Critical line of exponents, scattering theories for a weighted gradient system of semilinear wave equations

TL;DR

This work analyzes a weighted gradient system of semilinear wave equations with couplings $|u|^{\alpha}|v|^{\beta+2}u$ and $|u|^{\alpha+2}|v|^{\beta}v$, introducing a conserved weighted energy $E_w(u,v)$ and identifying sharp thresholds: a critical line $\alpha+\beta=2$ for $d=3$ and a critical point $(0,0)$ for $d=4$. The authors develop a framework combining Strichartz estimates, Besov-space methods, and energy arguments to prove local and global wellposedness across several Sobolev settings, obtain regularity in special cases, and establish scattering in energy-critical and energy-supercritical regimes. They show global existence and scattering for small data in the energy-critical subspaces and derive scattering operators in neighborhood of the origin, while also addressing local wellposedness and supercritical regimes via $s_c=\frac{d}{2}-\frac{2}{\alpha+\beta+2}$. The results delineate precise thresholds for global behavior and scattering in low dimensions, contributing to the understanding of coupled nonlinear wave systems with weighted gradient structure.

Abstract

In this paper, we consider the following Cauchy problem of a weighted gradient system of semilinear wave equations \begin{equation*} \left\{ \begin{array}{lll} u_{tt}-Δu=λ|u|^α|v|^{β+2}u,\quad v_{tt}-Δv=μ|u|^{α+2}|v|^βv,\quad x\in \mathbb{R}^d,\ t\in \mathbb{R},\\ u(x,0)=u_{10}(x),\ u_t(x,0)=u_{20}(x),\quad v(x,0)=v_{10}(x),\ v_t(x,0)=v_{20}(x),\quad x\in \mathbb{R}^d. \end{array}\right. \end{equation*} Here $d\geq 3$, $λ, μ\in \mathbb{R}$, $α, β\geq 0$, $(u_{10},u_{20})$ and $(v_{10},v_{20})$ belong to $H^1(\mathbb{R}^d)\oplus L^2(\mathbb{R}^d)$ or $\dot{H}^1(\mathbb{R}^d)\oplus L^2(\mathbb{R}^d)$ or $\dot{H}^γ(\mathbb{R}^d)\oplus H^{γ-1}(\mathbb{R}^d)$ for some $γ>1$. Under certain assumptions, we establish the local wellposedness of the $H^1\oplus H^1$-solution, $\dot{H}^1\oplus \dot{H}^1$-solution and $\dot{H}^γ\oplus \dot{H}^γ$-solution of the system with different types of initial data.