Cristina Marcelli

Travelling Wavefronts in Reaction‐Diffusion Equations with Convection Effects and Non‐Regular Terms

This paper deals with the appearance of monotone bounded travelling wave solutions for a parabolic reaction-diffusion equation which frequently meets both in chemical and biological systems. In particular, we prove the existence of monotone front type solutions for any wave speed c ≥ c* and give an estimate for the threshold value c*. Our model takes into account both of a density dependent diffusion term and of a non-linear convection effect. Moreover, we do not require the main non-linearity g to be a regular C1 function; in particular we are able to treat both the case when g′(0) = 0, giving rise to a degenerate equilibrium point in the phase plane, and the singular case when g′(0) = +∞. Our results generalize previous ones due to Aronson and Weinberger [Adv. Math. 30 (1978), pp. 33–76 ], Gibbs and Murray (see Murray [Mathematical Biology, Springer-Verlag, Berlin, 1993 ]) and McCabe, Leach and Needham [SIAM J. Appl. Math. 59 (1998), pp. 870–899 ]. Finally, we obtain our conclusions by means of a comparison-type technique which was introduced and developed in this framework in a recent paper by the same authors.

Finite Speed of Propagation in Monostable Degenerate Reaction-Diffusion-Convection Equations

Abstract We study the existence and properties of travelling wave solutions of the Fisher-KPP reaction-diffusion-convection equation ut + h(u)ux = [D(u)ux]x + g(u), where the diffusivity D(u) is simply or doubly degenerate. Both the cases when Ḋ(0) and Ḋ(1) are possibly zero real values or infinity, are treated. We discuss the effects, due to the presence of a convective term, concerning the property of finite speed of propagation. Moreover, in the doubly degenerate case we show the appearance of new types of profiles and provide their classification according to sharp relations between the nonlinear terms of the model. An application is also presented, concerning the evolution of a bacterial colony.

AGGREGATIVE MOVEMENT AND FRONT PROPAGATION FOR BI-STABLE POPULATION MODELS

Front propagation for the aggregation-diffusion-reaction equation is investigated, where f is a bi-stable reaction-term and D(v) is a diffusion coefficient with changing sign, modeling aggregating-diffusing processes. We provide necessary and sufficient conditions for the existence of traveling wave solutions and classify them according to how or if they attain their equilibria at finite times. We also show that the dynamics can exhibit the phenomena of finite speed of propagation and/or finite speed of saturation.

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

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.

Some results on the method of lower and upper solutions for non-compact domains

of the best-known existence results on method lower and upper solutions in initial value problems for systems of O.D.E. is the classical Miiller theorem [l]. According to this fundamental result, for every pair (u, v) of lower and upper solutions, such that u(t) 5 v(t) in [a, b]

A new estimate on the minimal wave speed for travelling fronts in reaction–diffusion–convection equations

In this paper we prove a new estimate of the threshold wave speed for travelling wavefronts of the reaction–diffusion–convection equations of the type vτ + h(v)vx = [D(v)vx]x + f (v) where h is a convective term, D is a positive (potentially degenerate) diffusive term and f stands for a monostable reaction term.