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Dynamics of the nonclassical diffusion equations. (English) Zbl 1154.35063

The authors consider the long-time behaviour of the solutions for a nonclassical diffusion equation used in fluid mechanics, solid mechanics and heat conduction theory.

It is different from the usual reaction diffusion equation, consequently it is possible to use the compact Sobolev embedding to obtain the key asymptotic compactness for both autonomous and nonautonomous cases.

For the autonomous case Sun and Yang prove by a decomposition technique the existence of a global attractor in H 0 1 only assuming that the forcing term is in H -1 ·

For the nonhomogeneous case under hypotheses that the forcing term is translation bounded is established that asymptotically the solutions are exponentially closing on approaching more regular solutions. As a consequence is proved the existence of a compact uniform attractor and moreover the existence of nonautonomous exponential attractors.

35K70Ultraparabolic equations, pseudoparabolic equations, etc.