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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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
Trans. Amer. Math. Soc. 361 (2009), 2567-2586 Request permission

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
  • 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