Regularity of Invariant Sets in Variable Internal Damped Wave Equations

Acta Mathematicae Applicatae Sinica, English Series - Tập 36 Số 4 - Trang 952-974 - 2020
Yue, Gao-cheng1, Liang, Yu-xin1, Yang, Jia-jia1
1Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, China

Tóm tắt

In this paper we prove that every compact invariant subset $$\mathscr{A}$$ associated with the semigroup Sn,k(t)t≥0 generated by wave equations with variable damping, either in the interior or on the boundary of the domain where Ω ⊂ ℝ3 is a smooth bounded domain, in H 0 1 (Ω) × L2(Ω) is in fact bounded in D(B0) × H 0 1 (Ω). As an application of our results, we obtain the upper-semicontinuity for global attractor of the weakly damped semilinear wave equation in the norm of H1(Ω) × L2(Ω) when the interior variable damping converges to the boundary damping in the sense of distributions.

Tài liệu tham khảo

citation_title=Attractors of evolution equations; citation_publication_date=1992; citation_id=CR1; citation_author=AV Babin; citation_author=MI Vishik; citation_publisher=Studies in Mathematics and its Applications citation_journal_title=Discrete and Continuous Dynamical Systems; citation_title=Global attractors for damped semilinear wave equations; citation_author=JM Ball; citation_volume=10; citation_publication_date=2004; citation_pages=31-52; citation_doi=10.3934/dcds.2004.10.31; citation_id=CR2 citation_title=An Introduction to Semilinear Evolution Equations; citation_publication_date=1988; citation_id=CR3; citation_author=T Cazenave; citation_author=A Haraux; citation_publisher=Clarendon Press citation_journal_title=Discrete and Continuous Dynamical Systems; citation_title=On the regularity of global attractors; citation_author=M Conti, V Pata; citation_volume=25; citation_publication_date=2009; citation_pages=1209-1217; citation_doi=10.3934/dcds.2009.25.1209; citation_id=CR4 citation_journal_title=SIAM Journal on Applied Mathematics; citation_title=Exponential decay of energy of evolution equations with locally distributed damping; citation_author=G Chen, SA Fulling, FJ Narcowich, S Sun; citation_volume=51; citation_publication_date=1991; citation_pages=266-301; citation_doi=10.1137/0151015; citation_id=CR5 citation_journal_title=Communications in Partial Differential Equations; citation_title=On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation; citation_author=I Chueshov, M Eller, I Lasiecka; citation_volume=27; citation_publication_date=2002; citation_pages=1901-1951; citation_doi=10.1081/PDE-120016132; citation_id=CR6 citation_journal_title=Communication in Partial Differential Equations; citation_title=The rate at which energy decays in a damped string; citation_author=S Cox, E Zuazua; citation_volume=19; citation_publication_date=1994; citation_pages=213-243; citation_doi=10.1080/03605309408821015; citation_id=CR7 citation_journal_title=Indiana University Mathematics Journal; citation_title=The rate at which energy decays in a string damped at one end; citation_author=S Cox, E Zuazua; citation_volume=44; citation_publication_date=1995; citation_pages=545-573; citation_doi=10.1512/iumj.1995.44.2001; citation_id=CR8 citation_journal_title=Turkish J. Math.; citation_title=On the convergence of attractors and exponential attractors for singularly perturbed hyperbolic equations; citation_author=A Eden, A Milani; citation_volume=19; citation_publication_date=1995; citation_pages=102-117; citation_id=CR9 citation_journal_title=Communications in Partial Differential Equations; citation_title=Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent; citation_author=E Feireisel, E Zuazua; citation_volume=18; citation_publication_date=1993; citation_pages=1539-1555; citation_doi=10.1080/03605309308820985; citation_id=CR10 citation_journal_title=Ann. Sc. Norm. Super. Pisa Cl. Sci.; citation_title=Regularity of the solutions of second order evolution equations and their attractors; citation_author=JM Ghidaglia, R Temam; citation_volume=14; citation_publication_date=1987; citation_pages=485-511; citation_id=CR11 Grasselli, M., Pata, V. On the damped semilinear wave equation with critical exponent. In: Dynamical Systems and Differential Equations, Wilmington, NC, 2002, Discrete Contin. Dyn. Syst. (Suppl.), 351–358 (2003) Hale, J.K. Asymptotic behavior of dissipative systems. Mathematical