×

On the convergence to stationary solutions for a semilinear wave equation with an acoustic boundary condition. (English) Zbl 1219.35143

The paper deals with solutions of the equation \[ u_{tt}+\omega u_{t}-\Delta u+u+f(u)=0 \text{ in }\Omega \times (0,+\infty ) \] with the boundary conditions \[ \delta _{tt}+\nu \delta _{t}+\delta =-u_{t}, \;\delta _{t}=\frac{\partial u}{\partial n}\text{ on }\partial \Omega \times (0,+\infty ). \] The problem is inspired by a model for acoustic wave motion of a fluid in a domain with locally reacting boundary surface, originally proposed by J.T. Beale and S. I. Rosencrans in [Bull. Am. Math. Soc. 80, 1276–1278 (1974; Zbl 0294.35045)]. Under some restrictions on \(f(u)\) the author proves that there exists a function \(u_{\infty }\) of the set of equilibria such that the solution \(u\) of the above problem starting from the given initial data converges to \(u_{\infty }\) as \(t\rightarrow +\infty .\) The result is obtained by use of Haraux’s and Jendoubi’s argument and the Simon-Łojasiewicz inequality.

MSC:

35L71 Second-order semilinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations

Citations:

Zbl 0294.35045
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beale, J. T. and Rosencrans, S. I., Acoustic boundary conditions. Bull. Amer. Math. Soc. 80 (1974), 1276 - 1278. · Zbl 0294.35045
[2] Beale, J. T., Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J. 25 (1976), 895 - 917. · Zbl 0325.35060
[3] Beale, J. T., Acoustic scattering from locally reacting surfaces. Indiana Univ. Math. J. 26 (1977), 199 - 222. · Zbl 0332.35053
[4] Casarino, V., Engel, K. J., Nagel, R. and Nickel, G., A semigroup approach to boundary feedback systems. Integral Equ. Oper. Theory 47 (2003), 289 - 306. · Zbl 1048.47054
[5] Casarino, V., Engel, K. J., Nickel, G. and Piazzera, S., Decoupling techniques for wave equations with dynamic boundary conditions. Discrete Contin. Dyn. Syst., Ser. B, 12 (2005), 761 - 772. · Zbl 1082.34048
[6] Frigeri, S., Convergence towards equilibria for a hyperbolic system arising in ferroelectricity. Adv. Math. Sci. Appl. 18 (2008), 169 - 184. · Zbl 1172.35328
[7] Frigeri, S., Attractors for semilinear damped wave equations with an acoustic boundary condition. J. Evol. Equ. 10 (2010), 29 - 58. · Zbl 1239.35025
[8] Frota, C. L. and Goldstein, J. A., Some nonlinear wave equations with acoustic boundary conditions. J. Diff. Equ. 164 (2000), 92 - 109. · Zbl 0979.35105
[9] Gal, C. G., Goldstein, G. R. and Goldstein, J. A., Oscillatory boundary con- ditions for acoustic wave equations. J. Evol. Equ. 3 (2003), 623 - 636. · Zbl 1058.35139
[10] Gal, C. G. and Grasselli, M., On the asymptotic behavior of the Caginalp sys- tem with dynamic boundary conditions. Commun. Pure Appl. Anal. 8 (2009), 689 - 710. 191 · Zbl 1171.35337
[11] Haraux, A. and Jendoubi, M. A., Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity. Calc. Var. Part. Diff. Equ. 9 (1999), 95 - 124. · Zbl 0939.35122
[12] Morse, P. M. and Ingard, K. U., Theoretical Acoustics. New York: McGraw-Hill 1968.
[13] Mugnolo, D., Abstract wave equations with acoustic boundary conditions. Math. Nachr. 279 (2006), 299 - 318. · Zbl 1109.47035
[14] Pata, V. and Zelik, S., A remark on the damped wave equation. Commun. Pure Appl. Anal. 5 (2006), 609 - 614. · Zbl 1140.35533
[15] Polá\check cik, P. and Rybakowski, K. P., Nonconvergent bounded trajectories in semilinear heat equations. J. Diff. Equ. 124 (1996), 472 - 494. · Zbl 0845.35054
[16] Polá\check cik, P. and Simondon, F., Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains. J. Diff. Equ. 186 (2002), 586 - 610. · Zbl 1024.35046
[17] Wu, H. and Zheng, S., Convergence to equilibrium for the Cahn-Hilliard equa- tion with dynamic boundary conditions. J. Diff. Equ. 204 (2004), 511 - 531. · Zbl 1068.35018
[18] Yeoul, P. J. and Ae, K. J., Some nonlinear wave equations with nonlinear memory source term and acoustic boundary conditions. Numer. Funct. Anal. Optim. 27 (2006), 889 - 903. · Zbl 1106.76062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.