Non decay of the total energy for the wave equation with the dissipative term of spatial anisotropy

Abstract

Abstract. We consider the behavior of the total energy for the wave equation with the dissipative term. When the dissipative term works well uniformly in every direction, several authors obtain uniform decay estimates of the total energy. On the other hand, if the dissipative term is small enough uniformly in every direction, it is known that there exists a solution whose total energy does not decay. We examine the case that the dissipative term vanishes only in a neighborhood of a half-line. We introduce a uniform decay property, which is a natural generalization of the uniform decay estimates, and show that this property does not hold in our case. We prove this by constructing asymptotic solutions supported in the place where the dissipative term vanishes.

Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain

“…such that lim t→∞ E(t; u − w) = 0 holds (see [42] for the case b = b(x) ∈ C ∞ 0 (R N ) and [27,24,29,15,35] for further improvements).…”

Sufficient conditions for decay estimates of the local energy and a behavior of the total energy of dissipative wave equations in exterior domains

“…Only for getting non-decaying property, it suffices to assume that ∞ 0 a(x 0 + sω)ds < ∞ for some x 0 ∈ Ω and ω ∈ S n−1 . Modifying the argument in Kawashita, Nakazawa and Soga [9], we can obtain this fact. Note that the half line x 0 + sω (s ≥ 0) is the ray of geometrical optics, and the above condition means that the amplitude of high frequency waves propagating along this ray does not decay as t → ∞.…”

Decay property of solutions to the wave equation with space‐dependent damping, absorbing nonlinearity, and polynomially decaying data

“…It is known that the so‐called geometric control condition (GCC) introduced by Rauch and Taylor 41 and Bardos et al 42 is sufficient for the energy decay of solutions with initial data in the energy space. For the problem () with $$$ \Omega &#x0003D;{\mathbb{R}}&#x0005E;n $$$, (GCC) is read as follows: There exist constants $$$ T&gt;0 $$$ and $$$ c&gt;0 $$$ such that for any , we have $$$$ \frac{1}{T}{\int}_0&#x0005E;Ta\left({x}_0&#x0002B;s{\xi}_0\right)\kern0.1em ds\ge c. $$$$ For this and related topics, we refer the reader to previous works 34,43–50 . We note that for $$$ a(x)&#x0003D;{\left\langle x\right\rangle}&#x0005E;{-\alpha } $$$ with $$$ \alpha &gt;0 $$$, (GCC) is not fulfilled.…”