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Volume 13
Issue 5
10.3390/math13050703
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Article

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

by
Carlo Bianca
1 and
Christian Dogbe
2,*
1
EFREI Research Lab, Université Paris-Panthéon-Assas, 30/32 Avenue de la République, 94800 Villejuif, France
2
UNICAEN, CNRS, LMNO, Normandie University, 14000 Caen, France
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(5), 703; https://doi.org/10.3390/math13050703
Submission received: 26 December 2024 / Revised: 28 January 2025 / Accepted: 17 February 2025 / Published: 21 February 2025
(This article belongs to the Section E4: Mathematical Physics)
Versions Notes

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

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MDPI and ACS Style

Bianca, C.; Dogbe, C. Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces. Mathematics 2025, 13, 703. https://doi.org/10.3390/math13050703

AMA Style

Bianca C, Dogbe C. Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces. Mathematics. 2025; 13(5):703. https://doi.org/10.3390/math13050703

Chicago/Turabian Style

Bianca, Carlo, and Christian Dogbe. 2025. "Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces" Mathematics 13, no. 5: 703. https://doi.org/10.3390/math13050703

APA Style

Bianca, C., & Dogbe, C. (2025). Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces. Mathematics, 13(5), 703. https://doi.org/10.3390/math13050703

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