Convergence to equilibrium for the damped semilinear wave equation with critical exponent and dissipative boundary condition
Authors:
Hao Wu and Songmu Zheng
Journal:
Quart. Appl. Math. 64 (2006), 167-188
MSC (2000):
Primary 35B40, 35Q99
DOI:
https://doi.org/10.1090/S0033-569X-06-01004-0
Published electronically:
January 24, 2006
MathSciNet review:
2211383
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: This paper is concerned with the asymptotic behavior of the solution to the following damped semilinear wave equation with critical exponent: \begin{equation} u_{tt} + u_t -\Delta u + f(x,u) = 0, \qquad (x,t) \in \Omega \times \mathbb {R}^+ \end{equation} subject to the dissipative boundary condition \begin{equation} \partial _\nu u+ u + u_t = 0, \qquad t > 0, \ x \in \Gamma \end{equation} and the initial conditions \begin{equation} u|_{t=0} = u_0(x),\quad u_t|_{t=0}=u_1(x), \qquad x \in \Omega , \end{equation} where $\Omega$ is a bounded domain in $\mathbb {R}^3$ with smooth boundary $\Gamma$ , $\nu$ is the outward normal direction to the boundary, and $f$ is analytic in $u$. In this paper convergence of the solution to an equilibrium as time goes to infinity is proved. While these types of results are known for the damped semilinear wave equation with interior dissipation and Dirichlet boundary condition, this is, to our knowledge, the first result with dissipative boundary condition and critical growth exponent.
- Sergiu Aizicovici, Eduard Feireisl, and Françoise Issard-Roch, Long-time convergence of solutions to a phase-field system, Math. Methods Appl. Sci. 24 (2001), no. 5, 277–287. MR 1818896, DOI https://doi.org/10.1002/mma.215
- A. V. Babin and M. I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992. Translated and revised from the 1989 Russian original by Babin. MR 1156492
- J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 31–52. Partial differential equations and applications. MR 2026182, DOI https://doi.org/10.3934/dcds.2004.10.31
- Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843
- H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562
- Ralph Chill, On the Łojasiewicz-Simon gradient inequality, J. Funct. Anal. 201 (2003), no. 2, 572–601. MR 1986700, DOI https://doi.org/10.1016/S0022-1236%2802%2900102-7
- Igor Chueshov, Matthias Eller, and Irena Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations 27 (2002), no. 9-10, 1901–1951. MR 1941662, DOI https://doi.org/10.1081/PDE-120016132
- Eduard Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Differential Equations 116 (1995), no. 2, 431–447. MR 1318582, DOI https://doi.org/10.1006/jdeq.1995.1042
- Eduard Feireisl, Françoise Issard-Roch, and Hana Petzeltová, Long-time behaviour and convergence towards equilibria for a conserved phase field model, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 239–252. Partial differential equations and applications. MR 2026193, DOI https://doi.org/10.3934/dcds.2004.10.239
- Maurizio Grasselli and Vittorino Pata, Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure Appl. Anal. 3 (2004), no. 4, 849–881. MR 2106302, DOI https://doi.org/10.3934/cpaa.2004.3.849
- Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371
- Alain Haraux, Semi-linear hyperbolic problems in bounded domains, Math. Rep. 3 (1987), no. 1, i–xxiv and 1–281. MR 1078761
- Alain Haraux and Mohamed Ali Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations 9 (1999), no. 2, 95–124. MR 1714129, DOI https://doi.org/10.1007/s005260050133
- Sen-Zhong Huang and Peter Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal. 46 (2001), no. 5, Ser. A: Theory Methods, 675–698. MR 1857152, DOI https://doi.org/10.1016/S0362-546X%2800%2900145-0
- Mohamed Ali Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal. 153 (1998), no. 1, 187–202. MR 1609269, DOI https://doi.org/10.1006/jfan.1997.3174
- Mohamed Ali Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Differential Equations 144 (1998), no. 2, 302–312. MR 1616964, DOI https://doi.org/10.1006/jdeq.1997.3392
- Irena Lasiecka, Mathematical control theory of coupled PDEs, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 75, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. MR 1879543
- I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations 6 (1993), no. 3, 507–533. MR 1202555
Li W. Li, Long-time convergence of solution to phase-field system with Neumann boundary conditions, to appear in Chinese Annals of Mathematics A.
