American Institute of Mathematical Sciences

Advanced
November  2020, 25(11): 4221-4255. doi: 10.3934/dcdsb.2020095

Fractional approximations of abstract semilinear parabolic problems

Flank D. M. Bezerra 1, , Alexandre N. Carvalho 2, and  Marcelo J. D. Nascimento 3,,

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

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  July 2019 Revised  November 2019 Published  November 2020 Early access  April 2020

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: Abstract parabolic equations, fractional power, sectorial operators, continuity, global attractors, wave equations.
Mathematics Subject Classification: Primary: 35K90, 35K58; Secondary: 35B41.
Citation: Flank D. M. Bezerra, Alexandre N. Carvalho, Marcelo J. D. Nascimento. Fractional approximations of abstract semilinear parabolic problems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4221-4255. doi: 10.3934/dcdsb.2020095
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.

show all references

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.

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

Download full-size image

Download as PowerPoint slide

[1]

Sergey Dashkovskiy, Oleksiy Kapustyan. Robustness of global attractors: Abstract framework and application to dissipative wave equations. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021054
[2]

Pengyu Chen, Xuping Zhang. Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4325-4357. doi: 10.3934/dcdsb.2020290
[3]

Antônio Luiz Pereira, Severino Horácio da Silva. Continuity of global attractors for a class of non local evolution equations. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 1073-1100. doi: 10.3934/dcds.2010.26.1073
[4]

John M. Ball. Global attractors for damped semilinear wave equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31
[5]

Valeria Danese, Pelin G. Geredeli, Vittorino Pata. Exponential attractors for abstract equations with memory and applications to viscoelasticity. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2881-2904. doi: 10.3934/dcds.2015.35.2881
[6]

Tomasz Dlotko, Tongtong Liang, Yejuan Wang. Critical and super-critical abstract parabolic equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1517-1541. doi: 10.3934/dcdsb.2019238
[7]

Wolf-Jürgen Beyn, Sergey Piskarev. Shadowing for discrete approximations of abstract parabolic equations. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 19-42. doi: 10.3934/dcdsb.2008.10.19
[8]

Yangrong Li, Shuang Yang, Guangqing Long. Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021303
[9]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks and Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[10]

Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015
[11]

Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure and Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605
[12]

Antonio Segatti. Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 801-820. doi: 10.3934/dcds.2006.14.801
[13]

Hong Lu, Jiangang Qi, Bixiang Wang, Mingji Zhang. Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 683-706. doi: 10.3934/dcds.2019028
[14]

Igor Chueshov, Alexander V. Rezounenko. Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1685-1704. doi: 10.3934/cpaa.2015.14.1685
[15]

A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119
[16]

Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1345-1377. doi: 10.3934/dcdsb.2021093
[17]

Mirelson M. Freitas, Anderson J. A. Ramos, Baowei Feng, Mauro L. Santos, Helen C. M. Rodrigues. Existence and continuity of global attractors for ternary mixtures of solids. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3563-3583. doi: 10.3934/dcdsb.2021196
[18]

Ali Akgül. Analysis and new applications of fractal fractional differential equations with power law kernel. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3401-3417. doi: 10.3934/dcdss.2020423
[19]

Bixiang Wang, Xiaoling Gao. Random attractors for wave equations on unbounded domains. Conference Publications, 2009, 2009 (Special) : 800-809. doi: 10.3934/proc.2009.2009.800
[20]

Benzion Shklyar. Exact null-controllability of interconnected abstract evolution equations with unbounded input operators. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 463-479. doi: 10.3934/dcds.2021124

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (287)
  • HTML views (260)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy