具有不定非线性的双分数抛物方程

IF 1.5 1区 数学 Q1 MATHEMATICS Advances in Mathematics Pub Date : 2024-08-20 DOI:10.1016/j.aim.2024.109891
Wenxiong Chen , Yahong Guo
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引用次数: 0

摘要

在本文中,我们考虑在 Rn×R 中涉及 Marchaud 分数时间导数的下列不定对偶分数抛物方程∂tαu(x,t)+(-Δ)su(x,t)=a(x)f(u(x,t)),其中 α,s∈(0,1),函数 a 和 f 是非递减函数。我们将证明不存在正界解。为此,我们首先证明所有正界解 u(⋅,t) 必须沿 a(x) 确定的方向严格单调递增。为了克服算子∂tα+(-Δ)s 的对偶非位置性所带来的挑战,我们引入了一些新思想和新技术。这些新方法不仅适用于手头的具体问题,还可扩展用于解决其他各种分式问题,无论是椭圆问题还是抛物问题,包括那些与马尔查时间导数相关的双重非局部性问题。
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Dual fractional parabolic equations with indefinite nonlinearities

In this paper, we consider the following indefinite dual fractional parabolic equation involving the Marchaud fractional time derivativetαu(x,t)+(Δ)su(x,t)=a(x)f(u(x,t))inRn×R, where α,s(0,1), and the functions a and f are nondecreasing. We prove that there is no positive bounded solutions. To this end, we first show that all positive bounded solutions u(,t) must be strictly monotone increasing along the direction determined by a(x). Then by mollifying the first eigenfunction for fractional Laplacian (Δ)s and constructing an appropriate subsolution for the Marchaud fractional operator tα1, we derive a contradiction and thus obtain the non-existence of solutions.

To overcome the challenges caused by the dual non-locality of the operator tα+(Δ)s, we introduce several new ideas and novel techniques. These novel approaches are not only applicable to the specific problem at hand but can also be extended to address various other fractional problems, be they elliptic or parabolic, including those featuring dual nonlocalities associated with the Marchaud time derivatives.

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来源期刊
Advances in Mathematics
Advances in Mathematics 数学-数学
CiteScore
2.80
自引率
5.90%
发文量
497
审稿时长
7.5 months
期刊介绍: Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.
期刊最新文献
Constructing smoothings of stable maps Non-smoothable homeomorphisms of 4-manifolds with boundary Affine dual Minkowski problems Editorial Board On exponential frames near the critical density
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