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Stochastic and deterministic non-autonomous reaction-diffusion equations

Davide A. Bignamini, Paolo De Fazio

TL;DR

The article develops a rigorous framework for non-autonomous deterministic and stochastic reaction–diffusion equations with polynomial nonlinearities in a Banach-space setting. By combining evolution-operator theory, Yosida approximations for dissipative nonlinearities, and a mild-solution approach, it establishes well-posedness for both deterministic and stochastic problems, including uniqueness and a priori estimates. A key technical contribution is a space–time regularity result for the non-autonomous stochastic convolution, obtained via a non-autonomous factorization method and Sobolev embeddings. The paper also furnishes concrete examples of operator families and nonlinearities showing the broad applicability of the abstract results to time-dependent diffusion operators and polynomial reactions, with implications for models in biology, physics, and beyond.

Abstract

In this paper we prove the well-posedness of non-autonomous deterministic and stochastic reaction-diffusion equations with a polynomial reaction term. Concerning the stochastic problem, we also prove a new result on the space-time regularity of the non-autonomous stochastic convolution.

Stochastic and deterministic non-autonomous reaction-diffusion equations

TL;DR

The article develops a rigorous framework for non-autonomous deterministic and stochastic reaction–diffusion equations with polynomial nonlinearities in a Banach-space setting. By combining evolution-operator theory, Yosida approximations for dissipative nonlinearities, and a mild-solution approach, it establishes well-posedness for both deterministic and stochastic problems, including uniqueness and a priori estimates. A key technical contribution is a space–time regularity result for the non-autonomous stochastic convolution, obtained via a non-autonomous factorization method and Sobolev embeddings. The paper also furnishes concrete examples of operator families and nonlinearities showing the broad applicability of the abstract results to time-dependent diffusion operators and polynomial reactions, with implications for models in biology, physics, and beyond.

Abstract

In this paper we prove the well-posedness of non-autonomous deterministic and stochastic reaction-diffusion equations with a polynomial reaction term. Concerning the stochastic problem, we also prove a new result on the space-time regularity of the non-autonomous stochastic convolution.

Paper Structure

This paper contains 16 sections, 17 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Left differentiable and dissipative mappings
  5. The Yosida approximations for dissipative functions
  6. Abstract evolution problems
  7. The continuous case
  8. Abstract parabolic problem
  9. Well-posedness of the deterministic problem \ref{['NAPP']}
  10. Approximating problems
  11. Well-posedness of \ref{['eq']}
  12. Well-posedness of the stochastic problem \ref{['stocpbintro']}
  13. Regularity of stochastic convolution
  14. Example
  15. Families of operators TEXT and TEXT
  16. ...and 1 more sections

Key Result

Proposition 2.3

Let $f:{\rm D}(f)\subseteq V\longrightarrow V$. $f$ is dissipative if and only if, for every $x,y\in {\rm D}(f)$ there exists $z^*\in\partial\left\lVert x-y\right\rVert_V$ such that If $V$ is a Hilbert space endowed with the scalar product $\langle\cdot,\cdot\rangle_V$, disban reads as

Theorems & Definitions (52)

  • Example 1.1: Non-autonomous reaction-diffusion equations
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Remark 2.8
  • ...and 42 more