Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Well-Posedness of Stochastic Chemotaxis System

Yunfeng Chen, Jianliang Zhai, Tusheng Zhang

Abstract

In this paper, we establish the existence and uniqueness of solutions of elliptic-parabolic stochastic Keller-Segel systems. The solution is obtained through a carefully designed localization procedure together with some a priori estimates. Both noise of linear growth and nonlinear noise are considered. The Lp Ito formula plays an important role.

Well-Posedness of Stochastic Chemotaxis System

Abstract

In this paper, we establish the existence and uniqueness of solutions of elliptic-parabolic stochastic Keller-Segel systems. The solution is obtained through a carefully designed localization procedure together with some a priori estimates. Both noise of linear growth and nonlinear noise are considered. The Lp Ito formula plays an important role.

Paper Structure

This paper contains 7 sections, 18 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Statement of the Main Result
  3. Existence and Uniqueness of Local Solutions
  4. Existence of Local Solutions
  5. Uniqueness of Local Solutions
  6. Global Existence
  7. The case of nonlinear noise

Key Result

Lemma 2.1

Let $\left(e^{-tA}\right)_{t\geq 0}$ be the Neumann heat semigroup in $\mathcal{O}$. Let $\nu_1$ denote the first nonzero eigenvalue of $A$ on $\mathcal{O}$ under Neumann boundary condition. Then, there exists a constant $C$, depending only on $\mathcal{O}$, such that:

Theorems & Definitions (33)

  • Lemma 2.1
  • Definition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.2
  • ...and 23 more