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April  2020, 19(4): 2219-2233. doi: 10.3934/cpaa.2020097

Attractors for semilinear wave equations with localized damping and external forces

1. 

Institute of Mathematical and Computer Sciences, University of São Paulo, São Carlos 13566-590, SP, Brazil

2. 

Faculty of Engineering, University of Ricardo Palma, Lima, Peru

*Corresponding author. Current affiliation: Department of Mathematics, University of Brasília, Brasília 70910-900, DF, Brazil

Dedicated to Professor Tomás Caraballo on occasion of his Sixtieth Birthday

Received  May 2019 Revised  September 2019 Published  January 2020

This paper is concerned with long-time dynamics of semilinear wave equations defined on bounded domains of $ \mathbb{R}^3 $ with cubic nonlinear terms and locally distributed damping. The existence of regular finite-dimensional global attractors established by Chueshov, Lasiecka and Toundykov (2008) reflects a good deal of the current state of the art on this matter. Our contribution is threefold. First, we prove uniform boundedness of attractors with respect to a forcing parameter. Then, we study the continuity of attractors with respect to the parameter in a residual dense set. Finally, we show the existence of generalized exponential attractors. These aspects were not previously considered for wave equations with localized damping.

Citation: To Fu Ma, Paulo Nicanor Seminario-Huertas. Attractors for semilinear wave equations with localized damping and external forces. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2219-2233. doi: 10.3934/cpaa.2020097
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.  Google Scholar

[2]

A. V. Babin and S. Yu Pilyugin, Continuous dependence of attractors on the shape of domain, J. Math. Sci., 87 (1997), 3304-3310.  doi: 10.1007/BF02355582.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications 25, North-Holland, Amsterdam, 1992.  Google Scholar

[4]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

[5]

M. BlairH. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1817-1829.  doi: 10.1016/j.anihpc.2008.12.004.  Google Scholar

[6]

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.  Google Scholar

[7]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[8]

I. ChueshovM. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.  Google Scholar

[9]

I. ChueshovI. Lasiecka and D. Toundykov, Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent, Discrete Contin. Dyn. Syst., 20 (2008), 459-509.  doi: 10.3934/dcds.2008.20.459.  Google Scholar

[10]

M. ContiT. F. MaE. M. Marchini and P. N. Seminario Huertas, Asymptotics of viscoelastic materials with nonlinear density and memory effects, J. Differential Equations, 264 (2018), 4235-4259.  doi: 10.1016/j.jde.2017.12.010.  Google Scholar

[11]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics 37, Masson, Paris. Wiley, Chichester, 1994.  Google Scholar

[12]

E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Comm. Partial Differential Equations, 18 (1993), 1539-1555.  doi: 10.1080/03605309308820985.  Google Scholar

[13]

X. FuJ. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control. Optim., 46 (2007), 1578-1614.  doi: 10.1137/040610222.  Google Scholar

[14]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, 1988.  Google Scholar

[15]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0.  Google Scholar

[16]

J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl., 154 (1989), 281-326.  doi: 10.1007/BF01790353.  Google Scholar

[17]

L. T. HoangE. J. Olson and J. C. Robinson, On the continuity of global attractors, Proc. Amer. Math. Soc., 143 (2015), 4389-4395.  doi: 10.1090/proc/12598.  Google Scholar

[18]

R. Joly and C. Laurent, Stabilization for the semilinear wave equation with geometric control condition, Analysis and PDE, 6 (2013), 1089-1119.  doi: 10.2140/apde.2013.6.1089.  Google Scholar

[19]

V. KalantarovA. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré, 17 (2016), 2555-2584.  doi: 10.1007/s00023-016-0480-y.  Google Scholar

[20]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[21]

I. LasieckaR. Triggiani and P.-F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235 (1999), 13-57.  doi: 10.1006/jmaa.1999.6348.  Google Scholar

[22]

I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: global uniqueness and observability in one shot, in, Differential Geometric Methods in the Control of Partial Differential Equations, 227–325, Contemp. Math. 268, Amer. Math. Soc., Providence, RI, 2000. doi: 10.1090/conm/268/04315.  Google Scholar

[23]

T. F. MaP. Marín-Rubio and C. Surco Chuño, Dynamics of wave equations with moving boundary, J. Differential Equations, 262 (2017), 3317-3342.  doi: 10.1016/j.jde.2016.11.030.  Google Scholar

[24]

X. Mei and C. Sun, Uniform attractors for a weakly damped wave equation with sup-cubic nonlinearity, Appl. Math. Letters, 95 (2019), 179-185.  doi: 10.1016/j.aml.2019.04.003.  Google Scholar

[25]

J. V. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math., 22 (1969), 807-823.  doi: 10.1002/cpa.3160220605.  Google Scholar

[26]

J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86.  doi: 10.1512/iumj.1974.24.24004.  Google Scholar

[27]

