Optimal decay rate for the wave equation on a square with constant damping on a strip

Abstract

We consider the damped wave equation with Dirichlet boundary conditions on the unit square parametrized by Cartesian coordinates x and y. We assume the damping a to be strictly positive and constant for x < σ and zero for x > σ. We prove the exact t −4/3 -decay rate for the energy of classical solutions. Our main result (Theorem 1) answers question (1) of [1, Section 2C.]. MSC2010: Primary 35B40, 47D06. Secondary 35L05, 35P20.

Semi-uniform stability of operator semigroups and energy decay of damped waves

“…If M=double-struckT2 is the 2-torus and b∈Lnormal∞false(Mfalse), then the energy decays with the rate Ofalse(t−12false), and the decay improves to O ( t −(1−ε) ) for certain b∈Wk,normal∞false(Mfalse), no matter how small the support of b ; see Anantharaman & Léautaud [97, Theorems 2.3 and 2.6]. On the other hand, for simple characteristic functions b , the optimal rate of decay is O ( t −2/3 ) [97,103], no matter how large the support of b . The fact that the regularity of b may play a more important role than its support is also nicely illustrated in [104–107].…”

Robustness of Polynomial Stability of Damped Wave Equations

“…In some cases of damping functions the estimate for the exponent of polynomial stability can be improved. For example, in [28] the exponent of polynomial stability for the damping function…”

Optimal Energy Decay for the Wave-Heat System on a Rectangular Domain

“…Another example is the following wave equation which is damped on one half of its rectangular domain but not on the other: Here Ω = (−1, 1) × (0, 1) and Ω + = (0, 1) × (0, 1) as in Section 1, and u 0 , v 0 are suitable functions defined on Ω. In this case the energy E z 0 (t) = 1 2 Ω |∇u(x, y, t)| 2 + |u t (x, y, t)| 2 d(x, y), t ≥ 0, of any classical solution u, with corresponding initial data z 0 = (u 0 , v 0 ), can be shown to satisfy E z 0 (t) = o(t −4/3 ) as t → ∞, and furthermore this estimate is sharp; see [3, Part IV.B], [17] and also [11,15]. Our methods can also be adapted to study the following wave equation on the square Ω − = (−1, 0)× (0, 1) subject to Dirichlet boundary conditions along three of its edges but with the coupled heat equation in (1.1) replaced by a dissipative boundary condition along the fourth edge:…”

“…Throughout Sections 2 and 3 we take advantage of the special rectangular geometry of our domain, which allows us to obtain optimal bounds by means of direct estimates as opposed, for instance, to indirect microlocal arguments. More specifically, we use separation of variables, thus decomposing the two-dimensional problem into a family of one-dimensional problems which can be dealt with using techniques akin to those developed in our earlier paper [9]; see also [11,17] for similar arguments in the context of damped wave equations.…”