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- u+tₜu=|u|ᵖ with time-dependent scale-invariant damping, where n 2, t 1, (0, 1) (1, 2] and p>1. This equation can be changed into the semilinear generalized Tricomi equation ₜ²u-tᵐ u=t^ (m) |u|ᵖ, where m=m () >0 and (m) R are two suitable constants. At first, for the more general semilinear Tricomi equation ₜ²v-tᵐ v=t^|v|ᵖ with any fixed constant m>0 and arbitrary parameter R, we shall show that in the case of -2, n 3 and p>1, the small data weak solution v exists globally; in the case of >-2, through determining the conformal exponent p₂₎₍₅ (n, m, ) >1, the global small data weak solution v exists when some extra restrictions of p p₂₎₍₅ (n, m, ) are given. Returning to the original equation ₜ²u- u+tₜu=|u|ᵖ, the corresponding global existence results on the small data solution u can be obtained.