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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The nonpositivity of solutions to pseudoparabolic equations

Authors: William Rundell and Michael Stecher
Journal: Proc. Amer. Math. Soc. 75 (1979), 251-254
MSC: Primary 35K25
MathSciNet review: 532145
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Abstract: Conditions are given on the nonnegative data $ \phi (x)$ and $ g(x,t)$ such that solutions of the pseudoparabolic inequality $ P[u] = (L - I){u_t} + Lu \leqslant 0$ in $ Dx(0,\tau )$

\begin{displaymath}\begin{array}{*{20}{c}} {u(x,0) = \phi (x),} \hfill & {x \in ... ...),} \hfill & {x \in D \times (0,\tau ),} \hfill \\ \end{array} \end{displaymath}

satisfy $ u(x,t) \geqslant 0$ in $ D \times (0,\tau )$. Here D is an open set in $ {{\mathbf{R}}^n}$ 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.

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Article copyright: © Copyright 1979 American Mathematical Society

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