No abstract

We consider reaction-diffusion equations of porous medium type, with different kind of reaction terms, and nonnegative bounded initial data. For all the reaction terms under consideration there are initial data for which the solution converges to 1 uniformly in compact sets for large times. We will characterize for which reaction terms this happens for all nontrivial nonnegative initial data, and for which ones there are also solutions converging uniformly to 0. Problems in this family have a unique (up to translations) travelling wave with a finite front and we will see how its speed gives the asymptotic velocity of all the solutions with compactly supported initial data. We will also prove in the one-dimensional case that solutions with bounded compactly supported initial data converging to 1 do so approaching a translation of this unique traveling wave. We will prove a similar result for non-compactly supported initial data in a certain class.2010 Mathematics Subject Classification. 35K55. 35B40. 35K65, 76S05

We prove existence, uniqueness and several qualitative properties for evolution equations that combine local and nonlocal diffusion operators acting in different subdomains and coupled in such a way that the resulting evolution equation is the gradient flow of an energy functional. We deal with the Cauchy, Neumann and Dirichlet problems, in the last two cases with zero boundary data. For the first two problems we prove that the model preserves the total mass. We also study the behaviour of the solutions for large times. Finally, we show that we can recover the usual heat equation (local diffusion) in a limit procedure when we rescale the nonlocal kernel in a suitable way.

We consider a family of singular Volterra integral equations that appear in the study of monotone travelling-wave solutions for a family of diffusionconvection-reaction equations involving the p-Laplacian operator. Our results extend the ones due to B. Gilding for the case p = 2. The fact that p = 2 modifies the nature of the singularity in the integral equation, and introduces the need to develop some new tools for the analysis. The results for the integral equation are then used to study the existence and properties of travellingwave solutions for doubly nonlinear diffusion-reaction equations in terms of the constitutive functions of the problem.

No abstract