From p₀(n) to p₀(n+2)

Original · EN

In this note we study the global existence of small data solutions to the Cauchy problem for the semi-linear wave equation with a not effective scale-invariant damping term, namely vtt- v + 21+tvₜ = |v|ᵖ, v(0,x)=v₀(x), vₜ(0,x)=v₁(x), where p>1, n≥ 2. We prove blow-up in finite time in the subcritical range p∈(1,p₂(n)] and an existence result for p>p₂(n), n=2,3. In this way we find the critical exponent for small data solutions to this problem. All these considerations lead to the conjecture p₂(n)=p₀(n+2) for n≥2, where p₀(n) is the Strauss exponent for the classical wave equation.