Positive probability of explosion for stochastic heat equation with superlinear accretive reaction term and polynomially growing multiplicative noise
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910270029299712 |
|---|---|
| author | Salins, Michael Zhang, Yuyang |
| author_facet | Salins, Michael Zhang, Yuyang |
| contents | This paper studies the finite time explosion of the stochastic heat equation $\frac{\partial u}{\partial t}(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)+(u(t,x))^β+σ(u(t,x))\dot{W}(t,x)$. We consider an interval $D=[-π,π]$ under periodic boundary condition where $\dot{W}(t,x)$ is a space-time white noise and $σ(u)\approx u^γ$ near $\infty$. Our results refine existing results by identifying behavior in a previously less understood regime, where we show that if $β\in(1,3),γ\in(\fracβ{2},\frac{β+3}{4})$ or $β>1,γ\in(0,\fracβ{2}]$ then mild solutions can explode with positive probability. This paper provides a partial characterization of the explosion behavior in an intermediate parameter regime, and contribute to the understanding of the interplay between the drift and diffusion terms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11319 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Positive probability of explosion for stochastic heat equation with superlinear accretive reaction term and polynomially growing multiplicative noise Salins, Michael Zhang, Yuyang Probability 60H15 This paper studies the finite time explosion of the stochastic heat equation $\frac{\partial u}{\partial t}(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)+(u(t,x))^β+σ(u(t,x))\dot{W}(t,x)$. We consider an interval $D=[-π,π]$ under periodic boundary condition where $\dot{W}(t,x)$ is a space-time white noise and $σ(u)\approx u^γ$ near $\infty$. Our results refine existing results by identifying behavior in a previously less understood regime, where we show that if $β\in(1,3),γ\in(\fracβ{2},\frac{β+3}{4})$ or $β>1,γ\in(0,\fracβ{2}]$ then mild solutions can explode with positive probability. This paper provides a partial characterization of the explosion behavior in an intermediate parameter regime, and contribute to the understanding of the interplay between the drift and diffusion terms. |
| title | Positive probability of explosion for stochastic heat equation with superlinear accretive reaction term and polynomially growing multiplicative noise |
| topic | Probability 60H15 |
| url | https://arxiv.org/abs/2605.11319 |