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On the local well-posedness of randomly forced reaction-diffusion equations with $L^2$ initial data and a superlinear reaction term

Mohammud Foondun, Davar Khoshnevisan, Eulalia Nualart

TL;DR

This work addresses local well-posedness of a parabolic SPDE driven by space-time white noise with diffusion coefficient bounded and Lipschitz, and drift with $L\log L$-type growth, for initial data $u_0\in L^2[0,1]$. The authors introduce a novel truncation/stopping-time strategy to control extreme behavior at early times and develop a tailored functional framework consisting of Banach spaces that capture small-time regularity of random fields. They prove existence, uniqueness, and regularity of a local random-field solution for a short time $t_0>0$, with sub-Gaussian moment bounds and convergence to $u_0$ as $t\to0+$, and they establish stability with respect to initial data. The results provide a strong affirmative answer to a long-standing open question and yield a robust local well-posedness theory that can potentially extend to broader SPDEs with similar growth and noise structures, paving the way for global-in-time results under additional structure.

Abstract

We consider a parabolic stochastic partial differential equation (SPDE) on $[0\,,1]$ that is forced with multiplicative space-time white noise with a bounded and Lipschitz diffusion coefficient and a drift coefficient that is locally Lipschitz and satisfies an $L\log L$ growth condition. We prove that the SPDE is well posed when the initial data is in $L^2[0\,,1]$. This solves a strong form of an open problem.

On the local well-posedness of randomly forced reaction-diffusion equations with $L^2$ initial data and a superlinear reaction term

TL;DR

This work addresses local well-posedness of a parabolic SPDE driven by space-time white noise with diffusion coefficient bounded and Lipschitz, and drift with -type growth, for initial data . The authors introduce a novel truncation/stopping-time strategy to control extreme behavior at early times and develop a tailored functional framework consisting of Banach spaces that capture small-time regularity of random fields. They prove existence, uniqueness, and regularity of a local random-field solution for a short time , with sub-Gaussian moment bounds and convergence to as , and they establish stability with respect to initial data. The results provide a strong affirmative answer to a long-standing open question and yield a robust local well-posedness theory that can potentially extend to broader SPDEs with similar growth and noise structures, paving the way for global-in-time results under additional structure.

Abstract

We consider a parabolic stochastic partial differential equation (SPDE) on that is forced with multiplicative space-time white noise with a bounded and Lipschitz diffusion coefficient and a drift coefficient that is locally Lipschitz and satisfies an growth condition. We prove that the SPDE is well posed when the initial data is in . This solves a strong form of an open problem.

Paper Structure

This paper contains 11 sections, 24 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Banach spaces of locally bounded functions
  3. Some preliminary integral inequalities
  4. On convolutions: Bounded case
  5. On stochastic convolutions: Bounded case
  6. The Lipschitz case with $L^2$ initial data
  7. Existence
  8. Regularity
  9. Uniqueness
  10. Stability
  11. Proof of Theorem \ref{['th:main']}

Key Result

Theorem 1.3

Suppose that $u_0\in L^2[0\,,1]$, and that Assumption cond-dif holds. Then, there exists a nonrandom number $t_0>0$ such that SHE:1 has a random-field solution $u=u(t\,,x)$ for all $(t\,,x)\in(0\,,t_0]\times[0\,,1]$ that satisfies the following: Finally, if $v$ is any other continuous random-field solution to SHE:1 on the time interval $(0\,,t_0]$ such that $\sup_{s\in(0,t_0]}(s^\alpha\|v(s)\|_{C

Theorems & Definitions (47)

  • Definition 1.2: Def. 1.3 of DKZ
  • Theorem 1.3
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Proposition \ref{['pr:norm<N']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 37 more