Asymptotic Behavior of Solutions to the Drift-Diffusion Equation with Critical Dissipation

Abstract

In this paper, the initial value problem for the drift-diffusion equation which stands for a model of a semiconductor device is studied. When the dissipative effect on the drift-diffusion equation is given by the half Laplacian, the dissipation balances to the extra force term. This case is called critical. The goal of this paper is to derive decay and asymptotic expansion of the solution to the drift-diffusion equation as time variable tends to infinity.

Asymptotic expansion of solutions to the drift–diffusion equation with fractional dissipation

“…Therefore, the perturbation methods would not work as discussed in more detail later. In [41], employing the energy method, the authors estimate the difference between the solution of (1.1) with θ = 1 and its second-order asymptotic expansion in L q (R n ) for 1 < q < ∞. But the cases q = 1 and q = ∞ are excepted.…”

“…For (1.14), well-posedness, global existence and decay of solutions for small initial data are proved. Particularly, for 1 ≤ q ≤ ∞, the decaying solution has the following asymptotic expansion as t → ∞ (see [15,41]):…”

On a Drift–Diffusion System for Semiconductor Devices

“…Considering the a priori estimates in the finite interval [0, T * ] we conclude (37) u 2 Ḣs + v 2 Ḣs ≤ C, ∀ t ≥ 0. Now the decay follows from interpolation (29).…”

“…Sugiyama, Yamamoto & Kato [33] studied local and global existence and uniqueness of the system with the fractional Laplacian, focusing primarily on the supercritical and critical cases 0 < α < 1 and α = 1, respectively. Yamamoto & Sugiyama [37,38] then derived lower bounds on the decay rates of a solution to the drift-diffusion system with the fractional Laplacian 0 < α ≤ 1 and obtained the asymptotic behavior of the solution as t → ∞. Similar systems arising in different contexts have been studied also by Li, Rodrigo & Zhang [27], Escudero [18], Bournaveas & Calvez [11], Biler & Karch [4], Biler & Wu [8], Biler, Karch & Woyczyński [5] Biler & Woyczyński [7,6], Zhao [40,39], Ascasibar, Granero-Belinchón & Moreno [1] and Burczak & Granero-Belinchón [13,14].…”

An Interdisciplinary Research on Students’ Employability in Technology Education to Advance Higher Education Enrollment Sustainability

“…Therefore, students' employability presents the individual competences in employment that belongs to the individualism perspective of SCT, the organizationism perspective of SCT was applied to discuss institutions' developmental sustainability, and the socializationism perspective of SCT was able to illustrate socialization media technology in higher education. Subsequently, not only was the Factor Analysis (FA) approach of quantitative analysis able to be first employed in the appraised execution of large-scale, weighted results from questionnaires completed from random university and college students and faculties with higher research validity and representativeness [26,27], but the Entropy Analysis (EA) model of qualitative analysis was also able to be secondly applied in the evaluated implementation of professional weighted results from the questionnaires completed from interdisciplinary experts in students' employability, institutions' developmental sustainability, and technology education [28,29] to enhance the research reliability, reproducibility, and accuracy, as illustrated in Figure 1.…”