论文标题
1D阻尼波方程的LP - 反应稳定性具有局部和非线性阻尼
Lp-asymptotic stability of 1D damped wave equations with localized and nonlinear damping
论文作者
论文摘要
在本文中，我们研究了一维非线性抑制波方程的$ l^p $ asymptotic稳定性，其$ p \ in（1，\ infty）$，在有限域中具有局部阻尼和dirichlet边界条件。我们首先解决了良好的问题。我们证明了[2，\ infty）$中的$ p \的弱解决方案的存在和独特性，以及[1，\ infty）$的所有$ p \ in [2，\ infty）$的存在和唯一性和唯一性。证据依赖于[4]与密度参数相结合的$ l^\ infty $框架中已经证明的良好性。然后，我们证明强溶液的能量成倍衰减至零。证明依赖于乘数方法与[8]中线性情况下完成的工作相结合。
In this paper, we study the $L^p$-asymptotic stability with $p\in (1,\infty)$ of the one-dimensional nonlinear damped wave equation with a localized damping and Dirichlet boundary conditions in a bounded domain $(0,1)$. We start by addressing the well-posedness problem. We prove the existence and the uniqueness of weak solutions for $p\in [2,\infty)$ and the existence and the uniqueness of strong solutions for all $p\in [1,\infty)$. The proofs rely on the well-posedness already proved in the $L^\infty$ framework by [4] combined with a density argument. Then we prove that the energy of strong solutions decays exponentially to zero. The proof relies on the multiplier method combined with the work that has been done in the linear case in [8].
