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

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 $d\ge 3$, $\lambda ,\mu \in \mathbb{R}$, $\alpha ,\beta \ge 0$, $(u_{10},u_{20})$ 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 ${\dot{H}}^{\gamma }({\mathbb{R}}^d)\oplus H^{\gamma -1}({\mathbb{R}}^d)$ for some $\gamma >1$. 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 ${\dot{H}}^{\gamma }\oplus {\dot{H}}^{\gamma }$-solution of the system with different types of initial data.