Global existence for wave equations with scale-invariant time-dependent damping and time derivative nonlinearity

This paper addresses the Cauchy problem for wave equations with scale-invariant time-dependent damping and nonlinear time-derivative terms, modeled as {∂_t²u−Δu+μ1+t∂_tu=f(∂_tu),x∈Rⁿ,t>0,u(x,0)=u₀(x),∂_tu(x,0)=u₁(x)x∈Rⁿ, where f(∂_tu)=|∂tu|^p or |∂_tu|^p−1∂_tu with p > 1 and μ > 0. We prove global existence of small data solutions in low dimensions 1 ≤ n ≤ 3 by using energy estimates in appropriate Sobolev spaces. Our primary contribution is an existence result for p>1+2/μ, in the one-dimensional case, when μ ≤ 2, which in conjunction with prior blow-up results from [1], establish that the critical exponent for small data solutions in one dimension is p_G(1,μ)=1+2/μ, when μ ≤ 2. To the best of our knowledge, this is the first identification of the critical exponent range for the time-dependent damped wave equations with scale-invariant and time-derivative nonlinearity.