On the support of solutions to nonlinear stochastic heat equations

2024/07/09 by Beom-Seok Han, Han, Beom-Seok, Kunwoo Kim +3 · 2 citations
Economics, Econometrics and Finance · Mathematics · #35R60 #60H15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2407.06827

openalex publication_date 2024/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the strict positivity and the compact support property of solutions to the one-dimensional nonlinear stochastic heat equation: ∂_t u(t,x) = (1)/(2)∂²_x u(t,x) + σ(u(t,x))W(t,x), (t,x)∈ R₊\timesR, with nonnegative and compactly supported initial data u₀, where W is the space-time white noise and σ:R → R is a continuous function with σ(0)=0. We prove that (i) if v/ σ(v) is sufficiently large near v=0, then the solution u(t,⋅) is strictly positive for all t>0, and (ii) if v/σ(v) is sufficiently small near v= 0, then the solution u(t,⋅) has compact support for all t>0. These findings extend previous results concerning the strict positivity and the compact support property, which were analyzed only for the case σ(u)≈ u^γ for γ>0. Additionally, we establish the uniqueness of a solution and the weak comparison principle in case (i).

Cited by

• L²-solutions to stochastic reaction-diffusion equations with superlinear drifts driven by space-time white noise^
• A stochastic heat equation with non-locally Lipschitz coefficients

Related

• The compact support property for solutions to stochastic heat equations with stable noise
• Comparison principle for stochastic heat equation on ℝ^d
• Initial measures for the stochastic heat equation
• The compact support property for solutions to the stochastic partial differential equations with colored noise
• The nonlinear stochastic heat equation with rough initial data:a summary of some new results