• Previous Article
    Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion
  • DCDS-B Home
  • This Issue
  • Next Article
    Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions
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  June 2018 Early access  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 and 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.

[1]

Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022

[2]

Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307

[3]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194

[4]

Shengfan Zhou, Linshan Wang. Kernel sections for damped non-autonomous wave equations with critical exponent. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 399-412. doi: 10.3934/dcds.2003.9.399

[5]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809

[6]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703

[7]

Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 991-1009. doi: 10.3934/dcds.2012.32.991

[8]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195

[9]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037

[10]

Penghui Zhang, Zhaosheng Feng, Lu Yang. Non-autonomous weakly damped plate model on time-dependent domains. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3319-3336. doi: 10.3934/dcdss.2021076

[11]

Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141

[12]

Shuang Yang, Yangrong Li. Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain. Evolution Equations and Control Theory, 2020, 9 (3) : 581-604. doi: 10.3934/eect.2020025

[13]

Radosław Czaja. Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021276

[14]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1997-2013. doi: 10.3934/cpaa.2020088

[15]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281

[16]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036

[17]

Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1299-1316. doi: 10.3934/dcdsb.2019221

[18]

Xudong Luo, Qiaozhen Ma. The existence of time-dependent attractor for wave equation with fractional damping and lower regular forcing term. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021253

[19]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887

[20]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (354)
  • HTML views (494)
  • Cited by (0)

Other articles
by authors

[Back to Top]