Existence and uniqueness results of a stochastic nonlinear heat equation with a constraint of codimension one
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909000717565952 |
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| author | Bawalia, Ashish Brzeźniak, Zdzisław Mohan, Manil T. |
| author_facet | Bawalia, Ashish Brzeźniak, Zdzisław Mohan, Manil T. |
| contents | In this work, we investigate the well-posedness of a stochastic heat equation with an arbitrary (but polynomial) nonlinearity in any dimension $d\geq 1$ perturbed by a multiplicative white noise in the Stratonovich form, subject to an $L^2-$norm constraint on the solution. In bounded smooth domains, we establish the existence of a martingale solution taking values in $H_0^1 \cap L^p$ for arbitrary $2 \le p < \infty$, using a modified Faedo-Galerkin scheme. By utilizing a sequence of self-adjoint operators which are bounded in $L^p$ for any $2 \le p < \infty$, we provide a novel proof of an Itô formula for the $L^p-$norm of the solution. Together with pathwise uniqueness of the martingale solution, the Yamada-Watanabe result then yields the existence of a strong solution and uniqueness in law. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26549 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence and uniqueness results of a stochastic nonlinear heat equation with a constraint of codimension one Bawalia, Ashish Brzeźniak, Zdzisław Mohan, Manil T. Probability In this work, we investigate the well-posedness of a stochastic heat equation with an arbitrary (but polynomial) nonlinearity in any dimension $d\geq 1$ perturbed by a multiplicative white noise in the Stratonovich form, subject to an $L^2-$norm constraint on the solution. In bounded smooth domains, we establish the existence of a martingale solution taking values in $H_0^1 \cap L^p$ for arbitrary $2 \le p < \infty$, using a modified Faedo-Galerkin scheme. By utilizing a sequence of self-adjoint operators which are bounded in $L^p$ for any $2 \le p < \infty$, we provide a novel proof of an Itô formula for the $L^p-$norm of the solution. Together with pathwise uniqueness of the martingale solution, the Yamada-Watanabe result then yields the existence of a strong solution and uniqueness in law. |
| title | Existence and uniqueness results of a stochastic nonlinear heat equation with a constraint of codimension one |
| topic | Probability |
| url | https://arxiv.org/abs/2604.26549 |