In this paper we shall investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation:\[ \begin{cases} \frac{\partial u}{\partial t}= Δ_{\mathbb{G},p}u+V(x)u^{p-1} & \text{in}\quad Ω\times (0, T), \quad 1<p<2, u(x,0)=u_{0}(x)\geq 0 & \text{in} \quadΩ, u(x,t)=0 & \text{on}\quad \partialΩ\times (0, T) \end{cases} \] where $ Δ_{\mathbb{G},p}$ is the $p$-sub-Laplacian on Carnot group $ \mathbb{G}$ and $V\in L_{\text{loc}}^1(Ω)$.
A Stein-Weiss type inequality on Carnot groups is established by proving the boundedness of an integral operator and the Hardy-Littlewood-Sobolev type inequality on Carnot groups is also derived.
We consider various types of Hardy-Sobolev inequalities on a Carnot-Carathéodory space $(\Om, d)$ associated to a system of smooth vector fields $X=\{X_1, X_2,...,X_m\}$ on $\RR^n$ satisfying the Hörmander's finite rank condition $rank Lie[X_1,...,X_m] \equiv n$. One of our main concerns is the trace inequality \int_{\Om}|ϕ(x)|^{p}V(x)dx\leq C\int_{\Om}|Xϕ|^{p}dx,\qquad ϕ\in C^{\infty}_{0}(\Om), where $V$ is a general weight, i.e., a nonnegative locally integrable function on $\Om$, and $1<p<+\infty$. Under sharp geometric assumptions on the domain $\Om\subset \Rn$ that can be measured equival…
We propose the analogues of boundary layer potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the single layer potential and establish the Plemelj type jump relations for the double layer potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton potential for the sub-Lap…
We give an elementary proof of the classical Hardy inequality on any Carnot group, using only integration by parts and a fine analysis of the commutator structure, which was not deemed possible until now. We also discuss the conditions under which this technique can be generalized to deal with hypoelliptic families of vector fields, which, in this case, leads to an open problem regarding the symbol properties of the gauge norm.
In this paper, we present the geometric Hardy inequalities on the starshaped sets in the Carnot groups. Also, we obtain the geometric Hardy inequalities on half-spaces for general vector fields.
Schauder estimates were a historical stepping stone for establishing uniqueness and smoothness of solutions for certain classes of partial differential equations. Since that time, they have remained an essential tool in the field. Roughly speaking, the estimates state that the Hölder continuity of the coefficient functions and inhomogeneous term implies the Hölder continuity of the solution and its derivatives. This document establishes pointwise Schauder estimates for second order "parabolic" equations of the form $$\partial_{t}u(x,t)-\sum_{i,j=1}^{m_1} a_{ij}(x,t)X_iX_ju(x,t)=f(x,t)$$ where …
In this paper we study various Hardy inequalities in the Heisenberg group $\mathbb H^n$, w.r.t. the Carnot-Carathéodory distance $δ$ from the origin. We firstly show that the optimal constant for the Hardy inequality is strictly smaller than $n^2 = (Q-2)^2/4$, where $Q$ is the homogenous dimension. Then, we prove that, independently of $n$, the Heisenberg group does not support a radial Hardy inequality, i.e., a Hardy inequality where the gradient term is replaced by its projection along $\nabla_{\mathbb H}δ$. This is in stark contrast with the Euclidean case, where the radial Hardy inequality…
By employing a Carnot parabolic maximum principle, we show existence-uniqueness of viscosity solutions to a class of equations modeled on the parabolic infinite Laplace equation in Carnot groups. We show stability of solutions within the class and examine the limit as t goes to infinity.
A classification of local asymptotic profiles and strong unique continuation properties are established for a class of fractional heat equations with a Hardy-type potential, via an Almgren-Poon monotonicity formula combined with a blow-up analysis.
We prove a Hardy-type inequality for the gradient of the Heisenberg Laplacian on open bounded convex polytopes on the first Heisenberg Group. The integral weight of the Hardy inequality is given by the distance function to the boundary measured with respect to the Carnot-Carathéodory metric. The constant depends on the number of hyperplanes, given by the boundary of the convex polytope, which are not orthogonal to the hyperplane $x_3=0$.
We consider the Hardy-Hénon parabolic equation $u_t-Δu =|x|^a |u|^{p-1}u$ with $p>1$ and $a\in {\mathbb R}$. We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial boundary value problem.
We show the existence of continuous viscosity solutions to the equation describing the flow of a graph in the Carnot group G x R according to its horizontal Gauss curvature. In doing so, we prove a comparison principle for degenerate parabolic equations of the form u_t + F(D_0u, (D_0^2u)^*) = 0 for u defined on G.