Survey, American Mathematical Society, 1988 citation_journal_title=J. Math. Pures Appl.; citation_title=Regularity, determining modes and Galerkin methods; citation_author=JK Hale, G Raugel; citation_volume=82; citation_publication_date=2003; citation_pages=1075-1136; citation_doi=10.1016/S0021-7824(03)00045-X; citation_id=CR14 citation_journal_title=J. Differential Equations; citation_title=Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation; citation_author=JK Hale, G Raugel; citation_volume=73; citation_publication_date=1988; citation_pages=197-214; citation_doi=10.1016/0022-0396(88)90104-0; citation_id=CR15 citation_journal_title=Portugaliae Mathematica; citation_title=Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps; citation_author=A Haraux; citation_volume=46; citation_publication_date=1989; citation_pages=246-257; citation_id=CR16 Haraux, A. Two remarks on hyperbolic dissipative problems. in: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vol. VII, Paris, 1983–1984, Pitman, Boston, 1985, 161–179. citation_journal_title=Journal of Differential Equations; citation_title=Convergence of the wave equation damped on the interior to the one damped on the boundary; citation_author=R Joly; citation_volume=229; citation_publication_date=2006; citation_pages=588-653; citation_doi=10.1016/j.jde.2006.01.006; citation_id=CR18 citation_journal_title=J. Fac. Sci. Univ. Tokyo Sect.; citation_title=Linear evolution equations of hyperbolic type; citation_author=T Kato; citation_volume=17; citation_publication_date=1970; citation_pages=241-258; citation_id=CR19 citation_journal_title=J. Math. Pures Appl.; citation_title=A direct method for the boundary stabilization of the wave equation; citation_author=V Komornik, E Zuazua; citation_volume=69; citation_publication_date=1990; citation_pages=33-45; citation_id=CR20 citation_journal_title=Journal of Mathematical Sciences; citation_title=Attractor for a semilinear wave equation with boundary damping; citation_author=IN Kostin; citation_volume=98; citation_publication_date=2000; citation_pages=753-764; citation_doi=10.1007/BF02355388; citation_id=CR21 citation_journal_title=Differential and Integral Equations; citation_title=Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping; citation_author=I Lasiecka, D Tataru; citation_volume=6; citation_publication_date=1993; citation_pages=507-533; citation_id=CR22 citation_title=Semigroups of linear operators and applications to partial differential equa- tions; citation_publication_date=1983; citation_id=CR23; citation_author=A Pazy; citation_publisher=Applied Mathematical Sciences citation_journal_title=J. Differential Equations; citation_title=Regularity of invariant sets in semilinear damped wave equations; citation_author=M Prizzi; citation_volume=247; citation_publication_date=2009; citation_pages=3315-3337; citation_doi=10.1016/j.jde.2009.08.011; citation_id=CR24 citation_journal_title=Topol. Methods Nonlinear Anal.; citation_title=Conley index continuation for singularly perturbed hyperbolic equations; citation_author=KP Rybakowski; citation_volume=22; citation_publication_date=2003; citation_pages=203-244; citation_doi=10.12775/TMNA.2003.037; citation_id=CR25 citation_journal_title=Archive of Rational Mechanics and Analysis; citation_title=Qualitative behavior of dissipative wave equations on bounded domains; citation_author=J Rauch; citation_volume=62; citation_publication_date=1976; citation_pages=77-85; citation_doi=10.1007/BF00251857; citation_id=CR26 citation_journal_title=Journal of Differential Equations; citation_title=Uniform decay rates and attractors for evolution PDEs with boundary dissipation; citation_author=D Tataru; citation_volume=121; citation_publication_date=1995; citation_pages=1-27; citation_doi=10.1006/jdeq.1995.1119; citation_id=CR27 citation_title=Infinite-dimensional systems in mechanics and physics; citation_publication_date=1997; citation_id=CR28; citation_author=R Temam; citation_publisher=Springer-Verlag citation_journal_title=Communications in Partial Differential Equations; citation_title=Exponential decay for the semilinear wave equation with locally distributed damping; citation_author=E Zuazua; citation_volume=15; citation_publication_date=1990; citation_pages=205-235; citation_doi=10.1080/03605309908820684; citation_id=CR29