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

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/su...

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Bibliographic Details
Published in:Mathematics (Basel) Vol. 13; no. 5; p. 703
Main Authors: Bianca, Carlo, Dogbe, Christian
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.03.2025
MDPI
Subjects:
Analysis
Differential equations
Hamilton–Jacobi equations
Hilbert space
Mathematics
nonlinear PDEs
Nonlinear Sciences
Numerical analysis
Operators (mathematics)
Optimal control
Optimization techniques
Partial differential equations
PDEs in infinite-dimensional Hilbert space
stationary equation
Uniqueness
Viscosity
viscosity solution
Germany
ISSN:2227-7390, 2227-7390
Online Access:Get full text
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Summary: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.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math13050703