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

Abstract

For the $2$-D semilinear wave equation with scale-invariant damping $\square u+\fracμ{t}\partial_tu=|u|^p$, where $t\geq 1$, $μ>0$ and $p>1$, it is conjectured that the global small data weak solution $u$ exists when $p>p_{s}(2+μ) =\frac{μ+3+\sqrt{μ^2+14μ+17}}{2(μ+1)}$ for $0<μ\leq 2$ and $p>p_f(2)=2$ for $μ\geq 2$. In our previous papers, the global small solution $u$ has been obtained for $p>p_{s}(2+μ)$ and $0<μ<2$ but $μ\not=1$. In the present paper, by the vector field method together with the delicate analysis on the Bessel functions, we will show the global existence of small solution $u$ for $p>2$ and $μ>2$. In forthcoming paper, for $μ=1$ and $p>p_{s}(2+μ)=p_{s}(3)=1+\sqrt 2$, the global solution $u$ is also obtained. Therefore, collecting our series of conclusions together with partial results from others, this open question has been solved completely.