Time—periodic weak solutions

Abstract

In continuing from previous papers, where we studied the existence and uniqueness of the global solution and its asymptotic behavior as time t goes to infinity, we now search for a time-periodic weak solution u(t) for the equation whose weak formulation in a Hilbert space H isddt(u′,v)+δ(u′,v)+αb(u,v)+βa(u,v)+(G(u),v)=(h,v)where: ′=d/dt; (′) is the inner product in H; b(u,v), a(u,v) are given forms on subspaces U⊂W, respectively, of H; δ>0, α≥0, β≥0 are constants and α+β>0; G is the Gateaux derivative of a convex functional J:V⊂H→[0,∞) for V=U, when α>0 and V=W when α=0, hence β>0; v is a test function in V; h is a given function of t with values in H.

Keywords

• periodic weak solution
• Gateaux derivative.