Global existence of small data weak solutions to the semilinear wave equations with time-dependent scale-invariant damping

In this paper, we are concerned with the global existence of small data weak solutions to the $n-$dimensional semilinear wave equation ∂ₜ²u-Δ u+μ/t∂ₜu=|u|ᵖ with time-dependent scale-invariant damping, where n≥ 2, t≥ 1, μ∈(0,1)∪(1,2] and $p>1$. This equation can be changed into the semilinear generalized Tricomi equation ∂ₜ²u-tᵐΔ u=tα(m)|u|ᵖ, where m=m(μ)>0 and α(m)∈ R are two suitable constants. At first, for the more general semilinear Tricomi equation ∂ₜ²v-tᵐΔ v=tα|v|ᵖ with any fixed constant $m>0$ and arbitrary parameter α∈ R, we shall show that in the case of α≤ -2, n≥ 3 and $p>1$, the small data weak solution v exists globally; in the case of α>-2, through determining the conformal exponent pconf(n,m,α)>1, the global small data weak solution v exists when some extra restrictions of p≥ pconf(n,m,α) are given. Returning to the original equation ∂ₜ²u-Δ u+μ/t∂ₜu=|u|ᵖ, the corresponding global existence results on the small data solution u can be obtained.