L. Robbiano and C. Zuily, Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients, Invent. Math., 131 (1998), 493-539.  doi: 10.1007/s002220050212.  Google Scholar

[28]

A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl., 71 (1992), 455–467.  Google Scholar

[29]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[30]

D. Toundykov, Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions, Nonlinear Anal., 67 (2007), 512-544.  doi: 10.1016/j.na.2006.06.007.  Google Scholar

[31]

Z. Yang and Z. Liu, Longtime dynamics of the quasi-linear wave equations with structural damping and supercritical nonlinearities, Nonlinearity, 30 (2017), 1120-1145.  doi: 10.1088/1361-6544/aa599f.  Google Scholar

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.  Google Scholar

[2]

A. V. Babin and S. Yu Pilyugin, Continuous dependence of attractors on the shape of domain, J. Math. Sci., 87 (1997), 3304-3310.  doi: 10.1007/BF02355582.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications 25, North-Holland, Amsterdam, 1992.  Google Scholar

[4]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

[5]

M. BlairH. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1817-1829.  doi: 10.1016/j.anihpc.2008.12.004.  Google Scholar

[6]

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.  Google Scholar

[7]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[8]

I. ChueshovM. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.  Google Scholar

[9]

I. ChueshovI. Lasiecka and D. Toundykov, Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent, Discrete Contin. Dyn. Syst., 20 (2008), 459-509.  doi: 10.3934/dcds.2008.20.459.  Google Scholar

[10]

M. ContiT. F. MaE. M. Marchini and P. N. Seminario Huertas, Asymptotics of viscoelastic materials with nonlinear density and memory effects, J. Differential Equations, 264 (2018), 4235-4259.  doi: 10.1016/j.jde.2017.12.010.  Google Scholar

[11]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics 37, Masson, Paris. Wiley, Chichester, 1994.  Google Scholar

[12]

E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Comm. Partial Differential Equations, 18 (1993), 1539-1555.  doi: 10.1080/03605309308820985.  Google Scholar

[13]

X. FuJ. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control. Optim., 46 (2007), 1578-1614.  doi: 10.1137/040610222.  Google Scholar

[14]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, 1988.  Google Scholar

[15]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0.  Google Scholar

[16]

J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl., 154 (1989), 281-326.  doi: 10.1007/BF01790353.  Google Scholar

[17]

L. T. HoangE. J. Olson and J. C. Robinson, On the continuity of global attractors, Proc. Amer. Math. Soc., 143 (2015), 4389-4395.  doi: 10.1090/proc/12598.  Google Scholar

[18]

R. Joly and C. Laurent, Stabilization for the semilinear wave equation with geometric control condition, Analysis and PDE, 6 (2013), 1089-1119.  doi: 10.2140/apde.2013.6.1089.  Google Scholar

[19]

V. KalantarovA. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré, 17 (2016), 2555-2584.  doi: 10.1007/s00023-016-0480-y.  Google Scholar

[20]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[21]

I. LasieckaR. Triggiani and P.-F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235 (1999), 13-57.  doi: 10.1006/jmaa.1999.6348.  Google Scholar

[22]

I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: global uniqueness and observability in one shot, in, Differential Geometric Methods in the Control of Partial Differential Equations, 227–325, Contemp. Math. 268, Amer. Math. Soc., Providence, RI, 2000. doi: 10.1090/conm/268/04315.  Google Scholar

[23]

T. F. MaP. Marín-Rubio and C. Surco Chuño, Dynamics of wave equations with moving boundary, J. Differential Equations, 262 (2017), 3317-3342.  doi: 10.1016/j.jde.2016.11.030.  Google Scholar

[24]

X. Mei and C. Sun, Uniform attractors for a weakly damped wave equation with sup-cubic nonlinearity, Appl. Math. Letters, 95 (2019), 179-185.  doi: 10.1016/j.aml.2019.04.003.  Google Scholar

[25]

J. V. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math., 22 (1969), 807-823.  doi: 10.1002/cpa.3160220605.  Google Scholar

[26]

J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86.  doi: 10.1512/iumj.1974.24.24004.  Google Scholar

[27]

L. Robbiano and C. Zuily, Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients, Invent. Math., 131 (1998), 493-539.  doi: 10.1007/s002220050212.  Google Scholar

[28]

A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl., 71 (1992), 455–467.  Google Scholar

[29]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[30]

D. Toundykov, Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions, Nonlinear Anal., 67 (2007), 512-544.  doi: 10.1016/j.na.2006.06.007.  Google Scholar

[31]

Z. Yang and Z. Liu, Longtime dynamics of the quasi-linear wave equations with structural damping and supercritical nonlinearities, Nonlinearity, 30 (2017), 1120-1145.  doi: 10.1088/1361-6544/aa599f.  Google Scholar

Figure 1.  The control region $ \omega $ satisfies (GCC). Any ray of geometric optics inside $ \Omega $ hits $ \omega $
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