Local well posedness, asymptotic behavior and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time
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- by A. N. Carvalho and J. W. Cholewa PDF
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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.References
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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
- 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
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 2567-2586
- MSC (2000): Primary 35G25, 35B33, 35B40, 35B41, 35B65
- DOI: https://doi.org/10.1090/S0002-9947-08-04789-2
- MathSciNet review: 2471929