Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, I

Abstract

There is an interesting open question: for the $n$-D ($n\ge 1$) semilinear wave equation with scale-invariant damping $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $t\ge 1$, $p>1$ and $μ>0$, the global small data weak solution $u$ will exist when $p>p_{crit}(n,μ)=\max\{p_s(n+μ), p_f(n)\}$ with $p_{s}(n+μ)=\frac{n+μ+1+\sqrt{(n+μ)^2+10(n+μ)-7}}{2(n+μ-1)}$ and $p_f(n)=1+\frac{2}{n}$. It is noticed that the weak solution $u$ can blow up in finite time when $1 0$). In forthcoming papers, we shall show the global existence of small solution $u$ for the remaining cases of $p>1$ and $μ>0$.