## Journal of the Mathematical Society of Japan

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

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

#### Abstract

In this paper we study the Cauchy problem to the linear damped wave equation ${u}_{tt}-\Delta u+2a{u}_{t}=0$ in $(0,\infty )$$\times {R}^{n}(n\ge 2)$. It has been asserted that the above equation has the diffusive structure as $t\to \infty $. We give the precise interpolation of the diffusive structure, which is shown by ${L}^{p}\text{-}{L}^{q}$ estimates. We apply the above ${L}^{p}\text{-}{L}^{q}$ estimates to the Cauchy problem for the semilinear damped wave equation ${u}_{tt}-\Delta u+$$2a{u}_{t}=|u{|}^{\sigma}u$ in $(0,\infty )$$\times {R}^{n}(2\le n\le 5)$. If the power $\sigma $ is larger than the critical exponent $2/n$(Fujita critical exponent) and it satisfies $\sigma \le 2/(n-2)$ when $n\ge 3$, 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

**Permanent link to this document**

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

**Keywords**

Damped wave equation $L^{p}-L^{q}$ estimate diffusive structure time global existence

#### 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