We make several remarks concerning properties of functions in parabolic De Giorgi classes of order $p$. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local Hölder continuity of these functions via Moser's ideas, thus avoiding De Giorgi's heavy machinery. We also seize this opportunity to give a transparent proof of a weak Harnack inequality for non-negative members of some super-class of De Giorgi, without any covering argument.
We give a unified proof of Hölder regularity of weak solutions for mixed local and nonlocal $p$-Laplace type parabolic equations with the full range of exponents $1<p<\infty$. Our proof is based on the expansion of positivity together with the energy estimate and De Giorgi type lemma.
We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Hölder inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions.
We consider mixed local and nonlocal quasilinear parabolic equations of $p$-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.
We investigate the mixed local and nonlocal parabolic $p$-Laplace equation \begin{align*} \partial_t u(x,t)-Δ_p u(x,t)+\mathcal{L}u(x,t)=0, \end{align*} where $Δ_p$ is the local $p$-Laplace operator and $\mathcal{L}$ is the nonlocal $p$-Laplace operator. Based on the combination of suitable Caccioppoli-type inequality and Logarithmic Lemma with a De Giorgi-Nash-Moser iteration, we establish the local boundedness and Hölder continuity of weak solutions for such equations.
Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations
This paper deals with the fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=Δu-χ\nabla \cdot (u\nabla v)+ξ\nabla\cdot(u \nabla w), \quad v_t=Δv-v+u, \quad w_t=Δw-w+u, \quad x \in Ω,\ t>0 \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω$ is an open ball in $\mathbb{R}^n$ ($n \ge 3$), $χ, ξ>0$ are constants. When $w=0$, finite-time blow-up in the corresponding Keller-Segel system has already been obtained. However, finite-time blow-up in the above attraction-repulsion chemotaxis system has not yet been established except for the …
Remarks on some quasi-linear fourth order parabolic equations arising in mathematical physics
Basque Center for Applied Mathematics（巴斯克应用数学中心）；Ikerbasque – Basque Foundation for Science（伊卡巴斯科-巴斯克科学基金会）；Université Claude Bernard Lyon 1, ICJ UMR5208（克洛德·贝尔纳里昂大学，ICJ UMR5208）；Universidad de Cantabria（坎塔布里亚大学）；University of Sevilla（塞维利亚大学）；Université Lyon 1, Centrale Lyon, INSA Lyon, Université Jean Monnet, CNRS, ICJ UMR5208（里昂第一大学，中央里昂学院，里昂国立应用科学学院，让·穆兰大学，法国国家科学研究中心，ICJ UMR5208）
These notes aim to provide a deeper insight on the specifics of two articles dealing with chemotaxis models with nonlinear production. More precisely, we are referring to the papers "Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with nonlinear signal production" by X. Tao, S. Zhou and M. Ding [J. Math. Anal. Appl. 474:1 (2019) 733-747] and "Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation" by S. Frassu and G. Viglialoro [Acta Appl. Math. 171:1 (2021), 19]. These works, independently published in these l…
In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition $u \to 0$ at infinity with respect to the spacial variables to a polynomial growth on $u$ by constructing auxiliary functions.Then we derive monotonicity for the solutions in a half space $\mathbb{R}_+^n \times \mathbb{R}$ and obtain some new …
Symmetrization for Linear and Nonlinear Fractional Parabolic Equations of Porous Medium Type
利用浓度比较的概念，建立了线性分数阶扩散方程$\partial_t u +(-Δ)^{σ/2}u=f$及其椭圆方程$h v +(-Δ)^{σ/2}v=f$，$h>0$解的对称化结果.结果扩展到非线性版本，$\partial_t u+(-Δ)^{σ/2}A(u)=f$，但只有当$A:\re_+\to\re_+$是一个凹函数。在椭圆情形下，当$B(v)$是一个凸非负函数，$v>0$与$B(0)=0$时，证明了$\,B(v)+(-Δ)^{σ/2}v=f$ \的完全对称化结果，当$B$是凹函数时，证明了部分对称化结果。当$A$为凸时，分别构造了抛物型方程的显着反例.当$B$是凹的时，这样的反例在标准扩散情形$σ=2$中不存在。
Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system
本文首先讨论了Boussinesq方程组的强解的高阶导数在通常的索伯列夫空间中的时空衰减性质。这里得到的衰减率是最佳的。证明是基于一个抛物型插值不等式，自举参数和一些加权估计。其次，我们给出了Boussinesq方程组的一个新的解积分公式，并利用该公式在标度不变函数空间中建立了强解的存在性.我们进一步研究了这些强解的渐近分布和衰减性质。我们的结果恢复和推广了Brandolese和Schonbek（Tran. A. M.S. 364，No.10，2012，5057-5090）。
In this paper, we consider the dual fractional parabolic problem in 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 $\partial_t^α$ and the fractional Laplacian $(-Δ)^s$. To overcome the difficulties caused by the dual nonlocality in space-time and by the remarkably weak assumptions on solutions, we introduced several …
