Fractional approximations of abstract semilinear parabolic problems

Flank D. M. Bezerra; Alexandre N. Carvalho; Marcelo J. D. Nascimento

doi:10.3934/dcdsb.2020095

Article Contents

2020, Volume 25, Issue 11: 4221-4255. Doi: 10.3934/dcdsb.2020095

This issue Previous Article A note on global stability in the periodic logistic map Next Article Extension, embedding and global stability in two dimensional monotone maps

Fractional approximations of abstract semilinear parabolic problems

1.
Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa, PB 58051-900, Brazil
2.
Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, São Carlos, SP Caixa Postal 668, 13560-970, Brazil
3.
Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP 13565-905, Brazil

^* 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.

Received: June 30, 2019

Revised: October 31, 2019

Early access: April 2020

Published: November 2020

Abstract Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

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.

Keywords:

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

Citation:

Full Text(HTML)

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

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[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. Bezerra, A. N. Carvalho, J. 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. Bezerra, A. N. Carvalho, T. 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. Bruschi, A. N. Carvalho, J. 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. Carbone, A. 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. Carvalho, J. A. Langa, J. 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. Cazenave, D. 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.