September  2022, 27(9): 4817-4835. doi: 10.3934/dcdsb.2021253

The existence of time-dependent attractor for wave equation with fractional damping and lower regular forcing term

School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

* Corresponding author

Received  April 2021 Revised  August 2021 Published  September 2022 Early access  October 2021

Fund Project: Luo is supported by NSF grant(11961059) and "Innovation Star" of Gansu Provincial Department of Education (2021CXZX-206)

We investigate the well-posedness and longtime dynamics of fractional damping wave equation whose coefficient $ \varepsilon $ depends explicitly on time. First of all, when $ 1\leq p\leq p^{\ast\ast} = \frac{N+2}{N-2}\; (N\geq3) $, we obtain existence of solution for the fractional damping wave equation with time-dependent decay coefficient in $ H_{0}^{1}(\Omega)\times L^{2}(\Omega) $. Furthermore, when $ 1\leq p<p^{*} = \frac{N+4\alpha}{N-2} $, $ u_{t} $ is proved to be of higher regularity in $ H^{1-\alpha}\; (t>\tau) $ and show that the solution is quasi-stable in weaker space $ H^{1-\alpha}\times H^{-\alpha} $. Finally, we get the existence and regularity of time-dependent attractor.

Citation: Xudong Luo, Qiaozhen Ma. The existence of time-dependent attractor for wave equation with fractional damping and lower regular forcing term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 4817-4835. doi: 10.3934/dcdsb.2021253
References:
[1]

J. ArrietaA. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 17 (1992), 841-866.  doi: 10.1080/03605309208820866.

[2]

A. V. Babin and M. I. Visik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., 62 (1983), 441-491. 

[3]

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.

[4]

V. V. ChepyzhovM. Conti and V. Pata, A minimal approach to the theory of global attractor, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088.  doi: 10.3934/dcds.2012.32.2079.

[5]

I. Chueshov and I. Lasiecka, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[6]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc. 2008.

[7]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512.  doi: 10.1007/s10884-004-4289-x.

[8]

M. Conti and V. Pata, On the time-dependent Cattaneo law in space dimension one, Appl. Math. Comput., 259 (2015), 32-44.  doi: 10.1016/j.amc.2015.02.039.

[9]

M. ContiV. Pata and R. Temam, Attractors for process on time-dependent space, application to wave equation, J. Differential Equations, 255 (2013), 1254-1277.  doi: 10.1016/j.jde.2013.05.013.

[10]

F. Di PlinioG. S. Duane and R. Temam, Time dependent attractor for the oscillon equation, Discrete Contin. Dyn. Syst., 29 (2011), 141-167.  doi: 10.3934/dcds.2011.29.141.

[11]

O. A. Ladyzhenskaya, Attractors of nonlinear evolution problems with dissipation, J. Sov. Math., 40 (1988), 632-640.  doi: 10.1007/BF01094189.

[12]

Q. MaJ. Wang and T. Liu, Time-dependent asymptotic behavior of the solution for wave equations with linear memory, Comput. Math. Appl., 76 (2018), 1372-1387.  doi: 10.1016/j.camwa.2018.06.031.

[13]

V. Pata and S. Zelik, A remark on the damped wave equation, Commun. Pure Appl. Anal., 5 (2006), 611-616.  doi: 10.3934/cpaa.2006.5.611.

[14]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.  doi: 10.1088/0951-7715/19/7/001.

[15]

A. Savostianov, Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains, Adv. Differential Equations, 20 (2015), 495-530. 

[16]

A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, Doctoral dissertation, 2015.

[17]

J. Simon, Compact sets in the space $L^{p}(0, T;B), $, Ann. Mat. Pur. Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[18]

H. F. Smith and C. D. Sogge, Global strichartz estimates for non-trapping perturbations of the laplacian, Comm. Partial Differential Equations, 25 (2000), 2171-2183.  doi: 10.1080/03605300008821581.

[19]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, New York, 1997.

[20]

Z. YangZ. Liu and N. Feng, Longtime behavior of the semilinear wave equation with gentle dissipation, Discrete Contin. Dyn. Syst., 36 (2016), 6557-6580.  doi: 10.3934/dcds.2016084.

show all references

References:
[1]

J. ArrietaA. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 17 (1992), 841-866.  doi: 10.1080/03605309208820866.

[2]

A. V. Babin and M. I. Visik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., 62 (1983), 441-491. 

[3]

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.

[4]

V. V. ChepyzhovM. Conti and V. Pata, A minimal approach to the theory of global attractor, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088.  doi: 10.3934/dcds.2012.32.2079.

[5]

I. Chueshov and I. Lasiecka, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[6]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc. 2008.

[7]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512.  doi: 10.1007/s10884-004-4289-x.

[8]

M. Conti and V. Pata, On the time-dependent Cattaneo law in space dimension one, Appl. Math. Comput., 259 (2015), 32-44.  doi: 10.1016/j.amc.2015.02.039.

[9]

M. ContiV. Pata and R. Temam, Attractors for process on time-dependent space, application to wave equation, J. Differential Equations, 255 (2013), 1254-1277.  doi: 10.1016/j.jde.2013.05.013.

[10]

F. Di PlinioG. S. Duane and R. Temam, Time dependent attractor for the oscillon equation, Discrete Contin. Dyn. Syst., 29 (2011), 141-167.  doi: 10.3934/dcds.2011.29.141.

[11]

O. A. Ladyzhenskaya, Attractors of nonlinear evolution problems with dissipation, J. Sov. Math., 40 (1988), 632-640.  doi: 10.1007/BF01094189.

[12]

Q. MaJ. Wang and T. Liu, Time-dependent asymptotic behavior of the solution for wave equations with linear memory, Comput. Math. Appl., 76 (2018), 1372-1387.  doi: 10.1016/j.camwa.2018.06.031.

[13]

V. Pata and S. Zelik, A remark on the damped wave equation, Commun. Pure Appl. Anal., 5 (2006), 611-616.  doi: 10.3934/cpaa.2006.5.611.

[14]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.  doi: 10.1088/0951-7715/19/7/001.

[15]

A. Savostianov, Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains, Adv. Differential Equations, 20 (2015), 495-530. 

[16]

A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, Doctoral dissertation, 2015.

[17]

J. Simon, Compact sets in the space $L^{p}(0, T;B), $, Ann. Mat. Pur. Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[18]

H. F. Smith and C. D. Sogge, Global strichartz estimates for non-trapping perturbations of the laplacian, Comm. Partial Differential Equations, 25 (2000), 2171-2183.  doi: 10.1080/03605300008821581.

[19]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, New York, 1997.

[20]

Z. YangZ. Liu and N. Feng, Longtime behavior of the semilinear wave equation with gentle dissipation, Discrete Contin. Dyn. Syst., 36 (2016), 6557-6580.  doi: 10.3934/dcds.2016084.

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