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International Journal of Mathematics and Mathematical Sciences
Volume 13, Issue 1, Pages 145-150

Time—periodic weak solutions

Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, Rio de Janeiro, Brazil

Received 30 December 1987

Copyright © 1990 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 UW, respectively, of H; δ>0, α0, β0 are constants and α+β>0; G is the Gateaux derivative of a convex functional J:VH[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.

Application is given to nonlinear initial-boundary value problems in a bounded domain of Rn.