Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces

Abstract

This paper is devoted to the uniqueness of solutions for a class of nonhomogeneous stationary partial differential equations related to Hamilton–Jacobi-type equations in infinite-dimensional Hilbert spaces. Specifically, the uniqueness of the viscosity solution is established by employing the inf/sup-convolution approach in a separable infinite-dimensional Hilbert space. The proof is based on the Faedo–Galerkin approximate method by assuming the existence of a Hilbert–Schmidt operator and by employing modulus continuity and Lipschitz arguments. The results are of interest regarding the stochastic optimal control problem.

Keywords

• nonlinear PDEs
• PDEs in infinite-dimensional Hilbert space
• Hamilton–Jacobi equations
• stationary equation
• viscosity solution