinear parabolic systems via Lyapunov type methods

Abstract

We consider semilinear parabolic systems of partial differential equations of the form

$$u_t \left( {t,x} \right) = D\Delta u\left( {t,x} \right) + f\left( {u\left( {t,x} \right)} \right)t > 0,x \in \Omega$$

((1))

with bounded initial data and homogeneous Neumann boundary conditions, where D is an m by m diagonal positive definite matrix, Ω is a smooth bounded region in R ⁿ and f:R ^m → R ^m is locally Lipschitz. We prove that if the vector field f satisfies a generalized Lyapunov type condition then either at least two components of the solution of (1) becomes unbounded in finite time or the solution exists for all t>0. Our result generalizes a recent result of Hollis, Martin, and Pierre [4], and the proof given is considerably simpler.