Existence and Asymptotic Behavior of Solutions to a Semilinear Hyperbolic-Parabolic Model of Chemotaxis

TL;DR

This paper proves the global existence and analyzes the long-term behavior of solutions to a complex hyperbolic-parabolic chemotaxis model, combining hyperbolic and parabolic analysis techniques.

Contribution

It establishes the existence and asymptotic properties of solutions for a multidimensional hyperbolic chemotaxis model, extending classical dissipative problem frameworks.

Findings

Global smooth solutions exist for the model.

Asymptotic decay rates of solutions are determined.

The model exhibits unique features not covered by classical dissipative frameworks.

Abstract

We consider a general hyperbolic model of chemotaxis in the multidimensional case. For this system we show the global existence of smooth solutions to the Cauchy problem and we determine their asymptotic behavior. Since this model does not enter in the classical framework of dissipative problems, we analyze it combining the features of the hyperbolic and the parabolic parts and using detailed decay estimates of the Green function.