On semilinear damped wave equations with initial data in homogeneous sobolev spaces

In this paper, we study semilinear damped equations $u_{tt}+u_t-Δu=|u|^p$ with the initial data in $({\dot{H}^{-γ}}\cap H^s)\times({\dot{H}^{-γ}}\cap L^2)$. Chen-Reissig (2023) studied the case $0<γ<\frac{n}{2}$ and showed that the exponent $p_{\mathrm{crit}}=1+\frac{4}{n+2γ}$ of $p$ distinguishes the time global existence and the blow-up of solution. In this paper, we discuss the case $γ\ge\frac{n}{2}$.

💡 Research Summary

In this paper the author investigates the semilinear damped wave equation
(u_{tt}+u_{t}-\Delta u = |u|^{p}) on (\mathbb{R}^{n}) with small amplitude initial data ((u_{0},u_{1}) = \varepsilon (u_{0},u_{1})), where the data belong to the intersection of a homogeneous Sobolev space of negative order and a standard Sobolev space, namely ((\dot H^{-\gamma}\cap H^{s})\times(\dot H^{-\gamma}\cap L^{2})). The parameter (\gamma>0) measures how far the data are from being square‑integrable; the case (\gamma\ge n/2) has not been covered in earlier work by Chen and Reissig (2023), which dealt with (0<\gamma< n/2).

The paper has two main objectives. First, it seeks a global‑existence result for the nonlinear problem when the exponent (p) is sufficiently large. Second, it aims to identify the range of (p) for which any weak solution must blow up in finite time, and to provide sharp upper bounds for the lifespan (T(\varepsilon)) as (\varepsilon\to0).

Linear theory and mild solutions.
The author writes down the fundamental solution (K(t,\xi)) of the linear damped wave operator in Fourier variables. Using this kernel, the mild solution is expressed as
(u(t)=K’(t)u_{0}+K(t)(u_{1}+u_{0})+\int_{0}^{t}K(t-\tau)*|u|^{p}(\tau),d\tau).
The explicit form of (K) shows a strong exponential decay for low frequencies and an oscillatory behavior for high frequencies, which is crucial for later time‑decay estimates.

Global existence (Theorem 1).
Assume (1\le n\le6) and (s\in(0,1]). Let (\gamma) satisfy
(\displaystyle \gamma\ge\min!\Bigl(\frac{n}{2},\sqrt{\frac{n^{2}}{16}+n}-\frac{n}{4}\Bigr)).
If the nonlinearity exponent fulfills
(\displaystyle p>\max!\Bigl(1+\frac{2}{n},;\sqrt{\frac{n^{2}}{16}+n}-\frac{n}{4}\Bigr))
and, when (n>2s), also (p\le\frac{n}{n-2s}), then for sufficiently small (\varepsilon) there exists a unique mild solution (u\in C(