Trajectory statistical solutions and Liouville type theorem for nonlinear wave equations with polynomial growth. (English) Zbl 1480.35056
Summary: This article investigates the following family of nonlinear wave equations
\[
\partial^2_t u+\gamma (-\Delta)^{\alpha}\partial_t u=\Delta u-f(u)+g(x), \quad x\in \Omega\subset \mathbb{R}^n, \,n\geqslant 1,
\]
with \(\alpha\in [0, 1/2)\) and
\[
|f(u)|\leqslant c_1 (1+|u|^{p-1}),
\]
where \(c_1 >0\) is a constant and \(p>2\) is arbitrary. We first prove the existence of trajectory attractor and then use the translation semigroup to construct the trajectory statistical solutions for above equations. Further we establish that the constructed trajectory statistical solutions possess an invariance property and satisfy a Liouville type theorem. Moreover, we reveal that the invariance property of the trajectory statistical solutions is a particular situation of the Liouville type theorem.
MSC:
| 35B41 | Attractors |
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
| 35L71 | Second-order semilinear hyperbolic equations |
| 35R11 | Fractional partial differential equations |
| 76F20 | Dynamical systems approach to turbulence |
References:
| [1] | C.D. Aliprentis and K.C. Border, “Infinite Dimensional Analysis,” A Hithhiker’s Guide, third edition, Springer-Verlag, New York, 2006. · Zbl 1156.46001 |
| [2] | J.M. Arrieta, A.N. Carvalho, and J.K. Hale, A damped hyperbolic equation with critical exponent, Comm. in Partial Differential Equations, 17 (1992), 841-866. · Zbl 0815.35067 |
| [3] | F. Boyer and P. Fabrie, “Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models,” Springer, New York, 2013. · Zbl 1286.76005 |
| [4] | J.M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 447-474. · Zbl 0958.35101 |
| [5] | J.M. Ball, Global attractors for damped semilinear wave equations, Discrete Cont. Dyn. Syst., 10 (2004), 31-52. · Zbl 1056.37084 |
| [6] | V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on R 3 , Discrete Cont. Dyn. Syst., 7 (2001), 719-735. · Zbl 1200.35032 |
| [7] | A. Bronzi, C.F. Mondaini, and R. Rosa, Trajectory statistical solutions for three-dimensional Navier-Stokes-like systems, SIAM J. Math. Anal., 46 (2014), 1893-1921. · Zbl 1298.76066 |
| [8] | A. Bronzi, C.F. Mondaini, and R. Rosa, Abstract framework for the theory of statistical solutions, J. Differential Equations, 260 (2016), 8428-8484. · Zbl 1334.76038 |
| [9] | A.N. Carvalho, J.W. Cholewa, and T. Dlotko, Damped wave equations with fast grow-ing dissipative nonlinearities, Distrete Cont. Dyn. Syst., 24 (2009), 1147-1165. · Zbl 1178.35264 |
| [10] | M. Chekroun and N.E. Glatt-Holtz, Invariant measures for dissipative dynamical sys-tems: Abstract results and applications, Comm. Math. Phys., 316 (2012), 723-761. · Zbl 1336.37060 |
| [11] | V.V. Chepyzhov and M.I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 664-913. · Zbl 0896.35032 |
| [12] | V.V. Chepyzhov, M.I. Vishik, and S.V. Zelik, Strong trajectory attractor for dissipative Euler equations, J. Math. Pures Appl., 96 (2011), 395-407. · Zbl 1230.35092 |
| [13] | V.V. Chepyzhov and M.I. Vishik, “Attractors for Equations of Mathematical Physics,” AMS Colloquium Publications, Vol. 49, AMS, Providence, R.I., 2002. · Zbl 0986.35001 |
| [14] | A. Cheskidov, Global attractors of evolutionary systems, J. Dyn. Differential Equa-tions, 21 (2009), 249-268. · Zbl 1176.35035 |
| [15] | I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 21 (2012), 1229-1262. · Zbl 1237.37053 |
| [16] | G. Duvaut and J.L. Lions, “Inequalities in Mechanics and Physics,” Springer-Verlag, Berlin/New York, 1976 · Zbl 0331.35002 |
| [17] | E. Feireisl, Global attractors for semilinear wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447. · Zbl 0819.35097 |
| [18] | C. Foias and G. Prodi, Sur les solutions statistiques equations de Navier-Stokes, Ann. Mat. Pura Appl., 111(1976), 307-330. · Zbl 0344.76015 |
| [19] | C. Foias, O. Manley, R. Rosa, and R. Temam, “Navier-Stokes Equations and Turbu-lence,” Cambridge University Press, Cambridge, 2001. · Zbl 0994.35002 |
| [20] | C. Foias, R. Rosa, and R. Temam, Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations, Annales de L’Institut Fourier, 63 (2013), 2515-2573. · Zbl 1304.35486 |
| [21] | V. Kalantarov and S.V. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. · Zbl 1183.35053 |
| [22] | V. Kalantarov, A. Savostianov, and S.V. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré, 17 (2016), 2555-2584. · Zbl 1356.35055 |
| [23] | C.B. Gentile Moussa, Invariant measures for multivalued semigroups, J. Math. Anal. Appl., 455 (2017), 1234-1248. · Zbl 1373.37014 |
