Longtime behavior of the semilinear wave equation with gentle dissipation

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November 2016, 36(11): 6557-6580. doi: 10.3934/dcds.2016084

Longtime behavior of the semilinear wave equation with gentle dissipation

Zhijian Yang ^1, , Zhiming Liu ^2, and Na Feng ^2,

1.	Department of Mathematics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001
2.	School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001, China, China

Received December 2015 Revised June 2016 Published August 2016

Abstract
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The paper investigates the well-posedness and longtime dynamics of the semilinear wave equation with gentle dissipation： $u_{tt}-\triangle u+\gamma(-\triangle)^{\alpha} u_{t}+f(u)=g(x)$, with $\alpha\in(0,1/2)$. The main results are concerned with the relationships among the growth exponent $p$ of nonlinearity $f(u)$ and the well-posedness and longtime behavior of solutions of the equation. We show that (i) the well-posedness and longtime dynamics of the equation are of characters of parabolic equations as $1 \leq p < p^* \equiv \frac{N + 4\alpha}{(N-2)^+}$; (ii) the subclass $\mathbb{G}$ of limit solutions has a weak global attractor as $p^* \leq p < p^{**}\equiv \frac{N+2}{N-2}\ (N \geq 3)$.

Keywords: gentle dissipation, exponential attractor., global attractor, well-posedness, Semilinear wave equation.

Mathematics Subject Classification: Primary: 35B41, 35B33; Secondary: 35B40, 35B65, 37L3.

Citation: Zhijian Yang, Zhiming Liu, Na Feng. Longtime behavior of the semilinear wave equation with gentle dissipation. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6557-6580. doi: 10.3934/dcds.2016084

References:

[1]	J. M. Ball, Continuity properties and attractors of generalized semiflows and the Navier-Stokes equations, Nonlinear Science, 7 (1997), 475-502. doi: 10.1007/s003329900037.
[2]	J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.
[3]	V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 7 (2001), 719-735. doi: 10.3934/dcds.2001.7.719.
[4]	N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical waves in $3D$ domains, J. of AMS, 21 (2008), 831-845. doi: 10.1090/S0894-0347-08-00596-1.
[5]	A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310. doi: 10.2140/pjm.2002.207.287.
[6]	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.
[7]	A. N. Carvalho and J. W. Cholewa, Regularity of solutions on the global attractor for a semilinear damped wave equation, J. Math. Anal. Appl., 337 (2008), 932-948. doi: 10.1016/j.jmaa.2007.04.051.
[8]	A. N. Carvalho, J. W. Cholewa and T. Dlotko, Strongly damped wave problems: Bootstrapping and regularity of solutions, J. Differential Equations, 244 (2008), 2310-2333. doi: 10.1016/j.jde.2008.02.011.
[9]	A. N. Carvalho, J. W. Cholewa and T. Dlotko, Damped wave equations with fast dissipative nonlinearities, Discrete Continuous Dynam. Systems - A, 24 (2009), 1147-1165. doi: 10.3934/dcds.2009.24.1147.
[10]	S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems, Lecture Notes in Math., Springer-Verlag, 1354 (1988), 234-256. doi: 10.1007/BFb0089601.
[11]	S. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15.
[12]	S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $0 <\alpha < 1/2$, Proceedings of the American Mathematical Society, 110 (1990), 401-415. doi: 10.2307/2048084.
[13]	I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, in Memories of AMS, 195, (Providence, RI: American Mathematical Society), 2008. doi: 10.1090/memo/0912.
[14]	I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106.
[15]	I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262. doi: 10.1016/j.jde.2011.08.022.
[16]	I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-posedness and Long Time Dynamics, Springer Science and Business Media, 2010. doi: 10.1007/978-0-387-87712-9.
[17]	I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. doi: 10.1007/978-3-319-22903-4.
[18]	E. Feireisl, Asymptotic behavior and attractors for a semilinear damped wave equation with supercritical exponent, Roy. Soc. Edinburgh Sect.- A, 125 (1995), 1051-1062. doi: 10.1017/S0308210500022630.
[19]	P. J. Graber and J. L. Shomberg, Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions, Nonlinearity, 29 (2016), 1171. doi: 10.1088/0951-7715/29/4/1171.
[20]	V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. doi: 10.1016/j.jde.2009.04.010.
[21]	V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré, (2016) DOI 10.1007/s00023-016-0480-y.
[22]	L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation, Comm. Partial Differential Equations, 20 (1995), 1303-1323. doi: 10.1080/03605309508821133.
[23]	V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. doi: 10.1007/s00220-004-1233-1.
[24]	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.
[25]	V. Pata and S. Zelik, A remark on the damped wave equation, Commun. Pure Appl. Anal., 5 (2006), 611-616.
[26]	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.
[27]	A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations, Mathemaica Bohemica, 139 (2014), 657-665.
[28]	A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping, Asymptot. Anal., 87 (2014), 191-221.
[29]	A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, Doctoral dissertation, University of Surrey, 2015.
[30]	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.
[31]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali diMatematica Pura ed Applicata, 146 (1986), 65-96. doi: 10.1007/BF01762360.
[32]	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[33]	Z. J. Yang, N. Feng and T. F. Ma, Global attracts of the generalized double dispersion, Nonlinear Analysis, 115 (2015), 103-106. doi: 10.1016/j.na.2014.12.006.

