A dissipative logarithmic type evolution of second order in time

TL;DR

This work analyzes a dissipative logarithmic-type evolution equation on $\mathbb{R}^N$ with non-local damping $L=\log(-\Delta+I)$, focusing on $L^2$-asymptotics and decay. Using a Fourier-space formulation, the authors obtain explicit solution formulas, derive energy identities and pointwise decay estimates, and establish an $L^2$-decay of order $t^{-\frac{N}{2}}$ for data in $L^1\cap L^2$. They identify an asymptotic profile in $L^2$ when $u_0=0$, showing the leading term is governed by $P_1\,\mathcal{F}^{-1}\Big( \frac{4}{\pi} e^{-\frac{t}{2}\log(1+|\xi|^2)}\sin(t\sqrt{\log(1+|\xi|^2)})\Big)$ with $P_1=\int u_1$, and prove the $L^2$-error decays like $t^{-\frac{N-2}{4}}$. Moreover, the paper proves that the $t^{-N/2}$ decay rate of the leading term is sharp by establishing matching upper and lower bounds for its $L^2$-norm. Together, these results provide a detailed, quantitative description of dissipation via logarithmic damping in a nonlocal setting and yield precise decay and asymptotic profiles for the solution.

Abstract

In this paper, we introduce a logarithmic-type second-order model with a non-local logarithmic damping mechanism in $R^N$. We present a motivation with a spectral approach to consider the equation, we consider the Cauchy problem associated with the model. More precisely, we study the asymptotic behavior of solutions as $t$ goes to infinity in $L^2$-sense; namely, we prove results on the asymptotic profile and optimal decay of solutions as time goes to infinity in $L^2$-sense.

Theorems & Definitions (23)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Proposition 2.6
• Theorem 2.7