## Differential and Integral Equations

- Differential Integral Equations
- Volume 32, Number 1/2 (2019), 1-36.

### The sharp estimate of the lifespan for semilinear wave equation with time-dependent damping

Masahiro Ikeda and Takahisa Inui

#### Abstract

We consider the following semilinear wave equation with time-dependent damping. \begin{align*} \left\{ \begin{array}{ll} \partial_t^2 u - \Delta u + b(t)\partial_t u = |u|^{p}, & (t,x) \in [0,T) \times \mathbb R^n, \\ u(0,x)=\varepsilon u_0(x), u_t(0,x)=\varepsilon u_1(x), & x \in \mathbb R^n, \end{array} \right. \end{align*} where $n \in \mathbb N$, $p > 1$, $\varepsilon>0$, and $b(t) \approx (t+1)^{-\beta}$ with $\beta \in [-1,1)$. It is known that small data blow-up occurs when $1 < p < p_F$ and, on the other hand, small data global existence holds when $p > p_F$, where $p_F:=1+2/n$ is the Fujita exponent. The sharp estimate of the lifespan was well studied when $1 < p < p_F$. In the critical case $p=p_F$, the lower estimate of the lifespan was also investigated. Recently, Lai and Zhou [15] obtained the sharp upper estimate of the lifespan when $p=p_F$ and $b(t)=1$. In the present paper, we give the sharp upper estimate of the lifespan when $p=p_F$ and $b(t) \approx (t+1)^{-\beta}$ with $\beta \in [-1,1)$ by the Lai--Zhou method.

#### Article information

**Source**

Differential Integral Equations, Volume 32, Number 1/2 (2019), 1-36.

**Dates**

First available in Project Euclid: 11 December 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1544497284

**Mathematical Reviews number (MathSciNet)**

MR3909977

**Zentralblatt MATH identifier**

07031707

**Subjects**

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 37D10: Invariant manifold theory 37K40: Soliton theory, asymptotic behavior of solutions 37K45: Stability problems

#### Citation

Ikeda, Masahiro; Inui, Takahisa. The sharp estimate of the lifespan for semilinear wave equation with time-dependent damping. Differential Integral Equations 32 (2019), no. 1/2, 1--36. https://projecteuclid.org/euclid.die/1544497284