Asymptotic behaviors of solutions to evolution equations in the presence of translation and scaling invariance

Abstract

There are wide classes of nonlinear evolution equations which possess invariant properties with respect to a scaling and translations. If a solution is invariant under the scaling then it is called a self-similar solution, which is a candidate for the asymptotic profile of general solutions at large time. In this paper we establish an abstract framework to find more precise asymptotic profiles by shifting self-similar solutions suitably.

Improvement of convergence rates for a parabolic system of chemotaxis in the whole space

“…Such asymptotic profiles of solutions such as (1.8) were also obtained in Witelski and Bernoff [21], Kim and Tzavaras [9], Kim and Ni [8], Miller and Bernoff [12], Yanagisawa [20], Nishihara [15], Kagei and Maekawa [4,5] and references therein. In particular, for Equation (1.1), in the case n D 1, Nishihara [15] .t/ exists.…”

On asymptotic behaviors of solutions to parabolic systems modelling chemotaxis

“…; see Remark 5 below and [17,Section 6.4]. The key structures of (1.1) underlying this spectral distribution are the conservation of mass, the translation invariance, and the scaling invariance.…”

“…As stated before, we will prove Theorem 2 by applying the abstract results in [17] which are based on the spectral properties of the linearized operator around the profile function U δ in connection with translation and scaling invariance. Our approach delineates the nature of the equation to yield the phenomenon as those in Theorem 2.…”

“…On the other hand, the authors [17] recently establish the abstract theory which gives the procedure to obtain accurate asymptotic profiles of solutions for some class of evolution equations possessing two symmetries: translation invariance and scaling invariance. Due to the presence of translation invariance, the abstract theory in [17] is indeed applicable for (1.1) and does present a more precise asymptotic profile than (1.4). But in order to verify this application we have to precisely track the behaviors of the spectrum of the linearized operators around the self-similar solutions.…”

“…The aim of this paper is thus to provide more accurate asymptotic profiles than (1.4) by applying the theory in [17], together with the detailed spectral analysis of the linearized operators around the self-similar solutions. For m, m > 0 set L 2 m = L 2 1 + |x| 2 m 2 dx ,…”