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Regularity of solutions to time fractional diffusion equations
Rate of attraction for a semilinear thermoelastic system with variable coefficients
Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa, PB, 58051-900, Brasil |
$ \begin{cases} \partial_t^2u-\partial_x(a_\varepsilon(x) \partial_xu)+\partial_x(m(x) \theta) = f(u)& \mbox{in}\ \ (0,l)\times(0,+\infty),\\ \partial_t\theta-\partial_x(\kappa_\varepsilon(x) \partial_x\theta)+m(x) \partial_{xt}u = 0& \mbox{in}\ \ (0,l)\times(0,+\infty), \end{cases} $ |
$ l>0 $ |
$ a_\varepsilon,\kappa_\varepsilon $ |
$ m $ |
$ f $ |
$ \varepsilon\to0^+ $ |
$ \|a_\varepsilon-a_0\|_{L^p(0,l)}+\|\kappa_\varepsilon-\kappa_0\|_{L^p(0,l)} $ |
$ p\geq 2 $ |
References:
[1] |
F. D. Araruna and F. D. M. Bezerra,
Rate of attraction for a semilinear wave equation with variable coefficients and critical nonlinearities, Pacific J. Math., 266 (2013), 257-282.
doi: 10.2140/pjm.2013.266.257. |
[2] |
J. M. Arrieta, F. D. M. Bezerra and A. N. Carvalho,
Rate of convergence of global attractors of some perturbed reaction-diffusion problems, Topological Methods in Nonlinear Analysis, 41 (2013), 229-253.
|
[3] |
J. M. Arrieta and A. N. Carvalho,
Abstract parabolic equations with critical nonlinearities and applications to Navier-Stokes and heat equations, Transactions of the AMS, 352 (2000), 285-310.
doi: 10.1090/S0002-9947-99-02528-3. |
[4] |
J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz,
Dynamics in dummbell domains Ⅲ. Continuity of attractors, J. Diff. Eqns., 247 (2009), 225-259.
doi: 10.1016/j.jde.2008.12.014. |
[5] |
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. |
[6] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and Its Applications, 25, New York, 1992. |
[7] |
A. V. Babin and M. I. Vishik,
Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., 62 (1983), 441-491.
|
[8] |
F. D. M. Bezerra and M. J. D. Nascimento,
Convergence estimates of the dynamics of a hyperbolic system with variable coefficients, Math. Methods Appl. Sci., 37 (2014), 663-675.
doi: 10.1002/mma.2823. |
[9] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[10] |
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. D. Diff. Eqns., 18 (2006), 767-814.
doi: 10.1007/s10884-006-9023-4. |
[11] |
V. L. Carbone, A. N. Carvalho and K. Schiabel-Silva,
Continuity of attractors for parabolic problems with localized large diffusion, Nonlinear Analysis, 68 (2008), 515-535.
doi: 10.1016/j.na.2006.11.017. |
[12] |
V. L. Carbone, M. J. D. Nascimento, K. Schiabel-Silva and R. P. Silva,
Pullback attractors for a singularly nonautonomous plate equation, Electronic Journal of Differential Equations, 77 (2011), 1-13.
|
[13] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer-Verlag, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[14] |
A. N. Carvalho and S. Piskarev,
A general approximation scheme for attractors of abstract parabolic problems, Numerical Functional Analysis and Optimization, 27 (2006), 785-829.
doi: 10.1080/01630560600882723. |
[15] |
H. Gao,
Global attractor for the semilinear thermoelastic problem, Math. Methods Appl. Sci., 26 (2003), 1255-1271.
doi: 10.1002/mma.416. |
[16] |
H. Gao and J. E. Muñoz Rivera,
Global existence and decay for the semilinear thermoelastic contact problem, J. Differential Equations, 186 (2002), 52-68.
