Existence and Uniqueness of Solutions for Non-Linear Stochastic Partial Differential Equations

Abstract

We state some results on existence and uniqueness for the solution of non linear stochastic PDEs with deviating arguments. In fact, we consider the equation dx(t) + (A(t; x(t)) + B(t; x(¿ (t))) + f(t)) dt = (C(t; x(½(t))) + g(t)) dwt ; where A(t; :) ; B(t; :) and C(t; :) are suitable families of non linear operators in Hilbert spaces, wt is a Hilbert valued Wiener process, and ¿ ; ½ are functions of delay. If A satisfies a coercivity condition and a monotonicity hypothesis, and if B ; C are Lipschitz continuous, we prove that there exists a unique solution of an initial value problem for the precedent equation. Some examples of interest for the applications are given to illustrate the results.

Solutions, Non–Linear Stochastic Partial Differential Equations