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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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