The critical power of short pulse initial data on the global existence or blowup of smooth solutions to 3-D semilinear Klein-Gordon equations
Jindou Shen, Huicheng Yin
TL;DR
This work identifies a sharp, data-dependent threshold for global existence versus finite-time blowup in 3-D cubic semilinear Klein-Gordon equations with short-pulse initial data. By employing a scaling transform and hyperboloidal energy methods, it proves global smooth solutions for $\nu> -\frac{1}{2}$ and constructs data leading to blowup for $\nu\le -\frac{1}{2}$, establishing $\nu = -\frac{1}{2}$ as the critical power. The results illuminate the interplay between pulse concentration, nonlinear cubic interactions, and relativistic dispersion, with explicit bounds expressed in the scaled framework. The methods combine energy estimates on hyperboloids, commutator calculus, and a bootstrap argument to propagate smallness, alongside a direct blow-up mechanism via a Lyapunov-type functional for the sharp nonlinearity. The findings have implications for understanding large-data dynamics in nonlinear dispersive equations and for guiding the design of initial data that drive global versus blow-up behavior in Klein-Gordon-type systems.
Abstract
It is well-known that there are global small data smooth solutions for the 3-D semilinear Klein-Gordon equations $\square u + u = F(u,{\partial u})$ with cubic nonlinearities. However, for the short pulse initial data $(u, \partial_tu)(0, x)=({δ^{ν+1}}{u_0}({\frac{x}δ}),{δ^ν}{u_1}({\frac{x}δ}))$ with $ν\in\Bbb R$ and $(u_0, u_1)\in C_0^{\infty}(\Bbb R)$, which are a class of large initial data, we establish that when $ν\le -\frac{1}{2}$, the solution $u$ can blow up in finite time for some suitable choices of $(u_0, u_1)$ and cubic nonlinearity $F(u,{\partial u})$; when $ν>-\frac{1}{2}$, the smooth solution $u$ exists globally. Therefore, $ν=-\frac{1}{2}$ is just the critical power corresponding to the global existence or blowup of smooth short pulse solutions for the cubic semilinear Klein-Gordon equations.