Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Kernel sections for damped non-autonomous wave equations with linear memory and critical exponent

Author: Shengfan Zhou
Journal: Quart. Appl. Math. 61 (2003), 731-757
MSC: Primary 37L30; Secondary 35L70, 35R10
DOI: https://doi.org/10.1090/qam/2019621
MathSciNet review: MR2019621
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the existence of kernel sections for the process generated by a non-autonomous wave equation with linear memory when there is nonlinear damping and the nonlinearity has a critically growing exponent; we also obtain a more precise estimate of upper bound of the Hausdorff dimension of the kernel sections. And we point out that in the case of autonomous systems with linear damping, the obtained upper bound of the Hausdorff dimension decreases as the damping grows for suitable large damping.

References [Enhancements On Off] (What's this?)

  • [1] V. Pata, A Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl. 11, 505-529 (2001) MR 1907454
  • [2] M. Grasselli, V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, Preprint MR 1944162
  • [3] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297-308 (1970) MR 0281400
  • [4] Z. Liu, S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. Appl. Math. 54, 21-31 (1996) MR 1373836
  • [5] M. Fabrizio, B. Lazzari, On the existence and asymptotic stability of solution for linear viscoelastic solids, Arch. Rational Mech. Anal. 116, 139-152 (1991) MR 1143437
  • [6] V. Chepyzhov, M. Vishik, A Hausdorff Dimension Estimate for Kernel Sections Of Non-autonomous Evolution Equations, Indiana Univ. Math. Journal 42, 1057-1076 (1993) MR 1254132
  • [7] J. Arrieta, A. N. Carvalho, and J. K. Hale, A damping wave equation with critical exponent, Commun. Partial Diff. Eq. 17, 841-866 (1992) MR 1177295
  • [8] E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Commun. Partial Diff. Eq. 18, 1539-1555 (1993) MR 1239923
  • [9] Zhou Shengfan, Dimension of the global attractor for damped semilinear wave equations with critical exponent, J. Math. Phys. 40, 4444-4451 (1999) MR 1708317
  • [10] Yu Huang, Zhao Yi, and Zhaoyang Yin, On the dimension of the global attractor for a damped semilinear wave equation with critical exponent, J. Math. Phys. 41, 4957-4966 (2000) MR 1765839
  • [11] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag, New York, 1983 MR 710486
  • [12] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sciences, 68, Springer-Verlag, New York, 1988 MR 953967

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 37L30, 35L70, 35R10

Retrieve articles in all journals with MSC: 37L30, 35L70, 35R10

Additional Information

DOI: https://doi.org/10.1090/qam/2019621
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society