Non-blowup of stochastic heat equations by noise
Abstract
Consider the heat equation: labeleq:00 du(t,x)=1/2Δu(t,x) dt+b(u(t,x)) dt, &t>0, x∈ D, [1mm] u(t,x)=0,&t≥0, x∈∂ D, u(0,x)=u0(x),&x∈ D. It is well known that superlinear growth of the coefficient b may lead to finite-time blow-up. In this paper, we show that, for every locally Lipschitz reaction coefficient b:R→R, there always exists a locally Lipschitz noise coefficient σ:R→R such that the stochastic heat equation labeleq:01 du(t,x)=1/2Δu(t,x) dt+b(u(t,x)) dt+σ(u(t,x)) dBt, &t>0, x∈ D, [1mm] u(t,x)=0,&t≥0, x∈∂ D, u(0,x)=u0(x),&x∈ D. has a unique global solution for every u0∈ Cc(D). Here (Bt)t≥0 is a real-valued Brownian motion. In particular, if b=0, the equation has a global solution for every locally Lipschitz σ, without any growth restriction at infinity. The proof uses reflected scalar barriers, a Lyapunov estimate, truncation, and comparison techniques.
Related
- Exact temporal variations and parameter estimation for space--time fractional stochastic heat equations driven by multiplicative noise
- Stochastic heat equations with non-Lipschitz coefficients driven by symmetric α-stable Lévy white noise
- Central Limit Theorem for Stochastic Nonlinear Heat Equation with Pure-Jump Lévy White Noise
- Central limit theorems for stochastic heat equation driven by Gaussian noise with heat-kernel covariance
- Stochastic fractional diffusion equations with spatial Gaussian noise in the Stratonovich regime
BibTeX
@misc{chen2026nonblowup,
title = {{Non-blowup of stochastic heat equations by noise}},
author = {Yunfeng Chen and Tusheng Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.13963},
eprint = {2609.13963},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13963},
}
RIS
TY - UNPB
TI - Non-blowup of stochastic heat equations by noise
AU - Yunfeng Chen
AU - Tusheng Zhang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.13963
UR - https://arxiv.org/abs/2609.13963
AB - Consider the heat equation: labeleq:00 du(t,x)=1/2Δu(t,x) dt+b(u(t,x)) dt, &t>0, x∈ D, [1mm] u(t,x)=0,&t≥0, x∈∂ D, u(0,x)=u_0(x),&x∈ D. It is well known that superlinear growth of the coefficient b may lead to finite-time blow-up. In this paper, we show that, for every locally Lipschitz reaction coefficient b:R→R, there always exists a locally Lipschitz noise coefficient σ:R→R such that the stochastic heat equation labeleq:01 du(t,x)=1/2Δu(t,x) dt+b(u(t,x)) dt+σ(u(t,x)) dB_t, &t>0, x∈ D, [1mm] u(t,x)=0,&t≥0, x∈∂ D, u(0,x)=u_0(x),&x∈ D. has a unique global solution for every u_0∈ C_c(D). Here (B_t)_t≥0 is a real-valued Brownian motion. In particular, if b=0, the equation has a global solution for every locally Lipschitz σ, without any growth restriction at infinity. The proof uses reflected scalar barriers, a Lyapunov estimate, truncation, and comparison techniques.
ER -