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June  2018, 23(4): 1645-1674. doi: 10.3934/dcdsb.2018068

Dynamics for the damped wave equations on time-dependent domains

a. 

College of Science, China University of Petroleum (East China), Qingdao, 266580, China

b. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

c. 

College of Science, Yanshan University, Qinhuangdao, 066004, China

* Corresponding author: Feng Zhou

Received  June 2017 Revised  August 2017 Published  January 2018

Fund Project: The first author is supported by NSFC (Grants No. 11601522) and the Fundamental Research Funds for the Central Universities of China (No. 17CX02036A), the second author is supported by NSFC (Grants Nos. 11471148,11522109)

We consider the asymptotic dynamics of a damped wave equations on a time-dependent domains with homogeneous Dirichlet boundary condition, the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. To this end, we establish the existence and uniqueness of strong and weak solutions satisfying energy inequality under the assumption that the spatial domains $\mathcal{O}_{t}$ in $\mathbb{R}^{3}$ are obtained from a bounded base domain $\mathcal{O}$ by a $C^{3}$-diffeomorphism $r(·, t)$. Furthermore, we establish the pullback attractor under a slightly weaker assumption that the measure of the spatial domains are uniformly bounded above.

Citation: Feng Zhou, Chunyou Sun, Xin Li. Dynamics for the damped wave equations on time-dependent domains. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1645-1674. doi: 10.3934/dcdsb.2018068
References:
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A. L. Amadori and J. L. Vazquez, Singular free boundary problem from image processing, Math. Models Methods Appl. Sci., 15 (2005), 689-715.  doi: 10.1142/S0218202505000509.  Google Scholar

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J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, J. Differential Equations, 199 (2004), 143-178.  doi: 10.1016/j.jde.2003.09.004.  Google Scholar

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

[4]

J. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅰ. Continuity of the set of equilibria, J. Differential Equations, 231 (2006), 551-597.  doi: 10.1016/j.jde.2006.06.002.  Google Scholar

[5]

J. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅱ. The limiting problem, J. Differential Equations, 247 (2009), 174-202.  doi: 10.1016/j.jde.2009.03.014.  Google Scholar

[6]

J. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅲ. Continuity of attractors, J. Differential Equations, 247 (2009), 225-259.  doi: 10.1016/j.jde.2008.12.014.  Google Scholar

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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.  Google Scholar

[8]

M. L. BernardiG. Guatteri and F. Luterotti, Abstract Schroedinger-type differential equations with variable domain, J. Math. Anal. Appl., 211 (1997), 84-105.  doi: 10.1006/jmaa.1997.5422.  Google Scholar

[9]

M. L. BernardiG. A. Pozzi and G. Savaré, Variational equations of Schroedinger-type in non-cylindrical domains, J. Differential Equations, 171 (2001), 63-87.  doi: 10.1006/jdeq.2000.3834.  Google Scholar

[10] H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983.   Google Scholar
[11]

P. CannarsaG. Da Prato and J. P. Zolesto, The damped wave equation in a moving domain, J. Differential Equations, 85 (1990), 1-16.  doi: 10.1016/0022-0396(90)90086-5.  Google Scholar

[12]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

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A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

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C. CarlosC. Nicolae and M. Arnaud, Controllability of the linear one-dimensional wave equation with inner moving forces, SIAM J. Control Optim., 52 (2014), 4027-4056.  doi: 10.1137/140956129.  Google Scholar

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X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986.  doi: 10.1137/S0036141002418388.  Google Scholar

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A. Cheskidov and S. S. Lu, Uniform global attractors for the nonautonomous 3D Navier-Stokes equations, Adv. Math., 267 (2014), 277-306.  doi: 10.1016/j.aim.2014.09.005.  Google Scholar

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J. Cooper, Scattering of plane waves by a moving obstacle, Arch. Rational Mech. Anal., 71 (1979), 113-141.  doi: 10.1007/BF00248724.  Google Scholar

[20]

J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent domain, J. Math. Anal. Appl., 42 (1973), 29-60.  doi: 10.1016/0022-247X(73)90120-0.  Google Scholar

[21]

A. D. D. Craik, The origins of water wave theory, Annu. Rev. Fluid Mech., 36 (2004), 1-28.  doi: 10.1146/annurev.fluid.36.050802.122118.  Google Scholar

[22]

H. CrauelP. E. Kloeden and M. H. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.  Google Scholar

[23]

L. CuiX. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.  Google Scholar

[24]

D. R. da CostaC. P. Dettmann and E. D. Leonel, Escape of particles in a time-dependent potential well, Phys. Rev. E, 83 (2011), 066211.  doi: 10.1103/PhysRevE.83.066211.  Google Scholar

[25] E. Dibenedetto, Degenerate Parabolic Equations, Springer-Verleg, New York, 1993.   Google Scholar
[26]

