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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.


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

  • [1] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.
    [2] J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, J. Differential Equations, 199 (2004), 143-178.  doi: 10.1016/j.jde.2003.09.004.
    [3] J. M. Arrieta and E. Santamaría, Estimates on the distance of inertial manifolds, Discrete Contin. Dyn. Syst., 34 (2014), 3921-3944.  doi: 10.3934/dcds.2014.34.3921.
    [4] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992.
    [5] A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., 62 (1983), 441-491. 
    [6] F. D. M. BezerraA. N. CarvalhoJ. W. Cholewa and M. J. D. Nascimento, Parabolic approximation of damped wave equations via fractional powers: Fast growing nonlinearities and continuity of the dynamics, J. Math. Anal. Appl., 450 (2017), 377-405.  doi: 10.1016/j.jmaa.2017.01.024.
    [7] F. D. M. BezerraA. N. CarvalhoT. Dlotko and M. J. D. Nascimento, Fractional Schrödinger equation; Solvability and connection with classical Schrödinger equation, J. Math. Anal. Appl., 457 (2018), 336-360.  doi: 10.1016/j.jmaa.2017.08.014.
    [8] S. M. BruschiA. N. CarvalhoJ. W. Cholewa and T. Dlotko, Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equations, J. Dynam. Differential Equations, 18 (2006), 767-814.  doi: 10.1007/s10884-006-9023-4.
    [9] V. L. CarboneA. N. Carvalho and K. Schiabel-Silva, Continuity of attractors for parabolic problems with localized large diffusion, Nonlinear Anal., 68 (2008), 515-535. 
    [10] A. N. Carvalho and J. W. Cholewa, Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc., 66 (2002), 443-463.  doi: 10.1017/S0004972700040296.
    [11] A. N. Carvalho, J. W. Cholewa and T. Dłotko, Equi-exponential attraction and rate of convergence of attractors with application to a perturbed damped wave equation., Proc. Roy. Soc. Edinburgh Sect. A 144 (2014) 13–51. doi: 10.1017/S0308210511001235.
    [12] A. N. Carvalho and T. Dlotko, Dynamics of the viscous Cahn-Hilliard equation, J. Math. Anal. Appl., 344 (2008), 703-725.  doi: 10.1016/j.jmaa.2008.03.020.
    [13] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, vol. 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.
    [14] A. N. CarvalhoJ. A. LangaJ. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations, 236 (2007), 570-603.  doi: 10.1016/j.jde.2007.01.017.
    [15] A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numer. Funct. Anal. Optim., 27 (2006), 785-829.  doi: 10.1080/01630560600882723.
    [16] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.
    [17] T. CazenaveD. Fang and Z. Han, Continuous dependence for NLS in fractional order spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 135-147.  doi: 10.1016/j.anihpc.2010.11.005.
    [18] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55.  doi: 10.2140/pjm.1989.136.15.
    [19] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Note Series, vol. 278, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404.
    [20] R. Czaja, Differential Equations with Sectorial Operator, Wydawnictwo Uniwersytetu Śląskiego, Katowice, 2002.
    [21] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988.
    [22] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981.
    [23] T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96.  doi: 10.3792/pja/1195524082.
    [24] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
    [25] S. G. Kre${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$n, Linear Differential Equations in Banach Space, American Mathematical Society, 1972.
    [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
    [27] H. Tanabe, Equations of Evolution, Monographs and Studies in Mathematics, vol. 6, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979.
    [28] M. E. Taylor, Partial Differential Equations. Basic Theory, Texts in Applied Mathematics, vol. 23, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.
    [29] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.
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