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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:
[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

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

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

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

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

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