Time—periodic weak solutions - Eliana Henriques de Brito (International Journal of Mathematics and Mathematical Sciences)
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.