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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Asymptotic behavior of solutions of hyperbolic inequalities


Author: Amy C. Murray
Journal: Trans. Amer. Math. Soc. 157 (1971), 279-296
MSC: Primary 35.19
MathSciNet review: 0274922
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Abstract: This paper discusses the asymptotic behavior of $ {C^2}$ solutions $ u = u(t,{x_1}, \ldots ,{x_v})$ of the inequality (1) $ \vert Lu\vert \leqq {k_1}(t,x)\vert u\vert + {k_2}(t,x)\vert\vert{\nabla _u}\vert\vert$, in domains in $ (t,x)$-space which grow unbounded in $ x$ as $ t \to \infty $. The operator $ L$ is a second order hyperbolic operator with variable coefficients. The main results establish the maximum rate of decay of nonzero solutions of (1). This rate depends on the asymptotic behavior of $ {k_1},{k_2}$, and the time derivatives of the coefficients of $ L$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0274922-5
PII: S 0002-9947(1971)0274922-5
Keywords: Hyperbolic partial differential inequality, maximum decay rate, a priori inequalities
Article copyright: © Copyright 1971 American Mathematical Society