Abstract
We prove the existence of the universal attractor for the strongly damped semilinear wave equation, in the presence of a quite general nonlinearity of critical growth. When the nonlinearity is subcritical, we prove the existence of an exponential attractor of optimal regularity, having a basin of attraction coinciding with the whole phase-space. As a byproduct, the universal attractor is regular and of finite fractal dimension. Moreover, we carry out a detailed analysis of the asymptotic behavior of the solutions in dependence of the damping coefficient.
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Communicated by P. Constantin
Research partially supported the Italian MIUR Research Projects Problemi di Frontiera Libera nelle Scienze Applicate, Aspetti Teorici e Applicativi di Equazioni a Derivate Parziali and Metodi Variazionali e Topologici nello Studio dei Fenomeni Nonlineari. The second author was also supported by the Istituto Nazionale di Alta Matematica “F. Severi” (INdAM).
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Pata, V., Squassina, M. On the Strongly Damped Wave Equation. Commun. Math. Phys. 253, 511–533 (2005). https://doi.org/10.1007/s00220-004-1233-1
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DOI: https://doi.org/10.1007/s00220-004-1233-1