Asymptotic profiles for a strongly damped plate equation with lower order perturbation

Abstract

We consider the Cauchy problem in R n for a strongly damped plate equation with a lower oder perturbation. We derive asymptotic profiles of solutions with weighted L 1,γ (R n ) initial velocity by using a new method introduced in [7].

Optimal decay rates and small global solutions to the dissipative Boussinesq equation

“…From the above argument, we find when taking = 1 in (11), = 2, = 1, = 1 in Equation (12), = 0, = 1 in Equation (13), respectively, they become Equation (8) with = 0. However, by comparing the results in the previous studies [17][18][19] with this paper, we find that the decay rate of the solutions, showed in this paper, are different from the ones in the previous studies. [17][18][19] When taking = 1 and = 1 in Equation (13), it becomes the double dispersion equation.…”

“…However, by comparing the results in the previous studies [17][18][19] with this paper, we find that the decay rate of the solutions, showed in this paper, are different from the ones in the previous studies. [17][18][19] When taking = 1 and = 1 in Equation (13), it becomes the double dispersion equation. Recently, D'Abbicco 20 studied the decay estimates on the small solutions for the Cauchy problem of the double dispersion equation in which the assumption on the initial data with negative order are avoided by assuming the initial data in real Hardy space.…”

“…Lemma 2. Let ( ) be defined by (23), G( , t) and K( , t) be the fundamental solutions of (19) given by (17) and (18) respectively. Then, the following pointwise estimates hold,…”

Asymptotic expansion of the L2-norm of a solution of the strongly damped wave equation

“…where α ≥ 0 and 0 ≤ θ ≤ 1. In their paper [7], Ikehata and Soga established asymptotic estimates for the squared L 2 -norm of the difference of the Fourier transform of the weak solution of (5.2) and the profile ν(t, ξ) found in this thesis and Ikehata's paper [4]. It is the opinion of the author that the asymptotic expansion may be obtained using similar methods to those found in this thesis.…”

Asymptotic Profiles and Convergence Rates of the Linearized Compressible Navier–Stokes– Korteweg System

“…In this section, our aim is to derive the decay properties of solution operators G. Since the solution operator G is given in term of G and H, therefore, we only study the decay properties of the solution operators G and H. The following estimate has been derived by applying the energy method in the Fourier space to the first equation in (11) (see [22][23][24]). Lemma 1.…”