doi: 10.1016/S0022-0396(02)00016-5. |
[17] |
J. K. Hale and A. Perissinotto,
Global attractor and convergence for one-dimensional semilinear thermoelasticity, Dynamic Systems and Applications, 2 (1993), 1-9.
|
[18] |
D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Springer-Verlag, 840 New York, 1981. |
[19] |
D. B. Henry, A. Perissinitto and O. Lopes,
On the essential spectrum of a semigroup of thermoelasticity, Applications, 21 (1993), 65-75.
doi: 10.1016/0362-546X(93)90178-U. |
[20] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, 1978. |
show all references
References:
[1] |
F. D. Araruna and F. D. M. Bezerra,
Rate of attraction for a semilinear wave equation with variable coefficients and critical nonlinearities, Pacific J. Math., 266 (2013), 257-282.
doi: 10.2140/pjm.2013.266.257. |
[2] |
J. M. Arrieta, F. D. M. Bezerra and A. N. Carvalho,
Rate of convergence of global attractors of some perturbed reaction-diffusion problems, Topological Methods in Nonlinear Analysis, 41 (2013), 229-253.
|
[3] |
J. M. Arrieta and A. N. Carvalho,
Abstract parabolic equations with critical nonlinearities and applications to Navier-Stokes and heat equations, Transactions of the AMS, 352 (2000), 285-310.
doi: 10.1090/S0002-9947-99-02528-3. |
[4] |
J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz,
Dynamics in dummbell domains Ⅲ. Continuity of attractors, J. Diff. Eqns., 247 (2009), 225-259.
doi: 10.1016/j.jde.2008.12.014. |
[5] |
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. |
[6] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and Its Applications, 25, New York, 1992. |
[7] |
A. V. Babin and M. I. Vishik,
Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., 62 (1983), 441-491.
|
[8] |
F. D. M. Bezerra and M. J. D. Nascimento,
Convergence estimates of the dynamics of a hyperbolic system with variable coefficients, Math. Methods Appl. Sci., 37 (2014), 663-675.
doi: 10.1002/mma.2823. |
[9] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[10] |
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. D. Diff. Eqns., 18 (2006), 767-814.
doi: 10.1007/s10884-006-9023-4. |
[11] |
V. L. Carbone, A. N. Carvalho and K. Schiabel-Silva,
Continuity of attractors for parabolic problems with localized large diffusion, Nonlinear Analysis, 68 (2008), 515-535.
doi: 10.1016/j.na.2006.11.017. |
[12] |
V. L. Carbone, M. J. D. Nascimento, K. Schiabel-Silva and R. P. Silva,
Pullback attractors for a singularly nonautonomous plate equation, Electronic Journal of Differential Equations, 77 (2011), 1-13.
|
[13] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer-Verlag, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[14] |
A. N. Carvalho and S. Piskarev,
A general approximation scheme for attractors of abstract parabolic problems, Numerical Functional Analysis and Optimization, 27 (2006), 785-829.
doi: 10.1080/01630560600882723. |
[15] |
H. Gao,
Global attractor for the semilinear thermoelastic problem, Math. Methods Appl. Sci., 26 (2003), 1255-1271.
doi: 10.1002/mma.416. |
[16] |
H. Gao and J. E. Muñoz Rivera,
Global existence and decay for the semilinear thermoelastic contact problem, J. Differential Equations, 186 (2002), 52-68.
doi: 10.1016/S0022-0396(02)00016-5. |
[17] |
J. K. Hale and A. Perissinotto,
Global attractor and convergence for one-dimensional semilinear thermoelasticity, Dynamic Systems and Applications, 2 (1993), 1-9.
|
[18] |
D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Springer-Verlag, 840 New York, 1981. |
[19] |
D. B. Henry, A. Perissinitto and O. Lopes,
On the essential spectrum of a semigroup of thermoelasticity, Applications, 21 (1993), 65-75.
doi: 10.1016/0362-546X(93)90178-U. |
[20] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, 1978. |
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