Global Existence for Semilinear Parabolic Systems

Global existence results are obtained for semilinear parabolic systems of partial differential equations of the form \[ u_t = DΔ u + (fu) {on }Ω × (0,T)\] with bounded initial data and various boundary conditions, where D is an m × m diagonal matrix with positive entries on the diagonal, Ω is a smooth bounded domain in Rⁿ, and f:Rᵐ → Rᵐ 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.