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

2024/01/01 by Song, Xianfa
#FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2401.00706

Abstract

In this paper, we consider the following Cauchy problem of a weighted gradient system of semilinear wave equations \ u_tt-Δu=λ|u|^α|v|^β+2u, v_tt-Δv=μ|u|^α+2|v|^βv, x∈ ℝ^d, t∈ ℝ,
u(x,0)=u₁₀(x), u_t(x,0)=u₂₀(x), v(x,0)=v₁₀(x), v_t(x,0)=v₂₀(x), x∈ ℝ^d.. Here d≥ 3, λ, μ∈ ℝ, α, β≥ 0, (u₁₀,u₂₀) and (v₁₀,v₂₀) belong to H¹(ℝ^d)⊕ L²(ℝ^d) or H¹(ℝ^d)⊕ L²(ℝ^d) or H^γ(ℝ^d)⊕ H^γ-1(ℝ^d) for some γ>1. Under certain assumptions, we establish the local wellposedness of the H¹⊕ H¹-solution, H¹⊕ H¹-solution and H^γ⊕ H^γ-solution of the system with different types of initial data.

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