Blowup for the multiplicative stochastic heat equation with superlinear drift
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918367089131520 |
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| author | Joseph, Mathew Ovhal, Shubham |
| author_facet | Joseph, Mathew Ovhal, Shubham |
| contents | We consider the stochastic heat equation with multiplicative white noise: $\partial_t u =\partial_x^2u + b(u) +σ(u) \dot W$, both on $[0,1]$ and $\mathbf{R}$. In the case of $[0,1]$ we show that the finite Osgood criterion on $b$ is a necessary and sufficient condition for finite-time blowup, under fairly general conditions on $σ$. In the case of $\mathbf{R}$ we show instantaneous explosion when we start with initial profile $u_0\equiv 1$, extending the work of [10] which dealt with bounded $σ$. The second result follows from the first by a comparison result which shows that the solution on $\mathbf{R}$ stays above the corresponding solution on $[0,1]$ with Dirichlet boundary conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23403 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Blowup for the multiplicative stochastic heat equation with superlinear drift Joseph, Mathew Ovhal, Shubham Probability We consider the stochastic heat equation with multiplicative white noise: $\partial_t u =\partial_x^2u + b(u) +σ(u) \dot W$, both on $[0,1]$ and $\mathbf{R}$. In the case of $[0,1]$ we show that the finite Osgood criterion on $b$ is a necessary and sufficient condition for finite-time blowup, under fairly general conditions on $σ$. In the case of $\mathbf{R}$ we show instantaneous explosion when we start with initial profile $u_0\equiv 1$, extending the work of [10] which dealt with bounded $σ$. The second result follows from the first by a comparison result which shows that the solution on $\mathbf{R}$ stays above the corresponding solution on $[0,1]$ with Dirichlet boundary conditions. |
| title | Blowup for the multiplicative stochastic heat equation with superlinear drift |
| topic | Probability |
| url | https://arxiv.org/abs/2511.23403 |