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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.

References [Enhancements On Off] (What's this?)

  • [1] M. H. Protter, Maximum principles, Maximum principles and eigenvalue problems in partial differential equations (Knoxville, TN, 1987) Pitman Res. Notes Math. Ser., vol. 175, Longman Sci. Tech., Harlow, 1988, pp. 1–14. MR 963455
  • [2] M. Stecher and W. Rundell, Maximum principles for pseudoparabolic partial differential equations, J. Math. Anal. Appl. 57 (1977), no. 1, 110–118. MR 0440202

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DOI: https://doi.org/10.1090/S0002-9939-1979-0532145-9
Article copyright: © Copyright 1979 American Mathematical Society