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Dynamics of a Leslie-Gower predator-prey system with hunting cooperation and prey harvesting
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 |
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.
References:
[1] |
J. Arrieta, A. 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. 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. |
[4] |
V. V. Chepyzhov, M. 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. Conti, V. 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 Plinio, G. 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. Ma, J. 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. Yang, Z. 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. Arrieta, A. 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. 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. |
[4] |
V. V. Chepyzhov, M. 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. Conti, V. 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 Plinio, G. 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. Ma, J. 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. Yang, Z. 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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