WIAS Preprint No. 1194, (2006)

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



Authors

  • Schimperna, Giulio
  • Segatti, Antonio

2010 Mathematics Subject Classification

  • 35K55, 35B40, 35B41

Keywords

  • Doubly nonlinear equation, singular potential, semiflow, global attractor, energy method, $omega$-limit

Abstract

A doubly nonlinear parabolic equation of the form $alpha(u_t)-Delta u +W'(u)ni f $, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions, is addressed. The two nonlinearities are given by the maximal monotone function $alpha$ and by the derivative $W'$ of a smooth but possibly nonconvex potential $W$; $f$ is a given known source. After defining a proper notion of solution and recalling a related existence result, we show that from any initial datum emanates at least one solution which gains further regularity for $t>0$. Such regularizing solutions contitute a semiflow $S$ for which unqueness is satisfied for strictly positive times and we can study long time behaviour properties,. In particular, we can prove existence of both global and exponential attractors and investigate the structure of $omega$-limits of single trajectories.

Appeared in

  • Asymptot. Anal., 56 (2008) pp. 61 - 86.

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