Double-periodic boundary value problem for non-linear dissipative hyperbolic equations

Let Z and R be the set of all integers and real numbers, respectively, and let Q = [0,27c] x [0, 2711. Let L’(Q) be the space of measurable real-valued functions U: Sz + R which are Lebesgue integrable over 52 with usual norm II.jI L1. Let L2(Q) be the space of measurable real-valued functions U: Sz + R which are Lebesgue square integrable over Q with usual inner product ( , ) and usual norm I/ . ljL2 and let L”(Q) be the space of measurable real-valued functions U: Q + R which are essentially bounded with the norm 1) u 11 Lm = ess sup 1 u( t, x) I . (I. x) E R Let C“(Q) be the space of all continuous functions u.: 52 --f R such that the partial derivatives up to order k with respect to both variables are continuous on 52, while C(Q) is used for Co(Q) with the usual norm 11. Iloo.