- S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Les Équations aux Dérivées Partielles (Paris, 1962) Éditions du Centre National de la Recherche Scientifique, Paris, 1963, pp. 87–89 (French). MR 0160856
- Stanislas Łojasiewicz, Sur la géométrie semi- et sous-analytique, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1575–1595 (French, with English and French summaries). MR 1275210
L1 S. Lojasiewicz, Ensemble semi-analytique. Bures-sur-Yvette: IHES (1965).
LM J. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems, Springer, Berlin, 1973.
- Hiroshi Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ. 18 (1978), no. 2, 221–227. MR 501842, DOI https://doi.org/10.1215/kjm/1250522572
- L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. With a chapter by E. Zehnder; Notes by R. A. Artino; Lecture Notes, 1973–1974. MR 0488102
- Peter Poláčik and Krzysztof P. Rybakowski, Nonconvergent bounded trajectories in semilinear heat equations, J. Differential Equations 124 (1996), no. 2, 472–494. MR 1370152, DOI https://doi.org/10.1006/jdeq.1996.0020
- P. Poláčik and F. Simondon, Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, J. Differential Equations 186 (2002), no. 2, 586–610. MR 1942223, DOI https://doi.org/10.1016/S0022-0396%2802%2900014-1
- Piotr Rybka and Karl-Heinz Hoffmann, Convergence of solutions to Cahn-Hilliard equation, Comm. Partial Differential Equations 24 (1999), no. 5-6, 1055–1077. MR 1680877, DOI https://doi.org/10.1080/03605309908821458
- R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs, vol. 49, American Mathematical Society, Providence, RI, 1997. MR 1422252
- James C. Robinson, Infinite-dimensional dynamical systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. An introduction to dissipative parabolic PDEs and the theory of global attractors. MR 1881888
- Leon Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 (1983), no. 3, 525–571. MR 727703, DOI https://doi.org/10.2307/2006981
- Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. MR 953967
- G. F. Webb, Compactness of bounded trajectories of dynamical systems in infinite-dimensional spaces, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), no. 1-2, 19–33. MR 549869, DOI https://doi.org/10.1017/S0308210500016930
- Hao Wu and Songmu Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions, J. Differential Equations 204 (2004), no. 2, 511–531. MR 2085545, DOI https://doi.org/10.1016/j.jde.2004.05.004
Ze T.I. Zelenyak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differentsial’nye Uravneniya, (1968), 17–22.
Zhang Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Comm. Pure Appl. Anal. 4, (2005), 683–693.
- Songmu Zheng, Nonlinear evolution equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 133, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2088362
ZC Songmu Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptotic Analysis, 45 (3,4) (2005), 301–312.
AFIS. Aizicovici, E. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Mathematical Methods in the Applied Sciences, 24, (2001), 277-287.
BV92 A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.
Ball J. Ball, Global attractors for damped semilinear wave equations, Discrete and Continuous Dynamical Systems, Vol. 10, No. 1&2(2004), 31-52.
BV V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, 1976.
BH H. Brézis, Opérateurs Maximaux Monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973.
CH R. Chill, On the Lojasiewicz-Simon gradient inequality, J. Funct. Anal., Vol. 201(2003), 572-601.
CELI. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. PDE. Vol. 27(2002), 1901-1951.
F95 E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, Journal of Differential Equations, Vol. 116 (1995), 431-447.
FIP E. Feireisl, F. Issard-Roch, and H. Petzeltova, Long-time behaviour and convergence towards equilibria for a conserved phase field model, Discrete and Continuous Dynamical Systems, Vol. 10, No. 1&2(2004), 239-252.
GP2 M. Grasselli, V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure Appl. Anal. 3(2004), 849–881.
Hale J.K. Hale, Asymptotic Behavior of Dissipative Systems, AMS Math. Surveys and Monographs, 25, Providence, Rhode Island, 1988.
