Continuous Dependence in Front Propagation for Convective Reaction-Diffusion Models with Aggregative Movements

Luisa Malaguti,Serena Matucci,Cristina Marcelli

doi:10.1155/2011/986738

Abstract

The paper deals with a degenerate reaction-diffusion equation, including aggregative movements and convective terms. The model also incorporates a real parameter causing the change from a purely diffusive to a diffusive-aggregative and to a purely aggregative regime. Existence and qualitative properties of traveling wave solutions are investigated, and estimates of their threshold speeds are furnished. Further, the continuous dependence of the threshold wave speed and of the wave profiles on a real parameter is studied, both when the process maintains its diffusion-aggregation nature and when it switches from it to another regime.

Highlights

This paper deals with the reaction-diffusion equation: vt h v vx D v vx x f v, t ≥ 0, x ∈ R, 1.1 which provides an interesting model in several frameworks such as the population dispersal, ecology, nerve pulses, chemical processes, epidemiology, cancer growth, chemotaxis processes etc
In Theorem 2.2 we prove that the threshold value c∗ k is always a lower semicontinuous function l.s.c. on the whole interval 0, 1 ; in particular, the lower semicontinuity holds when the process 1.7 switches from a purely diffusive to a diffusive-aggregative and from the latter to a purely aggregative behavior
We provide quite general conditions see Theorem 2.2 and Proposition 2.4 either guaranteeing that c∗ k is continuous on 0, 1 or that it fails to be continuous for k 0 or k 1

Summary

Introduction

This paper deals with the reaction-diffusion equation: vt h v vx D v vx x f v , t ≥ 0, x ∈ R, 1.1 which provides an interesting model in several frameworks such as the population dispersal, ecology, nerve pulses, chemical processes, epidemiology, cancer growth, chemotaxis processes etc. The extremal cases α 1 and α 0 respectively correspond to a purely diffusive and a purely aggregative term In the former case, that is, α 1, the presence of t.w.s. in these models and their main qualitative properties have been investigated since a long time, and we refer to 2, 7, 8 for details. Even in this possible ill-posedness context, the t.w.s. above defined are regular solutions for 1.1 , and this increases the interest in studying them As it is clear from the prototype equation 1.4 , very naturally these models include real parameters which frequently cause, on their varying, the transition of the process a diffusive to a diffusive-aggregative regime or from a diffusive-aggregative to a purely aggregative one.

Statements of the Main Results

Reduction to a Singular First-Order Equation

The Continuous Dependence Results