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

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

2006-11-15

arxiv.org/abs/math/0611464v1

Mathematics

Analysis of PDEs

Scientific paper

A doubly nonlinear parabolic equation of the form $\alpha(u_t)-\Delta u+W'(u)= f$, 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 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" constitute a semiflow $S$ for which uniqueness is satisfied for strictly positive times and we can study long time behavior properties. In particular, we can prove existence of both global and exponential attractors and investigate the structure of $\omega$-limits of single trajectories.

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