Nonlinear equations

Abstract

We consider a Cauchy problem in a Hilbert space H for the equation given A, linear operator; G, Gateaux derivative of a functional; real β, α>0,δgt;0, constants. We prove that if the equilibrium positions are isolated and if a(u(t))=(Au(t),u(t)) remains bounded for t≥0, then u(t) tends, as t goes to infinity, to an equilibrium position, in the energy norm. The proof relies on properties of the weakly continuous dynamical system defined by the equation. This technique was devejopped by Ball [1] , using the methods of Lions [4]. Application is given. In particular, a generalization of the damped extensible beam equation is considered, if G(u)= M(a(u))Au, for M a real function. Then it is quite simple to see that a(u(t)) remains bounded and, if all eigenvalues of αA²−λAu=0 are simple, the set of equilibrium positions is finite. For other applications, we refer to methods in Chow and Hale [3].

AMS (MOS):

• 35L20
• 35B40