Two-dimensional reaction-diffusion equations with memory

Abstract

In a two-dimensional space domain, we consider a reaction-diffusion equation whose diffusion term is a time convolution of the Laplace operator against a nonin-creasing summable memory kernel κ. This equation models several phenomena arising from many different areas. After rescaling κ by a relaxation time e > 0, we formulate a Cauchy-Dirichlet problem, which is rigorously translated into a similar problem for a semilinear hyperbolic integro-differential equation with nonlinear damping, for a particular choice of the initial data. Using the past history approach, we show that the hyperbolic equation generates a dynamical system which is dissipative provided that e is small enough, namely, when the equation is sufficiently close to the standard reaction-diffusion equation formally obtained by replacing κ with the Dirac mass at 0. Then, we provide an estimate of the difference between e-trajectories and 0-trajectories, and we construct a family of regular exponential attractors which is robust with respect to the singular limit e → 0. In particular, this yields the existence of a regular global attractor of finite fractal dimension. Convergence to equilibria is also examined. Finally, all the results are reinterpreted within the original framework.