| [24] | X. Li, W.X. Shen, and C.Y. Sun, Invariant measures for complex-valued dissipative dynamical systems and applications, Discrete Cont. Dyn. Syst.-B, 22 (2017), 2427-2446. · Zbl 1362.37154 |
| [25] | G. Lukaszewicz and J.C. Robinson, Invariant measures for non-autonomous dissipa-tive dynamical systems, Discrete Cont. Dyn. Syst., 34 (2014), 4211-4222. · Zbl 1351.37076 |
| [26] | G. Lukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Cont. Dyn. Syst.-B, 9 (2008), 643-659. · Zbl 1191.35067 |
| [27] | G. Lukaszewicz, J. Real, and J.C. Robinson, Invariant measures for dissipative dy-namical systems and generalised Banach limits, J. Dyn. Differential Equations, 23 (2011), 225-250. · Zbl 1230.37094 |
| [28] | V. Pata and S.V. Zelik, A remark on the damped wave equation, Comm. Pure Appl. Anal., 5 (2006), 611-616. · Zbl 1140.35533 |
| [29] | A. Savostianov, Strichartz estimates and smooth attractors for a sub-quinitic wave equation with fractional damping in bounded domains, Adv. Differential Equations, 20 (2015), 495-530. · Zbl 1316.35047 |
| [30] | A. Savostianov, Strichartz esitimates and smooth attractors for dissipative hyperbolic equations, Doctoral dissertation, University of Surrey, 2015. |
| [31] | R. Temam, “Infinite-dimensional Dynamical Dystems in Mechanics and Physics,” Springer-Verlag, New York, 1997. · Zbl 0871.35001 |
| [32] | M.I. Vishik and A.V. Fursikov, Translationally homogeneous statistical solutions and individual solutions with infinite energy of a system of Navier-Stokes equations, Siberian Mathematical Journal, 19 (1978), 710-729. · Zbl 0412.35078 |
| [33] | M.I. Vishik and V.V. Chepyzhov, Trajectory and global attractors of three-dimensional Navier-Stokes systems, Math. Notes, 77 (2002), 177-193. · Zbl 1130.37404 |
| [34] | M.I. Vishik and V.V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Math. Surveys, 4 (2011), 639-731. · Zbl 1277.37102 |
| [35] | J. Wang, C. Zhao, and T. Caraballo, Invariant measures for the 3D globally modified Navier-Stokes equations with unbounded variable delays, Comm. Nonlinear Sci. Numer Simu., 91 (2020), 105459. · Zbl 1454.35262 |
| [36] | X.M. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Cont. Dyn. Syst., 23 (2009), 521-540. · Zbl 1154.37370 |
| [37] | Z.J. Yang, Z.M. Liu, and N. Feng, Longtime behavior of the semilinear wave equation with gentle dissipation, Discrete Cont. Dyn. Syst., 26 (2016), 6557-6580. · Zbl 1362.35191 |
| [38] | Z.J. Yang, P.Y. Ding, and L. Li, Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlineaity, J. Math. Anal. Appl., 442 (2016), 485-510. · Zbl 1339.35050 |
| [39] | Z.J. Yang and Z.M. Liu, Upper semicontinuity of global attractors for a family of semilinear wave equations with gentle dissipation, Appl. Math. Lett., 69 (2017), 22-28. · Zbl 1378.35041 |
| [40] | Z.J. Yang and Z.M. Liu, Stability of exponential attractors for a family of semilinear wave equations with gentle dissipation, J. Differential Equations, 264 (2018), 3976-4005. · Zbl 1400.35036 |
| [41] | C. Zhao, Y. Li, and S. Zhou, Regularity of trajectory attractor and upper semiconti-nuity of global attractor for a 2D non-Newtonian fluid, J. Differential Equations, 247 (2009), 2331-2363. · Zbl 1182.35048 |
| [42] | C. Zhao, L. Kong, G. Liu, and M. Zhao, The trajectory attractor and its limiting behavior for the convective Brinkman-Forchheimer equations, Topological Meth. Nonl. Anal., 44 (2014), 413-433. · Zbl 1362.35056 |
| [43] | C. Zhao and , T. Caraballo, Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier-Stokes equations, J. Differential Equations, 266 (2019), 7205-7229. · Zbl 1411.35051 |
| [44] | C. Zhao, Y. Li, and T. Caraballo, Trajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications, J. Differential Equations, 269 (2020), 467-494. · Zbl 1436.35050 |
| [45] | C. Zhao, Y. Li, and G. Lukaszewicz, Statistical solution and partial degenerate regu-larity for the 2D non-autonomous magneto-micropolar fluids, Z. Angew. Math. Phys., 71 (2020), 1-24. · Zbl 1467.35267 |
| [46] | C. Zhao, Y. Li, and Y. Sang, Using trajectory attractor to construct trajectory statis-tical solutions for 3D incompressible micropolar flows, Z. Angew. Math. Mech., 100 (2020), e201800197. · Zbl 07800095 |
| [47] | C. Zhao, Z. Song, and T. Caraballo, Strong trajectory statistical solutions and Liouville type equations for dissipative Euler equations, Appl. Math. Lett., 99 (2020), 105981. · Zbl 1425.37047 |
| [48] | C. Zhao, Y. Li, and Z. Song, Trajectory statistical solutions for the 3D Navier-Stokes equations: The trajectory attractor approach, Nonlinear Anal.-RWA, 53 (2020), 103077. · Zbl 1433.35249 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
