A Stochastic Heat Equation with Nonlinear Dissipation on the Boundary

Abstract

In this paper, we discuss the existence and uniqueness of variational solutions for a stochastic heat equation in a bounded domain with both additive and laminar multiplicative noise and with a nonlinear dissipative condition on the boundary. Our construction includes, as a particular case, the Signorini-type boundary conditions.

A Variational Approach to Neumann Stochastic Semi-Linear Equations Modeling the Thermostatic Control

“…Recently, the same type of equation was studied in [7], and there is proved existence of a variational solution using different assumptions.…”

“…The interest of this work comes first from the fact that our assumptions are weaker than those needed in classical theory. On the other hand, the mild solution is obtained here without the growth assumptions from [7]. The main result is obtained by constructing an optimal control problem, which is equivalent to the equation when the cost functional applied to the optimal pair is equal to zero.…”