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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Local well posedness, asymptotic behavior and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time


Authors: A. N. Carvalho and J. W. Cholewa
Journal: Trans. Amer. Math. Soc. 361 (2009), 2567-2586
MSC (2000): Primary 35G25, 35B33, 35B40, 35B41, 35B65
Published electronically: November 4, 2008
MathSciNet review: 2471929
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Abstract: A class of semilinear evolution equations of the second order in time of the form $ u_{tt} + A u + \mu A u_t + A u_{tt}=f(u)$ is considered, where $ -A$ is the Dirichlet Laplacian, $ \Omega$ is a smooth bounded domain in $ \mathbb{R}^N$ and $ f\in C^1(\mathbb{R},\mathbb{R})$. A local well posedness result is proved in the Banach spaces $ W^{1,p}_0(\Omega)\times W^{1,p}_0(\Omega)$ when $ f$ satisfies appropriate critical growth conditions. In the Hilbert setting, if $ f$ satisfies an additional dissipativeness condition, the nonlinear semigroup of global solutions is shown to possess a gradient-like attractor. Existence and regularity of the global attractor are also investigated following the unified semigroup approach, bootstrapping and the interpolation-extrapolation techniques.


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Additional Information

A. N. Carvalho
Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
Email: andcarva@icmc.usp.br

J. W. Cholewa
Affiliation: Institute of Mathematics, Silesian University, 40-007 Katowice, Poland
Email: jcholewa@ux2.math.us.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04789-2
PII: S 0002-9947(08)04789-2
Keywords: Evolution equations of the second order in time, existence, uniqueness and continuous dependence of solutions on initial conditions, asymptotic behavior of solutions, attractors, regularity, critical exponents.
Received by editor(s): May 21, 2007
Published electronically: November 4, 2008
Additional Notes: This research was partially supported by grant # 300.889/92-5 CNPq and grant # 03/10042-0 FAPESP, Brazil
Article copyright: © Copyright 2008 American Mathematical Society