Longtime Behavior of Wave Equation with Kinetic Boundary Condition
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Longtime Behavior of Wave Equation with Kinetic Boundary Condition Xiaoyu Fu1 · Lingxia Kong2 Accepted: 22 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we consider a damped wave equation with mixed boundary conditions in a bounded domain. On one portion of the boundary, we have kinetic boundary condition: ∂ν y + m(x)ytt − T y = 0 with density function m(x), and on the other portion, we have homogeneous Neumann boundary condition: ∂ν y = 0. Based on the growth of the resolvent operator on the imaginary axis, solutions of the wave equations under consideration are proved to decay logarithmically. The proof of the resolvent estimation relies on the interpolation inequalities for an elliptic equation with Steklov type boundary conditions. Keywords Wave equation · Interpolation inequality · Kinetic boundary condition · Resolvent estimate
1 Introduction Let ⊂ lRn be a bounded domain with boundary of class C 2 , 0 , 1 be two non-empty open subsets of such that = 0 ∪ 1 and 0 ∩ 1 = ∅. Let ω be a nonempty open subset of with smooth boundary ∂ω = I ∪ 1 such that j ∩ I = ∅ ( j = 0, 1). We refer to [1, Finger 1] for the relationship of j ( j = 0, 1) and I. Denote l by χω the characteristic function of ω, c the complex conjugate of c ∈ C.
This work is partially supported by the NSF of China under Grants 11971333, 11931011.
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Lingxia Kong [email protected] Xiaoyu Fu [email protected]
1
School of Mathematics, Sichuan University, Chengdu 610064, China
2
School of Basic Science, Harbin University of Commerce, Harbin, China
123
Applied Mathematics & Optimization
Let m(·) ∈ L ∞ (0 ) be a non-negative bounded function such that m(x) ≥ m 0 > 0, a.e. x ∈ 1 .
(1.1)
Let us consider the following damped hyperbolic equation: ⎧ ytt − y + d(x)yt = 0 ⎪ ⎪ ⎪ ⎪ ⎨ ∂ν y + m(x)ytt − T y = 0 ∂ν y = 0 ⎪ ⎪ y(0, x) = y0 (x), yt (0, x) = z 0 (x) ⎪ ⎪ ⎩ yt (0, x) = w0 (x)
in lR+ × , on lR+ × 1 , on lR+ × 0 , in , on 1 ,
(1.2)
where d(x) = dχω with d > 0 is a constant, ν = (ν1 , · · · , νn ) is the unit outward normal along and T denotes the Laplace-Beltrami operator on 1 . In (1.2), we assume that y0 (x) ∈ H 1 (), z 0 (x) ∈ L 2 () and w0 (x) ∈ L 2 (1 ). In the case of m(x) = 0, the wave Eq. (1.2) possesses a Steklov type boundary condition on 1 . In the context of elliptic equations and the corresponding eigenvalue problems with such type of static boundary condition, we refer to [14] and the references therein. In the case when m(x) is not zero everywhere, we call such kind of boundary condition as the dynamical boundary condition. In the literature, there are numerous references on the PDEs problems with dynamical boundary conditions (such as, diffusion phenomena in thermodynamics, control theory), we refer the readers to the references [15,16] and references cited therein. It should be pointed out that the presence of two derivatives (in time) at the boundary 1 implies the presence of kinetic energy along 1 . In this respect, we refer th
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