New examples of damped wave equations with gradient like structure
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New examples of damped wave equations with gradient-like structure Romain JOLY Institut Fourier UMR 5582, Universite Joseph Fourier, CNRS 100, rue des Maths, BP74 F-38402 St Martin d'Heres, FRANCE Abstract : This article shows how to use perturbation methods to get new examples of evolutionary partial differential equations with gradient-like structure. In particular, we investigate the case of the wave equation with a variable damping satisfying the geometric control condition only, and the case of the wave equation with a damping of indefinite sign. Keywords : gradient structure, gradient-like systems, perturbation methods, damped wave equation, indefinite damping AMS Codes (2000) : 35B40, 35B41, 37L05, 37L45 1 Introduction A large class of physical problems lead to dissipative systems, that is physical systems which admit an energy decreasing along the trajectories and the trajectories of which asymptotically tend to equilibria. Such particular systems have been called gradient sys- tems or gradient-like systems (see Definition 2.1 below). The gradient structure plays an important role in the qualitative study of the dynamics of an equation, since, for example, 1
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New
examples of damped wave equations with gradient-like structure
Romain JOLY Institut Fourier UMR5582,UniversiteJosephFourier,CNRS 100, rue des Maths, BP74 F-38402StMartind’Heres,FRANCE Romain.Joly@ujf-grenoble.fr
Abstract :This article shows how to use perturbation methods to get new examples of evolutionarypartialdi erentialequationswithgradient-likestructure.Inparticular,we investigate the case of the wave equation with a variable damping satisfying the geometric controlconditiononly,andthecaseofthewaveequationwithadampingofindenitesign.
Keywords : gradient structure, gradient-like systems, perturbation methods, dampedwaveequation,indenitedamping
AMS Codes (2000) : 35B40, 35B41, 37L05, 37L45
1 Introduction
A large class of physical problems lead to dissipative systems, that is physical systems which admit an energy decreasing along the trajectories and the trajectories of which asymptotically tend to equilibria. Such particular systems have been called gradient sys-temsorgradient-likesystems(seeDenition2.1below).Thegradientstructureplaysan important role in the qualitative study of the dynamics of an equation, since, for example,
1
a gradient system does not contain any periodic orbit or homoclinic orbit. Here, we study the damped wave equation on a bounded regular domain
Rd(d= 1,2 or 3): utt(x, t) +(x)ut(x, t) = u(x, t) +f(x, u(x, t)) (x, t)∈R+ u(x, t () = 0x, t)∈∂R+(1.1) (u(x,0), ut(x,0))∈H01(
)L2(
) Conditions on the support of the dissipation(x)0 are known to imply the gradient-like structure for Equation (1.1), see [22] and [16], and also [17] for Neumann boundary conditions. The purpose of this paper is to enhance technics which show that the gradient-like structure is, in some sense, a property which is stable under small perturbations. We brie
y illustrate these technics with two examples : - In Section 2, we prove that, if the support of(x) satises the geometric control con-dition introduced by C.Bardos, G.Lebeau and J.Rauch in [3], Equation (1.1) generates a gradient-like dynamical system for a generic non-linearityf(x, u) (orf(u)). - In Section 3, we study the case where the damping(x) of (1.1) can be slightly negative on some part of
. Notice that, in this case, no explicit Lyapounov functional is known. However, we can prove the existence of a compact global attractor and exhibit a gradient-like structure for most of the cases.
Remarks : The damped wave equations are models for the propagation of waves in dissipative media. More generally, they are used to model propagation or invasion phenomena. For example, they arise in biology when studying the evolution of species populations (see [8] and [20]). Iurattoalsvtiynertahtgeoexeeptchtntrolconmetriccoussi]3[fonoitidethortfenci damped wave equation to be a gradient-like system. Indeed, the fact that the trajectory of any wave intersects the support of the damping should imply that any solution relaxes to an equilibrium. Theorem 2.3 is a slight progress in this direction, but the full result, the gradient-likestructurewithoutanyconditiononthenonlinearity,isstilladicultopen problem. Classically, the damping(x) of the wave equation is non-negative. Section 3, the In damping has an indenite sign. This can be explained as follows : the positive part of (x) is modelling a damping phenomena whereas the negative part is modelling a supply of energy given to the system. Therefore, the indenite damping is a basic model to study howasmalllocalizedsupplyofenergymodiesthedissipativestructureofasystem.To give a concrete example, in the biological model introduced in [20], a large birth rate of the species compared to the speed of di usion may be seen as a negative damping. efseg(inmpdateniroMaevahdutsapynsreprplateeqiedwaveotiihdneauitnows example[6],[7]or[18]).However,toourknowledge,itisthersttimethatthenonlinear equation is considered and that nonlinear properties as existence of a global attractor or gradient structure are obtained.
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