Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions

Abstract

We consider the Cauchy problem in R d for a class of semilinear parabolic partial dier- ential equations that arises in some stochastic control problems. We assume that the coecients are unbounded and locally Lipschitz, not necessarily dierentiable, with continuous data and local uniform ellipticity. We construct a classical solution by approximation with linear parabolic equations. The linear equations involved can not be solved with the traditional results. Therefore, we construct a classical solution to the linear Cauchy problem under the same hypotheses on the coecients for the semilinear equation. Our approach is using stochastic dierential equations and parabolic dierential equations in bounded domains. Finally, we apply the results to a stochastic optimal consumption problem. R