Perturbación y decaimiento en ecuaciones parabólicas no autónomas

Damos condiciones finas sobre el tamaño y la forma de una perturbación que consigue que un problema lineal no autónomo pase de ser neutralmente estable a exponencialmente estable. Estos resultados se aplican al estudio del comportamiento asintótico de ecuaciones logísticas no autónomas

Optimal existence classes and nonlinear-like dynamics in the heat equation in Rd

We analyse the behaviour of solutions of the linear heat equation in R d for initial data in the classes M ε (Rd) of Radon measures with ∫ R d e − ε | x | 2 d | u 0 | 0 M ε (Rd) consists precisely of those initial data for which the a solution of the heat equation can be given for all time using the heat kernel representation formula. After considering properties of existence, uniqueness, and regularity for such initial data, which can grow rapidly at infinity, we go on to show that they give rise to properties associated more often with nonlinear models. We demonstrate the finite-time blowup of solutions, showing that the set of blowup points is the complement of a convex set, and that given any closed convex set there is an initial condition whose solutions remain bounded precisely on this set at the ‘blowup time’. We also show that wild oscillations are possible from non-negative initial data as t →∞ (in fact we show that this behaviour is generic), and that one can prescribe the behaviour of u (0 ,t ) to be any real-analytic function γ ( t ) on [0 , ∞ )

Linear non-local diffusion problems in metric measure spaces

The aim of this paper is to provide a comprehensive study of some linear non-local diffusion problems in metric measure spaces. These include, for example, open subsets in ℝN, graphs, manifolds, multi-structures and some fractal sets. For this, we study regularity, compactness, positivity and the spectrum of the stationary non-local operator. We then study the solutions of linear evolution non-local diffusion problems, with emphasis on similarities and differences with the standard heat equation in smooth domains. In particular, we prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods

Asymptotic behaviour for a phase field model in higher order Sobolev spaces

In this paper we analyze the long time behavior of a phase field model by showing the existence of global compact attractors in the strong norm of high order Sobolev spaces

The sub-supertrajectory method. Application to the nonautonomous competition Lotka-Volterra model

In this paper we study in detail the pullback and forwards attractions to non-autonomous competition Lotka-Volterra system. In particular, under some conditions on the parameters, we prove the existence of a unique non-degenerate global solution for these models, which attracts any other complete bounded trajectory. For that we present the sub-supertrajectory tool as a generalization of the now classical subsupersolution method

On the loss of mass for the heat equation in an exterior domain with general boundary conditions

In this work, we study the decay of mass for solutions to the heat equation in exterior domains, i.e., domains which are the complement of a compact set in $\mathbb{R}^N$. Different homogeneous boundary conditions are considered, including Dirichlet, Robin, and Neumann conditions. We determine the exact amount of mass loss and identify criteria for complete mass decay, in which the dimension of the space plays a key role. Furthermore, the paper provides explicit mass decay rates.Comment: Changed a spelling error in an author's surnam