scholarly journals Global solution to Cauchy problem of fractional drift diffusion system with power-law nonlinearity

2021 ◽  
Author(s):  
Caihong Gu ◽  
Yanbin Tang
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Decay Rate ◽  
Power Law ◽  
Mild Solutions ◽  
Diffusion System ◽  
Heat Semigroup ◽  
Drift Diffusion ◽  
Continuous Mappings ◽  
Time Space

In this paper we consider the global existence, regularizing-decay rate and asymptotic behavior of mild solutions to the Cauchy problem of fractional drift diffusion system with power-law nonlinearity. Using the properties of fractional heat semigroup and the classical estimates of fractional heat kernel, we first prove the global-in-time existence and uniqueness of the mild solutions in the frame of mixed time-space Besov space with multi-linear continuous mappings. Then we show the asymptotic behavior and regularizing-decay rate estimates of the solution to equations with power-law nonlinearity by the method of multi-linear operator and the classical Hardy-Littlewood-Sobolev inequality.

scholarly journals On the uniqueness of mild solutions for the parabolic-elliptic Keller-Segel system in the critical $ L^{p} $-space$ ^\dagger $

2021 ◽  
Vol 4 (6) ◽  
pp. 1-14
Author(s):  
Lucas C. F. Ferreira ◽  
Keyword(s):  
Decay Rate ◽  
Weak Solutions ◽  
Lebesgue Space ◽  
Mild Solutions ◽  
Hölder’S Inequality ◽  
Heat Semigroup ◽  
Well Posedness ◽  
Bilinear Estimate ◽  
Bilinear Term ◽  
Weighted Norms

<abstract><p>We are concerned with the uniqueness of mild solutions in the critical Lebesgue space $ L^{\frac{n}{2}}(\mathbb{R}^{n}) $ for the parabolic-elliptic Keller-Segel system, $ n\geq4 $. For that, we prove the bicontinuity of the bilinear term of the mild formulation in the critical weak-$ L^{\frac{n}{2}} $ space, without using Kato time-weighted norms, time-spatial mixed Lebesgue norms (i.e., $ L^{q}((0, T);L^{p}) $-norms with $ q\neq\infty $), and any other auxiliary norms. Our proofs are based on Yamazaki's estimate, duality and Hölder's inequality, as well as an adapted Meyer-type argument. Since they are different from those of Kozono, Sugiyama and Yahagi [J. Diff. Eq. 253 (2012)] and it is not clear whether mild solutions are weak solutions in the critical $ C([0, T);L^{\frac{n}{2}}) $, our results complement theirs in a twofold way. Moreover, the bilinear estimate together heat semigroup estimates yield a well-posedness result whose dependence with respect to the decay rate $ \gamma $ of the chemoattractant is also analyzed.</p></abstract>

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO DRIFT-DIFFUSION SYSTEM WITH GENERALIZED DISSIPATION

2009 ◽  
Vol 19 (06) ◽  
pp. 939-967 ◽  
Author(s):  
TAKAYOSHI OGAWA ◽  
MASAKAZU YAMAMOTO
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Asymptotic Expansion ◽  
Initial Data ◽  
Global Existence ◽  
Asymptotic Behavior Of Solutions ◽  
Large Initial Data ◽  
Drift Diffusion ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem

We show the global existence and asymptotic behavior of solutions for the Cauchy problem of a nonlinear parabolic and elliptic system arising from semiconductor model. Our system has generalized dissipation given by a fractional order of the Laplacian. It is shown that the time global existence and decay of the solutions to the equation with large initial data. We also show the asymptotic expansion of the solution up to the second terms as t → ∞.

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

scholarly journals On a Drift–Diffusion System for Semiconductor Devices

2016 ◽  
Vol 17 (12) ◽  
pp. 3473-3498 ◽  
Author(s):  
Rafael Granero-Belinchón
Keyword(s):  
Semiconductor Devices ◽  
Diffusion System ◽  
Drift Diffusion
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