## A Foias-Saut type of expansion for dissipative wave equations.(English)Zbl 0963.35123

The author considers the nonlinear wave equation $$u_{tt}-\Delta u+au_t+bu=f(u_t,u)$$ with the homogeneous Dirichlet boundary condition in a bounded domain, where $$a$$ is a positive constant and $$f(u_t,u)$$ is flat at $$(0,0).$$ It is proved that each small solution $$u(t)$$ to this problem admits a Foias-Saut type of expansion $$\sum_{\mu\in\Pi(L)}e^{-\mu t}w_{\mu}(t),$$ where $$\Pi(L)$$ is the additive semigroup generated by the spectrum of the infinitesimal generator $$L$$ of the linear operator semigroup associated to this problem, and for all $$w_\mu(t)$$ is a polynomial in $$t$$ whose coefficients are functions of $$x$$ in a Sobolev space, such for all $$\Lambda>0$$, $$u(t)-\sum_{\mu\in\Pi(L),R\mu}\leq\Lambda e^{-\mu t}w_\mu(t)= o(e^{-\Lambda t})$$ as $$t\to +\infty.$$ Moreover, if the spectrum of $$L$$ satisfies a nonresonance condition, then all functions $$\omega_\mu$$ are independent of $$t.$$

### MSC:

 35L70 Second-order nonlinear hyperbolic equations 35C10 Series solutions to PDEs 47D06 One-parameter semigroups and linear evolution equations 35L20 Initial-boundary value problems for second-order hyperbolic equations
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### References:

 [1] Adams, R.S. 1975. ”Sobolev spaces”. New York: Academic Press. · Zbl 0314.46030 [2] Arnold, V.I. ”Geometical method in the theory of ordinary differential equations”. [3] Bardos C, C.R.Acad. Sci 313 pp 757– (1991) [4] Bardos C, Asymptotic Anal 4 pp 285– (1991) [5] C. Bardos, G. Lebeau and J. Rauch, Contrôle et stabilisation dans les problèmes hyperboliques, Appendice I1 in: J.L. Lions [29] · Zbl 0673.93037 [6] DOI: 10.1137/0330055 · Zbl 0786.93009 [7] Biler P, Appl. Anal 55 pp 1– (1994) [8] DOI: 10.1137/0319008 · Zbl 0461.93036 [9] DOI: 10.1080/03605309408821015 · Zbl 0818.35072 [10] DOI: 10.1512/iumj.1995.44.2001 · Zbl 0847.35078 [11] Debussche A, Differential and Inegral Equations 4 pp 897– (1991) [12] Foias C, C. R. Acad. Sci 295 pp 325– (1982) [13] DOI: 10.1512/iumj.1984.33.33025 · Zbl 0565.35087 [14] DOI: 10.1512/iumj.1984.33.33049 · Zbl 0572.35081 [15] Foias C, Ann. Inst. Henri Poincaré 4 pp 1– (1987) [16] DOI: 10.1512/iumj.1991.40.40015 · Zbl 0739.35066 [17] DOI: 10.1016/0022-0396(86)90121-X · Zbl 0549.35102 [18] DOI: 10.1080/00036819108840034 · Zbl 0724.35015 [19] I.C. Gohberg and M.G. Krejn, Introduction to the theory of linear nonselfadjoint operators, American Mathematical Society, 1969 [20] DOI: 10.1007/BF00281421 · Zbl 0187.05901 [21] DOI: 10.1007/BF02099268 · Zbl 0763.35058 [22] L. Hsiao, Nonlinear diffusive phenomena of solutions for quasilinear hyperbolic systems, in Tatsien Li, M. Mimura et al:[28] · Zbl 0960.35061 [23] DOI: 10.1137/0321004 · Zbl 0512.93014 [24] Lebeau G, Math. Phys. Stud 19 pp 73– (1996) [25] DOI: 10.1215/S0012-7094-97-08614-2 · Zbl 0884.58093 [26] Li Tatsien, RMA 32 (1994) [27] Tatsien Li, Nonlinear heat conduction with finite speed of propogation, in Tatsien Li, M. Mimura et al:[28] · Zbl 0959.35088 [28] Li Tatsien, World Scientific [29] Lions J.L, perturbations et stabilisation des systèmes distribués 8 (1988) [30] Lions, J.L and Magenes, E. 1972. ”Non-homogeneous boundary value problems and applications”. Vol. 1, New York: Springer-Verlag. · Zbl 0223.35039 [31] DOI: 10.1023/A:1022696614020 · Zbl 0970.34045 [32] Phung Kim-Dang, C. R. Acad. Sci 323 pp 169– (1996) [33] Phung Kim-Dang, C. R. Acad. Sci 320 pp 187– (1995) [34] Temam, R. 1988. ”Infinite-dimentional dynamical systems in mechanics and physics”. New York: Springer-Verlag. · Zbl 0662.35001 [35] Han Yang and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves in low space dimensions, to appear · Zbl 0959.35022
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