Stability of higher-level energy norms of strong solutions to a wave equation with localized nonlinear damping and a nonlinear source term

• Autorzy: Lasiecka I. , Toundykov D.
• Języki publikacji: EN

Abstrakty

EN

We derive global in time a priori bounds on higher-level energy norms of strong solutions to a semilinear wave equation: in particular, we prove that despite the influence of a nonlinear source, the evolution of a smooth initial state is globally bounded in the strong topology ∼ H2 x H1. And the bound is uniform with respect to the corresponding norm of the initial data. It is known that an m-accretive semigroup generator moriotoni-cally propagates smoothness of the initial condition; however, this result does not hold in general for Lipschitz perturbations of monotone systems where higher order Sobolev norms of the solution may blowup asymptotically as t → ∞. Due to nonlinearity of the system, the only a priori global-in-time bound that follows from classical methods is that on finite energy: ∼ H1 x L2. We show that under some correlation between growth rates of the damping and the source, the norms of topological order above the finite energy level remain globally bounded. Moreover, we also establish this result when damping exhibits sublinear or superlinear growth at the origin, or at infinity, which has immediate applications to asymptotic estimates on the decay rates of the finite energy. The approach presented in the paper is not specific to the wave equation, and can be extended to other hyperbolic systems: e.g. plate, Maxwell, and Schrodinger equations.

Słowa kluczowe

EN

wave equation; localized nonlinear damping; nonlinear source; stability; uniform boundedness; higher energy; strong solutions

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2007
• Tom: Vol. 36, no 3
• Strony: 681--710
• Opis fizyczny: Bibliogr. 20 poz.

Twórcy

autor

Lasiecka I.

autor

Toundykov D.

• Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA

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