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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.