【摘要】 研究具有阻尼的半线性波动方程的初边值问题u_tt-△u+βu_t=|u|^p-1u,x∈Ω,t>0u（x,0）=u₀（x）,u_t（x,0）=u₁（x）,x∈Ωu|_（?）Ω=0,t≥0其中γ为正常数,Ω■Rⁿ为有界域,当n≥3时,1<p≤（n+2）/（n-2）;当n=1,2时,1<p<∞.首先利用紧致性方法和位势井方法证明了此问题整体弱解的存在性.而后,利用位势井族方法证明了,当时间t→+∞时,此解依t的指数形式衰减于零.结果从根本上改进了已有结果.更多还原

【Abstract】 In this paper we study the initial-boundary value problem of damped semilinearwave equation u_tt-Δu+γu_t=|u|^p-1u,x∈Ω,t>0u（x,0）=u₀（x）,u_t（x,0）=u₁（x）,x∈Ω u|αΩ=0,t≥0 whereγis a positive constant,Ω■Rⁿ is a bounded domain,1<p≤（n+2）/（n-2）for n≥3;1<p<∞for n=1,2.First by using compactness method and potential well method weprove the global existence of weak solution.Then by introducing a family of potential wells,we prove that when time t→+∞the global weak solution decays to zero exponently.更多还原