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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 |
$ \frac{du}{dt}+Au = f(u), $ |
$ \frac{du}{dt} + A^{\alpha}u = f(u), $ |
$ X $ |
$ A:D(A) \subset X \to X $ |
$ f:X^\alpha\to X $ |
$ X^\alpha: = D(A^\alpha $ |
$ \alpha $ |
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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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