Sharp polynomial decay rates for the damped wave equation with Hölder-like damping

Abstract

We study decay rates for the energy of solutions of the damped wave equation on the torus. We consider dampings invariant in one direction and bounded above and below by multiples of x β near the boundary of the support and show decay at rate 1/t β+2 β+3 . In the case where W vanishes exactly like x β this result is optimal by [Kle19]. The proof uses a version of the Morawetz multiplier method.

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

“…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].…”

Sharp Polynomial Decay for Waves Damped from the Boundary in Cylindrical Waveguides

“…Also note the related non-concentration results in [BZ04, BZ05,BHW07,BZ12]. The relation between polynomial decay and regularity of interior damping on T 2 has been well explained in [ALN14, Kle19a,Kle19b,DK20].…”

“…We use a monotonicity argument to improve the regularity of a, b by one order without tarnishing the optimality of the polynomial decay. This is surprising, since in [ALN14,DK20] we know that sharp polynomial decay depends on the Hölder regularity of interior damping. See Remark 3.4(1).…”

Robustness of Polynomial Stability of Damped Wave Equations

“…was shown to be α = 3/2. Moreover, additional differentiability assumptions on d(•, •) improve the rate of polynomial decay, as shown in [5,11,14].…”