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In this paper we determine \(L^{p}-L^{q}\) estimates, \(1\leq p\leq q\leq \infty \) , for the solution to the Cauchy problem for the damped wave equation \(\displaystyle \begin {cases} u_{tt}-\varDelta u + (-\varDelta )^{\frac \theta 2} u_{t}=0, & t\geq 0, x \in \mathbb {R}^{n}\\ u(0,x)=0,\,u_t(0,x)=u_1(x); \end {cases} \) in the case of very strong damping, \(\theta >2\) .

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\(L^p-L^q\) Estimates for Very Strongly Damped Wave Equations

  • 会议论文

  • 发表日期：

    2025年

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  • Marcello D’Abbicco,
  • Antonio Lagioia

摘要

In this paper we determine \(L^{p}-L^{q}\) estimates, \(1\leq p\leq q\leq \infty \) , for the solution to the Cauchy problem for the damped wave equation \(\displaystyle \begin {cases} u_{tt}-\varDelta u + (-\varDelta )^{\frac \theta 2} u_{t}=0, & t\geq 0, x \in \mathbb {R}^{n}\\ u(0,x)=0,\,u_t(0,x)=u_1(x); \end {cases} \) in the case of very strong damping, \(\theta >2\) .