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

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

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.
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. 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.

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. 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.

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