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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MDPI AG
2025-02-01
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| author | Carlo Bianca Christian Dogbe |
| author_facet | Carlo Bianca Christian Dogbe |
| author_sort | Carlo Bianca |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-a2c181a24f24453c93bf64f38400af40 |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-a2c181a24f24453c93bf64f38400af402025-08-20T02:59:15ZengMDPI AGMathematics2227-73902025-02-0113570310.3390/math13050703Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert SpacesCarlo Bianca0Christian Dogbe1EFREI Research Lab, Université Paris-Panthéon-Assas, 30/32 Avenue de la République, 94800 Villejuif, FranceUNICAEN, CNRS, LMNO, Normandie University, 14000 Caen, FranceThis 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.https://www.mdpi.com/2227-7390/13/5/703nonlinear PDEsPDEs in infinite-dimensional Hilbert spaceHamilton–Jacobi equationsstationary equationviscosity solution |
| spellingShingle | Carlo Bianca Christian Dogbe Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces Mathematics nonlinear PDEs PDEs in infinite-dimensional Hilbert space Hamilton–Jacobi equations stationary equation viscosity solution |
| title | Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces |
| title_full | Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces |
| title_fullStr | Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces |
| title_full_unstemmed | Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces |
| title_short | Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces |
| title_sort | uniqueness results of semilinear parabolic equations in infinite dimensional hilbert spaces |
| topic | nonlinear PDEs PDEs in infinite-dimensional Hilbert space Hamilton–Jacobi equations stationary equation viscosity solution |
| url | https://www.mdpi.com/2227-7390/13/5/703 |
| work_keys_str_mv | AT carlobianca uniquenessresultsofsemilinearparabolicequationsininfinitedimensionalhilbertspaces AT christiandogbe uniquenessresultsofsemilinearparabolicequationsininfinitedimensionalhilbertspaces |