Abstract
In this paper, we survey our recent results [10, 11] with Motohiro Sobajima, which give blow-up results and upper estimates of lifespan for certain small data to semilinear wave equations with the scaling invariant space-dependent or time-dependent damping:
\equation{\tagDW$_{V_0}$} \cases{
\partial_t^2 u - \Delta u + \dfrac{V_0}{|x|} \partial_t u = |u|^{p_V_0}}, \quad (t, x) \in (0, T) \times \mathbb{R}^N,
(u, \partial_tu)|_{t=0} = \varepsilon (u_0, u_1), \quad x \in \mathbb{R}^N,
\equation{\tagDW$_{\mu}$} \cases{
\partial_t^2 v - \Delta v + \dfrac{\mu}{1 + t} \partial_t v = |v|^{p_\mu}}, \quad (t, x) \in (0,T) \times \mathbb{R}^N,
(v, \partial_tv)|_{t=0} = \varepsilon (v_0, v_1), \quad x \in \mathbb{R}^N,
where $N \in \mathbb{N}$, $V_0 \in [0,(N - 1)^2/(N + 1))$, $p_{V_0} \in (N/(N - 1), p_S(N + V_0)]$, $\mu \in [0, (N^2 + N + 2)/(N + 2))$, $p_{\mu} \in (p_F(N), p_S(N + \mu)]$. Here $p_S$ is the Strauss exponent and $p_F$ is the Fujita exponent. And $(u_0, u_1)$ and $(v_0, v_1)$ are pairs of given functions, whose support is compact, and $\varepsilon \gt 0$ is a small parameter.
Information
Digital Object Identifier: 10.2969/aspm/08510163