H87 A. Haraux, Semilinear hyperbolic problems in bounded domains, Mathematical Reports, Vol. 3, Harwood Gordon Breach, New York, 1987.
HM A. Haraux and M.A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. vol. 9 (1999), 95-124.
HT01 S.Z. Huang and P. Tak$\acute {a}\breve {c}$, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Analysis, 46 (2001), 675–698.
J981 M.A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187–202.
J982 M.A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, Journal of Differential Equations, 144 (1998), 302–312.
LI I. Lasiecka, Mathematical Control Theory of Coupled PDE’s, CBMS-NFS Lecture Notes. SIAM, Philadelphia, 2002.
LI93 I. Lasiecka, D. Tataru, Uniform boundary stabilization of a semilinear wave equation with nonlinear boundary damping, Differential and Integral Equations, vol. 6 (1993), 507-533.
Li W. Li, Long-time convergence of solution to phase-field system with Neumann boundary conditions, to appear in Chinese Annals of Mathematics A.
L2 S. Lojasiewicz, Une propri$\acute {e}$t$\acute {e}$ topologique des sous-ensembles analytiques réels. Colloques Internationaux du C.N.R.S. #117, (1963), 87–89.
L3 S. Lojasiewicz, Sur la géométrie semi- et sous-analytique . Ann. Inst. Fourier (Grenoble) 43, (1963), 1575–1595.
L1 S. Lojasiewicz, Ensemble semi-analytique. Bures-sur-Yvette: IHES (1965).
LM J. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems, Springer, Berlin, 1973.
M78 H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ. 18-2 (1978), 221–227.
Ni L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Science, New York, 1974.
PR96 P. Polac̆ik and K.P. Rybakowski, Nonconvergent bounded trajectories in semilinear heat equations, J. Diff. Eqs., 124 (1996), 472–494.
PS P. Polac̆ik and F. Simondon, Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, J. Diff. Eqs., 186 (2002), 586–610.
RH P. Rybka and K.H. Hoffmann, Convergence of solutions to Cahn-Hilliard equation, Comm. PDEs, 24 (5&6), (1999), 1055–1077.
SR R. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, AMS, Providence, 1997.
SY01 G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002.
S83 L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math., 118 (1983), 525–571.
T R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., 68, Springer-Verlag, New York, 1988.
Webb G.F. Webb, Compactness of bounded trajectories of dynamical systems in infinite-dimensional spaces, Proc. Roy. Soc. Edinburgh, 84 A(1979), 19-34.
WZ Hao Wu and Songmu Zheng, Convergence to equibrium for the Cahn-Hilliard equation with dynamic boundary conditions, Journal of Differential Equations, Vol. 204, (2004), 511-531.
Ze T.I. Zelenyak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differentsial’nye Uravneniya, (1968), 17–22.
Zhang Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Comm. Pure Appl. Anal. 4, (2005), 683–693.
Z1 Songmu Zheng, Nonlinear Evolution Equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, 133, Chapman & Hall/CRC, Boca Raton, Florida, 2004.
ZC Songmu Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptotic Analysis, 45 (3,4) (2005), 301–312.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2000):
35B40,
35Q99
Retrieve articles in all journals
with MSC (2000):
35B40,
35Q99
Additional Information
Hao Wu
Affiliation:
Institute of Mathematics, Fudan University, 200433 Shanghai, P.R. China
Email:
haowufd@yahoo.com
Songmu Zheng
Affiliation:
Institute of Mathematics, Fudan University, 200433 Shanghai, P.R. China
Email:
songmuzheng@yahoo.com
Keywords:
Semilinear wave equation,
critical growth exponent,
dissipative boundary condition,
Simon-Lojasiewciz inequality
Received by editor(s):
July 18, 2005
Published electronically:
January 24, 2006
Additional Notes:
The authors are supported by the NSF of China under grant No. 10371022 and by the Ministry of Education in China under grant No. 20050246002, and by Key Laboratory of Mathematics for Nonlinear Sciences in Fudan University sponsored by the Ministry of Education in China.
Article copyright:
© Copyright 2006
Brown University