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Lower bounds for solutions of hyperbolic inequalities in unbounded regions


Author: Amy C. Murray
Journal: Proc. Amer. Math. Soc. 38 (1973), 127-134
MSC: Primary 35B45; Secondary 35L10
DOI: https://doi.org/10.1090/S0002-9939-1973-0312048-X
MathSciNet review: 0312048
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Abstract: This paper considers $ {C^2}$ solutions $ u = u(t,x)$ of the differential inequality $ \vert Lu\vert \leqq {k_1}(t,x)\vert u\vert + {k_2}(t,x)\vert\vert\nabla u\vert\vert$. The coefficients of the hyperbolic operator $ L$ depend on both $ t$ and $ x$. Explicit lower bounds are given for the energy of $ u$ in a region of $ x$-space expanding at least as fast as wave-fronts for $ L$. These bounds depend on the asymptotic behavior of $ {k_1},{k_2}$, and the coefficients of $ L$. They do not require boundary conditions on $ u$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0312048-X
Keywords: Hyperbolic inequalities, maximal rates of decay, a priori inequalities
Article copyright: © Copyright 1973 American Mathematical Society

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