A new critical exponent for semi-linear damped wave equations with the initial data from Sobolev spaces of negative order
Dinh Van Duong, Tuan Anh Dao
TL;DR
The paper analyzes the semilinear damped wave equation with initial data in Sobolev spaces of negative order, deriving a sharp critical exponent $p_c(m,\gamma,n)=1+\frac{2m}{n+m\gamma}$ that governs global existence versus blow-up. Employing a Duhamel-based fixed-point framework together with precise linear kernel decay estimates and nonlinear controls in weighted Sobolev spaces, the authors prove global existence for $p\ge p_c$ (including the critical case) and finite-time blow-up for $1<p<p_c$, along with lifespan estimates in the subcritical regime. The work extends Fujita-type thresholds to negative-order data, showing that the nonlinearity $|u|^p$ decays sufficiently fast to permit global solutions at the critical exponent, and provides sharp decay rates in multiple norms. These results advance understanding of the interplay between damping, diffusion, and nonlinear effects for rough initial data, and offer quantitative lifespan information essential for applications and further analysis.
Abstract
In this paper, we would like to study the critical exponent for semi-linear damped wave equations with power nonlinearity and the initial data belonging to Sobolev spaces of negative order $\dot{H}_m^{-γ}$. Precisely, we obtain a new critical exponent $$p_{\rm c}(m,γ,n): = 1 + \frac{2m}{n+mγ} $$ for $m \in (1, 2], \,γ\in \big[0, \frac{n(m-1)}{m}\big)$ by proving the global (in time) existence of small data Sobolev solutions when $p \geq p_{\rm c}(m,γ,n)$ and the blow-up result for weak solutions in finite time even for small data if $1 < p < p_{\rm c}(m,γ,n)$. In addition, a novelty of this paper is that the critical value $p= p_{\rm c}(m,γ,n)$ belongs to the global existence range. Furthermore, we are going to provide lifespan estimates for solutions when a blow-up phenomenon occurs.