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Fractional approximations of abstract semilinear parabolic problems

  • * Corresponding author: Marcelo J. D. Nascimento

    * Corresponding author: Marcelo J. D. Nascimento

The first author is supported by FAPESP # 2014/03686-3, Brazil.
The second author is supported by CNPq # 303929/2015-4 and by FAPESP # 2003/10042-0, Brazil.
The third author is supported by FAPESP # 2017/06582-2, Brazil.

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  • In this paper we study the abstract semilinear parabolic problem of the form

    $ \frac{du}{dt}+Au = f(u), $

    as the limit of the corresponding fractional approximations

    $ \frac{du}{dt} + A^{\alpha}u = f(u), $

    in a Banach space $ X $, where the operator $ A:D(A) \subset X \to X $ is a sectorial operator in the sense of Henry [22]. Under suitable assumptions on nonlinearities $ f:X^\alpha\to X $ ($ X^\alpha: = D(A^\alpha $)), we prove the continuity with rate (with respect to the parameter $ \alpha $) for the global attractors (as seen in Babin and Vishik [4] Chapter 8, Theorem 2.1). As an application of our analysis we consider a fractional approximation of the strongly damped wave equations and we study the convergence with rate of solutions of such approximations.

    Mathematics Subject Classification: Primary: 35K90, 35K58; Secondary: 35B41.

    Citation:

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  • Figure 1.  $\Gamma = \Gamma_1\cup\Gamma_2\cup\Gamma_3 $, ($\Gamma = -\mathcal{G}$)

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