Global existence for semilinear parabolic systems on one-dimensional bounded domains (Q1176413)

The author is considering the following semilinear parabolic system \[ u_ t=Du_{xx}+Cu_ x+f(u), 0<x<1, \quad u(t,0)=u(t,1)=0, \quad u(0,x)=u_ 0(x), \leqno (1.1) \] where the unknown function \(u\) is vector valued and the nonlinear term \(f\) is a smooth map from \(\mathbb{R}^ m\) into \(\mathbb{R}^ m\) and \(D\) and \(C\) are matrices satisfying certain parabolicity conditions. To consider the global existence of solutions, the author assumes a kind of Lyapunov structure on the nonlinear term \(f\). That is, there exists a Lyapunov function for the following ordinary differential equation (vector field on some region of \(\mathbb{R}^ m\)), \[ dv/dt=f(v). \leqno (1.2) \] Recently, such type of methods has been employed by several people (see the references). In this paper, the author gives a generalization of his earlier work to cover the equations including a convection term. He gives conditions on the nonlinear term which ensures the global unique existence of the solution to the initial boundary value problem to (1.1).