L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.  Google Scholar

[27]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246.   Google Scholar

[28]

C. He and L. Hsiao, Two-dimensional Euler equations in a time dependent domain, J. Differential Equations, 163 (2000), 265-291.  doi: 10.1006/jdeq.1999.3702.  Google Scholar

[29]

A. K. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.  doi: 10.1016/j.jmaa.2005.05.031.  Google Scholar

[30]

N. JamesA. Ilyasse and D. Stevan, Control of parabolic PDEs with time-varying spatial domain: Czochralski crystal growth process, Internat. J. Control, 86 (2013), 1467-1478.  doi: 10.1080/00207179.2013.786187.  Google Scholar

[31]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.  Google Scholar

[32]

P. E. KloedenP. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.  doi: 10.1016/j.jde.2007.10.031.  Google Scholar

[33]

P. E. KloedenJ. Real and C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Equations, 246 (2009), 4702-4730.  doi: 10.1016/j.jde.2008.11.017.  Google Scholar

[34]

E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.  doi: 10.1007/s10440-014-9993-x.  Google Scholar

[35] O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer, 1985.  doi: 10.1007/978-1-4757-4317-3.  Google Scholar
[36]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Nonlineaires, (French) Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[37]

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

[38]

L. MorinoB. K. BharadvajM. I. Freedman and K. Tseng, Boundary integral equation for wave equation with moving boundary and applications to compressible potential aerodynamics of airplanes and helicopters, Comput. Mech., 4 (1989), 231-243.  doi: 10.1007/BF00301382.  Google Scholar

[39]

J. V. Pereira and R. P. Silva, Reaction-diffusion equations in a noncylindrical thin domain, Bound. Value Probl. , 2013 (2013), 10pp. doi: 10.1186/1687-2770-2013-248.  Google Scholar

[40] J. C. Robinson, Infinite-Dimensional Dyanamical Systems, Cambridage Univ. Press, Cambridge, 2001.   Google Scholar
[41]

S. E. Shreve and H. M. Soner, A free boundary problem related to singular stochastic control: Parabolic case, Comm. Partial Differential Equations, 16 (1991), 373-424.  doi: 10.1080/03605309108820763.  Google Scholar

[42]

J. Stefan, Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere, Ann. Phys. Chem., 278 (1891), 269-286.  doi: 10.1002/andp.18912780206.  Google Scholar

[43]

C. Y. SunD. M. Cao and J. Q. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665.   Google Scholar

[44]

C. Y. Sun and Y. B. Yuan, $L^{p}$-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052.  doi: 10.1017/S0308210515000177.  Google Scholar

[45]

P. K. C. Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theor. Appl., 65 (1990), 331-362.  doi: 10.1007/BF01102351.  Google Scholar

[46]

S. V. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Comm. Pure Appl. Anal., 3 (2004), 921-934.  doi: 10.3934/cpaa.2004.3.921.  Google Scholar

show all references

References:
[1]

A. L. Amadori and J. L. Vazquez, Singular free boundary problem from image processing, Math. Models Methods Appl. Sci., 15 (2005), 689-715.  doi: 10.1142/S0218202505000509.  Google Scholar

[2]

J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, J. Differential Equations, 199 (2004), 143-178.  doi: 10.1016/j.jde.2003.09.004.  Google Scholar

[3]

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

[4]

J. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅰ. Continuity of the set of equilibria, J. Differential Equations, 231 (2006), 551-597.  doi: 10.1016/j.jde.2006.06.002.  Google Scholar

[5]

J. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅱ. The limiting problem, J. Differential Equations, 247 (2009), 174-202.  doi: 10.1016/j.jde.2009.03.014.  Google Scholar

[6]

J. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅲ. Continuity of attractors, J. Differential Equations, 247 (2009), 225-259.  doi: 10.1016/j.jde.2008.12.014.  Google Scholar

[7]

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.  Google Scholar

[8]

M. L. BernardiG. Guatteri and F. Luterotti, Abstract Schroedinger-type differential equations with variable domain, J. Math. Anal. Appl., 211 (1997), 84-105.  doi: 10.1006/jmaa.1997.5422.  Google Scholar

[9]

M. L. BernardiG. A. Pozzi and G. Savaré, Variational equations of Schroedinger-type in non-cylindrical domains, J. Differential Equations, 171 (2001), 63-87.  doi: 10.1006/jdeq.2000.3834.  Google Scholar

[10] H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983.   Google Scholar
[11]

P. CannarsaG. Da Prato and J. P. Zolesto, The damped wave equation in a moving domain, J. Differential Equations, 85 (1990), 1-16.  doi: 10.1016/0022-0396(90)90086-5.  Google Scholar

[12]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[13]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[14]

