Journal of the Mathematical Society of Japan

$L^{p}-L^{q}$ estimates for damped wave equations and their applications to semi-linear problem

Takashi NARAZAKI

Abstract

In this paper we study the Cauchy problem to the linear damped wave equation $u_{tt}-\Delta u+2au_{t}=0$ in $(0,\infty)$$\times R^{n}(n\geq 2)$. It has been asserted that the above equation has the diffusive structure as $t\rightarrow\infty$. We give the precise interpolation of the diffusive structure, which is shown by $L^{p}- L^{q}$ estimates. We apply the above $L^{p}- L^{q}$ estimates to the Cauchy problem for the semilinear damped wave equation $u_{tt}-\Delta u+$$2au_{t}=|u|^{\sigma}u$ in $(0,\infty)$$\times R^{n}(2\leq n\leq 5)$. If the power $\sigma$ is larger than the critical exponent $2/n$(Fujita critical exponent) and it satisfies $\sigma\leq 2/(n-2)$ when $n\geq 3$, then the time global existence of small solution is proved, and the decay estimates of several norms of the solution are derived.

Article information

Source
J. Math. Soc. Japan, Volume 56, Number 2 (2004), 585-626.

Dates
First available in Project Euclid: 3 October 2007

https://projecteuclid.org/euclid.jmsj/1191418647

Digital Object Identifier
doi:10.2969/jmsj/1191418647

Mathematical Reviews number (MathSciNet)
MR2048476

Zentralblatt MATH identifier
1059.35073

Subjects
Primary: 35L05: Wave equation
Secondary: 35B45: A priori estimates 35B40: Asymptotic behavior of solutions

Citation

NARAZAKI, Takashi. $L^{p}-L^{q}$ estimates for damped wave equations and their applications to semi-linear problem. J. Math. Soc. Japan 56 (2004), no. 2, 585--626. doi:10.2969/jmsj/1191418647. https://projecteuclid.org/euclid.jmsj/1191418647