The nonpositivity of solutions to pseudoparabolic equations
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Conditions are given on the nonnegative data (x) and g(x, t) such that solutions of the pseudoparabolic inequality P[u] = (L - I)ut + Lu 0 in Dx (0, t) u(x, 0) = (x), x D, u(x, t)=g(x, t), x D g(x, t), satisfy u(x, t) 0 in D (0, t). Here D is an open set in Rn and L is a second order elliptic differential operator. A counterexample is provided to show that this condition is in a sense necessary. The result implies that solutions P[u] = 0 do not in general satisfy a maximum principle. 1979 American Mathematical Society.