### Strongly damped wave equation and its Yosida approximations

DOI: http://dx.doi.org/10.12775/TMNA.2015.059

#### Abstract

#### Keywords

#### References

H. Amann, Linear and Quasilinear Parabolic Problems, Abstract Linear Theory, Monographs in Mathematics, Volume I, Birkhauser, 1995.

A. V. Babin and M. I. Vishik, Attractors in Evolutionary Equations, Studies in Mathematics and its Applications 25, North-Holland Publishing Co., Amsterdam, 1992.

G. Bachman and L. Narici, Functional Analysis, Dover Publications, Mineola, NY, 2000.

I. L. Bogolubsky, Some examples of inelastic soliton interaction, Comput. Phys. Comm. 13 (1977), 149-155.

A. N. Carvalho, J. A. Langa and J.C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Springer, 2013.

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.

A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math. 207 (2002), 287-310.

A. N. Carvalho and J. W. Cholewa, Local well poshness, asymptotic behaviour and asymptotic bootstrapping for a class of semilinear evolution equations of second order in time, Trans. Amer. Math. Soc. (2007), 2567-2586.

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.

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results, Commun. Pure Appl. Anal. 12 (2013), 3047-3071.

S. Chunyou, Y. Lu and D. Jinqiao, Asymptotic behaviour for a semilinear second order evolution equation, Trans. Amer. Math. Soci. 363 (2011), 6085-6109.

M. Conti and V. Pata, On the regularity of global attractors, Discrete Contin. Dyn. Syst. 25 (2009), 1209-1217.

D. E. Edmunds and H. Triebel, Function spaces, entropy numbers and differential operators, Cambridge University Press, 1996.

J. Garcia-Luengo, P. Marin-Rubio and J. Real, Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations, Nonlinearity 25 (2012), no. 4, 905-930.

J. M. Ghidaglia and A. Marzocchi, Longtime behavior of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal. 22 (1991), 879-895.

J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. 154 (4) (1989), 281-326.

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.

D. Henry, Lecture Notes - Invariant Manifolds Near a Fixed Point.

T. Kano and T. Nishida, A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves, Osaka J. Math. 23 (1986), 389-413.

V. G. Makhankov, On stationary solutions of the Schrodinger equation with a self-consistent potential satisfying Boussinesq's equation, Phys. Lett. A 50 (1974), 42-44.

V. G. Makhankov, Dynamics of classical solitons (in non-integrable systems), Physics Reports, Phys. Lett. C 35 (1978), 1-128.

V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys. 253 (2005), 511-533.

V. Pata and S.V. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity 19 (2006), 1495-1506.

C. Sun, L. Yang and J. Duan, Asymptotic behavior for a semilinear second order evolution equation, Trans. Amer. Math. Soc.

S. Wang and G. Chen, The Cauchy problem for the generalized IMBq equation in $W^{s,p}({Bbb R}^n)$, J. Math. Anal. Appl. 266 (2002), 38-54.

G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math. 32 (1980), 631-643.

M. Yang and C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, Trans. Amer. Math. Soc. 361 (2009), 1069-1101.

M. Yang and C. Sun, Exponential attractors for the strongly damped wave equations, Nonlinar Anal., Real World Appl. 11 (2010), 913-919.

### Refbacks

- There are currently no refbacks.