Attractors for the semiflow associated with a class of doubly nonlinear parabolic equations

Abstract

We address a parabolic equation of the form α(u_t)−Δu+W′(u)=f, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions. The “double nonlinearity” is due to the simultaneous presence of the maximal monotone function α and of the derivative W′ of a smooth, but possibly nonconvex, potential W; f is a source term. After recalling an existence result for weak solutions, we show that, among all the weak solutions, at least one for each admissible choice of the initial datum “regularizes” for t>0. Moreover, the class of “regularizing” solutions constitutes a semiflow 𝒮 for which we prove unique continuation for strictly positive times. Finally, we address the long time behavior of 𝒮. In particular, we can prove existence of both global and exponential attractors and investigate the structure of ω-limits of single trajectories.