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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.00706 |
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Table of Contents:
- 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.