show all references

References:

[1]	J. M. Ball, Continuity properties and attractors of generalized semiflows and the Navier-Stokes equations, Nonlinear Science, 7 (1997), 475-502. doi: 10.1007/s003329900037.
[2]	J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.
[3]	V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 7 (2001), 719-735. doi: 10.3934/dcds.2001.7.719.
[4]	N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical waves in $3D$ domains, J. of AMS, 21 (2008), 831-845. doi: 10.1090/S0894-0347-08-00596-1.
[5]	A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310. doi: 10.2140/pjm.2002.207.287.
[6]	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.
[7]	A. N. Carvalho and J. W. Cholewa, Regularity of solutions on the global attractor for a semilinear damped wave equation, J. Math. Anal. Appl., 337 (2008), 932-948. doi: 10.1016/j.jmaa.2007.04.051.
[8]	A. N. Carvalho, J. W. Cholewa and T. Dlotko, Strongly damped wave problems: Bootstrapping and regularity of solutions, J. Differential Equations, 244 (2008), 2310-2333. doi: 10.1016/j.jde.2008.02.011.
[9]	A. N. Carvalho, J. W. Cholewa and T. Dlotko, Damped wave equations with fast dissipative nonlinearities, Discrete Continuous Dynam. Systems - A, 24 (2009), 1147-1165. doi: 10.3934/dcds.2009.24.1147.
[10]	S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems, Lecture Notes in Math., Springer-Verlag, 1354 (1988), 234-256. doi: 10.1007/BFb0089601.
[11]	S. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15.
[12]	S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $0 <\alpha < 1/2$, Proceedings of the American Mathematical Society, 110 (1990), 401-415. doi: 10.2307/2048084.
[13]	I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, in Memories of AMS, 195, (Providence, RI: American Mathematical Society), 2008. doi: 10.1090/memo/0912.
[14]	I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106.
[15]	I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262. doi: 10.1016/j.jde.2011.08.022.
[16]	I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-posedness and Long Time Dynamics, Springer Science and Business Media, 2010. doi: 10.1007/978-0-387-87712-9.
[17]	I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. doi: 10.1007/978-3-319-22903-4.
[18]	E. Feireisl, Asymptotic behavior and attractors for a semilinear damped wave equation with supercritical exponent, Roy. Soc. Edinburgh Sect.- A, 125 (1995), 1051-1062. doi: 10.1017/S0308210500022630.
[19]	P. J. Graber and J. L. Shomberg, Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions, Nonlinearity, 29 (2016), 1171. doi: 10.1088/0951-7715/29/4/1171.
[20]	V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. doi: 10.1016/j.jde.2009.04.010.
[21]	V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré, (2016) DOI 10.1007/s00023-016-0480-y.
[22]	L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation, Comm. Partial Differential Equations, 20 (1995), 1303-1323. doi: 10.1080/03605309508821133.
[23]	V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. doi: 10.1007/s00220-004-1233-1.
[24]	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.
[25]	V. Pata and S. Zelik, A remark on the damped wave equation, Commun. Pure Appl. Anal., 5 (2006), 611-616.
[26]	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.
[27]	A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations, Mathemaica Bohemica, 139 (2014), 657-665.
[28]	A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping, Asymptot. Anal., 87 (2014), 191-221.
[29]	A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, Doctoral dissertation, University of Surrey, 2015.
[30]	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.
[31]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali diMatematica Pura ed Applicata, 146 (1986), 65-96. doi: 10.1007/BF01762360.
[32]	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[33]	Z. J. Yang, N. Feng and T. F. Ma, Global attracts of the generalized double dispersion, Nonlinear Analysis, 115 (2015), 103-106. doi: 10.1016/j.na.2014.12.006.

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