The Cauchy Problem For Nonlinear Coupled Equations

In this paper, We study the following Cauchy Problem for a class of nonlinear coupledequationswhere f(u,w) and g(u,w) are given nonlinear functions,αis a constant, u(x,t) and u(x,t) areunknown functions; u₀(x), u₁(x),w₀(x) and w₁(x) are given initial value functions. We studythe existence and uniqueness of the local solution to the Cauchy problem (0.1) by the contraction mapping principle. Under the assumptions for nonlinear terms, we prove the existence anduniqueness of the global solution, and give sufficient conditions of blow up of the solution infinite time by convex methods. At last, we give several examples satisfying assumptions.