Luisa Malaguti

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.

On a two-point boundary value problem for the second order ordinary differential equations with singularities

V clanku jsou nalezeny postacujici podminky pro řesitelnost dvoubodove okrajove ulohy pro singularni obycejne diferencialni rovnice druheho řadu.

Nonlocal semilinear evolution equations without strong compactness: theory and applications

A semilinear multivalued evolution equation is considered in a reflexive Banach space. The nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. No strong compactness is assumed, neither on the evolution operator generated by the linear part, or on the nonlinear term. A wide family of nonlocal associated boundary value problems is investigated by means of a fixed point technique. Applications are given to an optimal feedback control problem, to a nonlinear hyperbolic integro-differential equation arising in age-structure population models, and to a multipoint boundary value problem associated to a parabolic partial differential equation.MSC:34G25, 34B10, 34B15, 47H04, 28B20, 34H05.

Floquet Boundary Value Problems for Differential Inclusions: a Bound Sets Approach

A technique is developed for the solvability of the Floquet boundary value problem associated to a differential inclusion. It is based on the usage of a not necessarily C-class of Liapunov-like bounding functions. Certain viability arguments are applied for this aim. Some illustrating examples are supplied.

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.

Bounded solutions of Carathéodory differential inclusions: a bound sets approach

A bound sets technique is developed for Floquet problems of Caratheodory differential inclusions. It relies on the construction of either continuous or locally ipschitzian Lyapunov-like bounding functions. Proceeding sequentially, the existence of bounded trajectories is then obtained. Nontrivial examples are supplied to illustrate our approach.

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.

An approximation solvability method for nonlocal differential problems in Hilbert spaces

A new approach is developed for the solvability of nonlocal problems in Hilbert spaces associated to nonlinear differential equations. It is based on a joint combination of the degree theory with the approximation solvability method and the bounding functions technique. No compactness or condensivity condition on the nonlinearities is assumed. Some applications of the abstract result to the study of nonlocal problems for integro-differential equations and systems of integro-differential equations are then showed. A generalization of the result by using nonsmooth bounding functions is given.

Nonsmooth feedback controls of nonlocal dispersal models

The paper deals with a nonlocal diffusion equation which is a model for biological invasion and disease spread. A nonsmooth feedback control term is included and the existence of controlled dynamics is proved, satisfying different kinds of nonlocal condition. Jump discontinuities appear in the process. The existence of optimal control strategies is also shown, under suitably regular control functionals. The investigation makes use of techniques of multivalued analysis and is based on the degree theory for condensing operators in Hilbert spaces.