C. CarlosC. Nicolae and M. Arnaud, Controllability of the linear one-dimensional wave equation with inner moving forces, SIAM J. Control Optim., 52 (2014), 4027-4056.  doi: 10.1137/140956129.  Google Scholar

[15]

X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986.  doi: 10.1137/S0036141002418388.  Google Scholar

[16]

C. ChenL. Jiang and B. Bian, Free boundary and American options in a jump-diffusion model, European J. Appl. Math., 17 (2006), 95-127.  doi: 10.1017/S0956792505006340.  Google Scholar

[17]

A. Cheskidov and S. S. Lu, Uniform global attractors for the nonautonomous 3D Navier-Stokes equations, Adv. Math., 267 (2014), 277-306.  doi: 10.1016/j.aim.2014.09.005.  Google Scholar

[18] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, New York, 2015.  doi: 10.1007/978-3-319-22903-4.  Google Scholar
[19]

J. Cooper, Scattering of plane waves by a moving obstacle, Arch. Rational Mech. Anal., 71 (1979), 113-141.  doi: 10.1007/BF00248724.  Google Scholar

[20]

J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent domain, J. Math. Anal. Appl., 42 (1973), 29-60.  doi: 10.1016/0022-247X(73)90120-0.  Google Scholar

[21]

A. D. D. Craik, The origins of water wave theory, Annu. Rev. Fluid Mech., 36 (2004), 1-28.  doi: 10.1146/annurev.fluid.36.050802.122118.  Google Scholar

[22]

H. CrauelP. E. Kloeden and M. H. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.  Google Scholar

[23]

L. CuiX. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.  Google Scholar

[24]

D. R. da CostaC. P. Dettmann and E. D. Leonel, Escape of particles in a time-dependent potential well, Phys. Rev. E, 83 (2011), 066211.  doi: 10.1103/PhysRevE.83.066211.  Google Scholar

[25] E. Dibenedetto, Degenerate Parabolic Equations, Springer-Verleg, New York, 1993.   Google Scholar
[26]

L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.  Google Scholar

[27]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246.   Google Scholar

[28]

C. He and L. Hsiao, Two-dimensional Euler equations in a time dependent domain, J. Differential Equations, 163 (2000), 265-291.  doi: 10.1006/jdeq.1999.3702.  Google Scholar

[29]

A. K. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.  doi: 10.1016/j.jmaa.2005.05.031.  Google Scholar

[30]

N. JamesA. Ilyasse and D. Stevan, Control of parabolic PDEs with time-varying spatial domain: Czochralski crystal growth process, Internat. J. Control, 86 (2013), 1467-1478.  doi: 10.1080/00207179.2013.786187.  Google Scholar

[31]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.  Google Scholar

[32]

P. E. KloedenP. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.  doi: 10.1016/j.jde.2007.10.031.  Google Scholar

[33]

P. E. KloedenJ. Real and C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Equations, 246 (2009), 4702-4730.  doi: 10.1016/j.jde.2008.11.017.  Google Scholar

[34]

E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.  doi: 10.1007/s10440-014-9993-x.  Google Scholar

[35] O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer, 1985.  doi: 10.1007/978-1-4757-4317-3.  Google Scholar
[36]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Nonlineaires, (French) Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[37]

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

[38]

L. MorinoB. K. BharadvajM. I. Freedman and K. Tseng, Boundary integral equation for wave equation with moving boundary and applications to compressible potential aerodynamics of airplanes and helicopters, Comput. Mech., 4 (1989), 231-243.  doi: 10.1007/BF00301382.  Google Scholar

[39]

J. V. Pereira and R. P. Silva, Reaction-diffusion equations in a noncylindrical thin domain, Bound. Value Probl. , 2013 (2013), 10pp. doi: 10.1186/1687-2770-2013-248.  Google Scholar

[40] J. C. Robinson, Infinite-Dimensional Dyanamical Systems, Cambridage Univ. Press, Cambridge, 2001.   Google Scholar
[41]

S. E. Shreve and H. M. Soner, A free boundary problem related to singular stochastic control: Parabolic case, Comm. Partial Differential Equations, 16 (1991), 373-424.  doi: 10.1080/03605309108820763.  Google Scholar

[42]

J. Stefan, Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere, Ann. Phys. Chem., 278 (1891), 269-286.  doi: 10.1002/andp.18912780206.  Google Scholar

[43]

C. Y. SunD. M. Cao and J. Q. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665.   Google Scholar

[44]

C. Y. Sun and Y. B. Yuan, $L^{p}$-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052.  doi: 10.1017/S0308210515000177.  Google Scholar

[45]

P. K. C. Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theor. Appl., 65 (1990), 331-362.  doi: 10.1007/BF01102351.  Google Scholar

[46]

S. V. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Comm. Pure Appl. Anal., 3 (2004), 921-934.  doi: 10.3934/cpaa.2004.3.921.  Google Scholar

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