Two-dimensional reaction-diffusion equations with memory

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 nonincreasing summable memory kernel k. This equation models several phenomena arising from many different areas. After rescaling k by a relaxation time ε>0, we formulate the corresponding 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 ε is small enough, namely, when the equation is sufficiently ``close" to thestandard reaction-diffusion equation formally obtained by replacing k with the Dirac mass at 0. Then, we provide an estimate of the difference between ε-trajectories and 0-trajectories, and we construct a family of regular exponential attractors which is robust with respect to the singular limit ε→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.