Global Existence for Semilinear Parabolic Systems

Global existence results are obtained for semilinear parabolic systems of partial differential equations of the form \[ u_t = D\Delta u + (fu)\quad {\text{on }}\Omega \times (0,T)\] with bounded initial data and various boundary conditions, where D is an $m \times m$ diagonal matrix with positive entries on the diagonal, $\Omega $ is a smooth bounded domain in $R^n $, and $f:R^m \to R^m $ is locally Lipschitz. These results are based on f satisfying a Lyapunov-type condition, and generalize a previous result of l Iollis, Martin, and Pierre [SIAM J. Math. Anal., 18 (1987), pp. 744–761]. This theory is applied to some specific reaction-diffusion and nerve conduction problems.