The critical power of short pulse initial data on the global existence or blowup of smooth solutions to 3-D semilinear Klein-Gordon equations

It is well-known that there are global small data smooth solutions for the 3-D semilinear Klein-Gordon equations u + u = F(u,∂ u) with cubic nonlinearities. However, for the short pulse initial data (u, ∂ₜu)(0, x)=(δν+1u₀(x/δ),δ^ν u₁(x/δ)) with ν∈ R and (u₀, u₁)∈ C₀∞( R), which are a class of large initial data, we establish that when ν≤ -1/2, the solution u can blow up in finite time for some suitable choices of (u₀, u₁) and cubic nonlinearity F(u,∂ u); when ν>-1/2, the smooth solution u exists globally. Therefore, ν=-1/2 is just the critical power corresponding to the global existence or blowup of smooth short pulse solutions for the cubic semilinear Klein-Gordon equations.