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THE INITIAL VALUE PROBLEM
ASYMPTOTICS IN SOME
6 other sections not shown
a.e. in Q addition almost-periodic solution applied asymptotic attractor Au(t Banach space bounded domain bounded operators bounded set Cauchy sequence Chapter consequence of Proposition consequence of Theorem continuous imbedding convergence Corollary deduce defined denoted dense differential dx dt establish Eu(t evolution equations example exists a unique Fatou's lemma following result follows easily follows immediately formula fulfilled g satisfies hence Hilbert space hyperbolic problems hypotheses of Theorem immediate consequence implies inequality initial data initial value problem inner product integrating isometry Let Q Liapunov function lim sup linear Lipschitz continuous non-decreasing non-linear norm obtain oscillation properties particular periodic solutions precompact Proof Let proof of Proposition proof of Theorem Proposition III.2.2.1 purely dissipative real Hilbert space real numbers Remark replaced semi-group sequence Sobolev spaces solution of V.0.1 Step theory trajectories unique solution valid vector VreR VteR VueK wave equation weak solutions