Global Existence of a Chemotactic Movement with Singular Sensitivity by Two Stimuli

Heping Ma,Shiping Wen

doi:10.1155/2021/9323338

Abstract

In this study, we deal with the chemotaxis system with singular sensitivity by two stimuli under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Under appropriate regularity assumptions on the initial data, we show that the system possesses global classical solution. Our results generalize and improve previously known ones.

Highlights

When τ1, τ2 > 0, D(u) > 0 for u ≥ 0 and D(u) ≥ dum− 1 with d > 0 and m ≥ 1 for all u > 1, Li et al [22] proved that the corresponding initial boundary value problem possesses a unique global bounded classical solution for m > 2 − (2/N)
Inspired by the arguments in previous studies [8, 13, 14, 26, 27], we mainly investigate the global classical solution in a chemotactic movement with singular sensitivity by two stimuli. eorem
Under the framework of fixed point theorem, we will prove the local existence of classical solution to system (3) in the following lemma. e proof is quite standard, and a more detailed display of a similar reasoning in a related circumstance can be found in [14]

Summary

Introduction

Chemotaxis is a well-known biological phenomenon describing the collective motion of cells or the evolution of density of bacteria driven by chemicals, including embryo development, skin wound healing, cancer invasion, and metastasis. e pioneering works of the chemotaxis model was introduced by Keller and Segel in [1], describing the aggregation of cellular slime mold toward a higher concentration of a chemical signal, which reads. Winkler [8] proved that if initial data satisfy appropriate regularity assumptions, system (2) possesses at least one global generalized solution in two-dimensional bounded domains. When u is [13] showed replaced by that for any sufficiently smooth initial data, system (2) admits a global classical solution when either N 1 and α < 2 or N ≥ 2 and α < 1 − (N/4). When τ1, τ2 > 0, D(u) > 0 for u ≥ 0 and D(u) ≥ dum− 1 with d > 0 and m ≥ 1 for all u > 1, Li et al [22] proved that the corresponding initial boundary value problem possesses a unique global bounded classical solution for m > 2 − (2/N).

Preliminaries and Main Result

Proof of Theorem 1