Asymptotic method of moving planes for fractional parabolic equations

Chen, Wenxiong; Wang, Pengyan; Niu, Yahui; Hu, Yunyun

doi:10.1016/j.aim.2020.107463

Cited by 34 publications

(18 citation statements)

References 30 publications

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“…Very recently, many substantial advances have been made in the symmetry, monotonicity, and non-existence of positive solutions for fractional parabolic equations of type (1.1) with the usual local time derivative ∂ t u(x, t), based on the method of moving planes (cf. [18,20,21,30,37,38] and the references therein).…”

Section: Introductionmentioning

confidence: 98%

Qualitative properties of solutions for dual fractional nonlinear parabolic equations

Chen¹,

Ma²

2023

Preprint

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In this paper, we consider the dual fractional parabolic problemis the right half space. We prove that the positive solutions are strictly increasing in x 1 direction without assuming the solutions be bounded.So far as we know, this is the first paper to explore the monotonicity of possibly unbounded solutions for the nonlocal parabolic problem involving both the fractional time derivative ∂ α t and the fractional Laplacian (−∆) s . To overcome the difficulties caused by the dual nonlocality in space-time and by the remarkably weak assumptions on solutions, we introduced several new ideas and our approaches are quite different from those in the previous literature. We first establish an unbounded narrow region principle without imposing any decay and boundedness assumptions on the antisymmetric functions at infinity by estimating the nonlocal operator ∂ α t + (−∆) s along a sequence of suitable auxiliary functions at their minimum points, which is an essential ingredient to carry out the method of moving planes at the starting point. Then in order to remove the decay or bounded-ness assumption on the solutions, we develop a new novel approach lies in establishing the averaging effects for such nonlocal operator and apply these averaging effects twice to guarantee that the plane can be moved all the way to infinity to derive the monotonicity of solutions.We believe that the new ideas and techniques developed here will become very useful tools in studying the qualitative properties of solutions, in particular of those unbounded solutions, for a wide range of fractional elliptic and parabolic problems.Mathematics Subject classification (2020): 35R11; 35B50; 35K58; 26A33. Keywords: dual nonlocal parabolic equations; fractional time-diffusion; narrow region principle in unbounded domains; averaging effects; direct method of moving planes; monotonicity.

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Section: Introductionmentioning

confidence: 98%

Qualitative properties of solutions for dual fractional nonlinear parabolic equations

Chen¹,

Ma²

2023

Preprint

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“…They introduced a generalized weighted average inequality and the maximum principle in unbounded domains to derive the symmetry and monotonicity of fractional parabolic equations. In addition, other results involving nonlocal parabolic operators, we can refer to [4,13,15]. The aim of this paper is to derive the one-dimensional symmetry and monotonicity of positive solutions for the fractional uniformly parabolic equation (1.1).…”

Section: Introductionmentioning

confidence: 99%

Sliding method for equations with uniformly parabolic fractional operators

Tang

2023

Preprint

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In this paper, we obtain the one-dimensional symmetry and monotonicity of entire solutions for equations involving uniformly parabolic fractional operators via a sliding method. We first establish a generalized weighted average inequality and the maximum principle in unbounded domains, then using the sliding method, we derive the symmetry and monotonicity of entire solutions to fractional uniformly parabolic equations in the whole space. MSC 2020: 35B50, 35R11, 35K60.

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“…Chen etc. [7] introduced the asymptotic method of moving planes to study the following fractional parabolic problem…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by the work of [7] and [8], we study the monotonicity of the solutions to the fractional parabolic equation (1.1) by using the method of moving planes. Meanwhile, we find the connection between the existence of solutions of (1.1) and that of the following equation…”

Section: Introductionmentioning

confidence: 99%

Monotonicity of solutions for fractional parabolic equations

Liu

2023

Preprint

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In this paper, we consider the following fractional parabolic equation with a gradient nonlinear term on the upper half space \begin{equation*} \begin{cases} \frac{\partial u}{\partial t}(x, t)+(-\Delta)^s u(x, t)=f(t,x,u(x,t),\triangledown u(x,t)),& (x, t) \in \mathbb{R}_+^{n} \times \mathbb{R}, \\u(x, t) = 0,& (x, t) \notin \mathbb{R}_+^{n} \times \mathbb{R}. \end{cases} \end{equation*} Without assuming any asymptotic decay of $u$ near infinity, we prove that~$u(x,t)$~is strictly increasing with respect to $x_1$ by the method of moving planes. Meanwhile, we show the existence of positive solutions related to the fractional parabolic equations on the whole space $\mathbb{R}^{n-1}.$

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Asymptotic method of moving planes for fractional parabolic equations

Cited by 34 publications

References 30 publications

Qualitative properties of solutions for dual fractional nonlinear parabolic equations

Qualitative properties of solutions for dual fractional nonlinear parabolic equations

Sliding method for equations with uniformly parabolic fractional operators

Monotonicity of solutions for fractional parabolic equations

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