Blowup for the multiplicative stochastic heat equation with superlinear drift

2025/11/28 by Mathew Joseph, Joseph, Mathew, Shubham Ovhal +1
Economics, Econometrics and Finance · Engineering · Mathematics · #FOS: Mathematics #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2511.23403

openalex publication_date 2025/11/28 · openalex created_date 2025/12/03 · openalex updated_date 2026/07/28

Abstract

We consider the stochastic heat equation with multiplicative white noise: ∂_t u =∂_x²u + b(u) +σ(u) W, both on [0,1] and 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 R we show instantaneous explosion when we start with initial profile u₀≡ 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 R stays above the corresponding solution on [0,1] with Dirichlet boundary conditions.

Citations

• On the local well-posedness of randomly forced reaction-diffusion equations with L² initial data and a superlinear reaction term
• Instantaneous blowup for interacting SDEs with superlinear drift
• Global solution for superlinear stochastic heat equation on ℝ^d under Osgood-type conditions
• Instantaneous everywhere-blowup of parabolic SPDEs
• Global solutions for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity
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• Existence and uniqueness for the mild solution of the stochastic heat equation with non-Lipschitz drift on an unbounded spatial domain
• The Osgood condition for stochastic partial differential equations
• Stochastic heat equations with logarithmic nonlinearity
• An approximation result for a class of stochastic heat equations with\n colored noise
• Strong invariance and noise-comparison principles for some parabolic\n stochastic PDEs

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• Instantaneous everywhere-blowup of parabolic SPDEs
• Blow-up for the one dimensional stochastic wave equations