6533b823fe1ef96bd127f7d4

RESEARCH PRODUCT

An implicit non-linear time dependent equation has a solution

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has a solution (u, u, w). The operators &s(l) and a(t) are maximal monotone from a real Hilbert space V to its dual such that &(r) + 9?(r) are V-coercive and a(r) are not degenerate. A linear compact injection i embeds V to a real Banach space W and each d(r) is the strongly monotone subdifferential of a continuous convex function #(I, ) on W. The function f is square integrable. The functions W(r): V+ W* are Lipschitzian as V*-valued functions. Section 3 contains the theorems. The main result is Theorem 2. Theorems 3 and 4 demonstrate the smoothing effect on the initial condition. Their proofs are given in Section 4. They exploit the methods of di Benedetto and Showalter, [4], who studied this equation in the case where .d and a do not depend on time and V 0, but .d can be degenerate. Grange and Mignot studied a similar autonomous equation without V [7]. Saguez introduced the operator W [ 111, exploiting their methods, by which he later proved together with Bermudez and Durany the solvability if 9J does not depend on time [ 131. In Section 6 there is an example. More examples, in the time independent case, as remarks and counterexamples on the uniqueness of the solution, can be found in [4].