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April, 2004 Lp-Lq estimates for damped wave equations and their applications to semi-linear problem
Takashi NARAZAKI
J. Math. Soc. Japan 56(2): 585-626 (April, 2004). DOI: 10.2969/jmsj/1191418647

Abstract

In this paper we study the Cauchy problem to the linear damped wave equation utt-Δu+2aut=0 in (0,)×Rn(n2). It has been asserted that the above equation has the diffusive structure as t. We give the precise interpolation of the diffusive structure, which is shown by Lp-Lq estimates. We apply the above Lp-Lq estimates to the Cauchy problem for the semilinear damped wave equation utt-Δu+2aut=|u|σu in (0,)×Rn(2n5). If the power σ is larger than the critical exponent 2/n(Fujita critical exponent) and it satisfies σ2/(n-2) when n3, then the time global existence of small solution is proved, and the decay estimates of several norms of the solution are derived.

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Takashi NARAZAKI. "Lp-Lq estimates for damped wave equations and their applications to semi-linear problem." J. Math. Soc. Japan 56 (2) 585 - 626, April, 2004. https://doi.org/10.2969/jmsj/1191418647

Information

Published: April, 2004
First available in Project Euclid: 3 October 2007

zbMATH: 1059.35073
MathSciNet: MR2048476
Digital Object Identifier: 10.2969/jmsj/1191418647

Subjects:
Primary: 35L05
Secondary: 35B40 , 35B45

Keywords: $L^{p}-L^{q}$ estimate , damped wave equation , diffusive structure , time global existence

Rights: Copyright © 2004 Mathematical Society of Japan

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Vol.56 • No. 2 • April, 2004
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