We consider the Cauchy problem for the Hardy parabolic equation $\partial_t u-Δu=|x|^{-γ}u^p$ with initial data $u_0$ singular at some point $z$. Our main results show that, if $z\neq 0$, then the optimal strength of the singularity of $u_0$ at $z$ for the solvability of the equation is the same as that of the Fujita equation $\partial_t u-Δu=u^p$. Moreover, if $z=0$, then the optimal singularity for the Hardy parabolic equation is weaker than that of the Fujita equation. We also obtain analogous results for a fractional case $\partial_t u+(-Δ)^{θ/2} u=|x|^{-γ}u^p$ with $0<θ<2$.
We study the Cauchy problem for the semilinear heat equation with the singular potential, called the Hardy-Sobolev parabolic equation, in the energy space. The aim of this paper is to determine a necessary and sufficient condition on initial data below or at the ground state, under which the behavior of solutions is completely dichotomized. More precisely, the solution exists globally in time and its energy decays to zero in time, or it blows up in finite or infinite time. The result on the dichotomy for the corresponding Dirichlet problem is also shown as a by-product via comparison principle…
Well-posedness and global existence for Hardy--Hénon parabolic equations in Herz spaces
Laboratory of Functional Analysis and Geometry of Spaces, Faculty of Mathematics and Informatics, Department of Mathematics, M’sila University（穆西拉大学数学与信息学院功能分析与空间几何实验室）
In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality \[ |\operatorname{div} (A(x,t) \nabla u) - u_t| \leq \frac{M}{|x|^{2}} |u|,\ \ \ \ \] where the coefficient matrix $A$ is Lipschitz continuous in $x$ and $t$. Our main result sharpens a previous one of Vessella concerned with the subcritical case as well as extends a recent result of one of us with Garofalo and Manna for the heat operator.
We prove some Hardy type inequalities related to quasilinear second order degenerate elliptic differential operators L_p(u):=-\nabla_L^*(\abs{\nabla_L u}^{p-2}\nabla_L u). If ϕis a positive weight such that -L_pϕ>= 0, then the Hardy type inequality c\int_Ω\frac{\abs u^p}{ϕ^p}\abs{\nabla_L ϕ}^p dξ\le \int_Ω\abs{\nabla_L u}^p dξholds. We find an explicit value of the constant involved, which, in most cases, results optimal. As particular case we derive Hardy inequalities for subelliptic operators on Carnot Groups.
The paper concentrates on the application of the following Hardy inequality \begin{equation*} \int_Ω\ |ξ(x)|^p ω_{1 }(x)dx\le \int_Ω|\nabla ξ(x)|^pω_{2 }(x)dx, \end{equation*} to the proof of existence of weak solutions to degenerate parabolic problems of the type \begin{equation*} \left\{\begin{array}{ll} u_t-div(ω_2(x)|\nabla u|^{p-2} \nabla u )= λW(x) |u|^{p-2}u& x\inΩ, u(x,0)=f(x)& x\inΩ, u(x,t)=0& x\in\partialΩ,\ t>0,\\ \end{array}\right. \end{equation*} on an open subset $Ω\subseteq\mathbb{R}^n$, not necessarily bounded, where \[W(x)\leq \min\{m,ω_1(x)\},\qquad m\in\mathbb{R}_+.\]
In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-Δ)^\fracθ{2} u=|x|^{-γ} u^p ,\quad x\in{\bf R}^N, t>0, \qquad u(0)=μ\quad \mbox{in} \quad {\bf R}^N, \] where $N\ge 1$, $0<θ\le2$, $p>1$, $γ>0$ and $μ$ is a nonnegative Radon measure on ${\bf R}^N$. Using these conditions, we attempt to identify the optimal strength of the singularity of $μ$ for the existence of solutions to this problem.
The Cauchy problem for the Hardy-Hénon parabolic equation is studied in the critical and subcritical regime in weighted Lebesgue spaces on the Euclidean space $\mathbb{R}^d$. Well-posedness for singular initial data and existence of non-radial forward self-similar solution of the problem are previously shown only for the Hardy and Fujita cases ($γ\le 0$) in earlier works. The weighted spaces enable us to treat the potential $|x|^γ$ as an increase or decrease of the weight, thereby we can prove well-posedness to the problem for all $γ$ with $-\min\{2,d\}<γ$ including the Hénon case ($γ>0$). As …
Unconditional uniqueness of Hardy--Hénon parabolic equations on Herz spaces
Graduate School of Information Science and Technology, The University of Osaka（大阪大学信息理工学研究科）；Center for Advanced Intelligence Project RIKEN（理化学研究所先进智能项目中心）
We study the Hardy-Hénon parabolic equations on $\mathbb{R}^{N}$ ($N=2, 3$) under the effect of an additive fractional Brownian noise with Hurst parameter $H>\max\left(1/2, N/4\right).$ We show local existence and uniqueness of a mid $L^{q}$-solution under suitable assumptions on $q$.
We give necessary and sufficient conditions for the existence of weak solutions of a parabolic problem corresponding to the Kolmogorov operators perturbed by a multipolar inverse square potential with respect to the Gaussian probability measure which is the unique invariant measure for Ornstein-Uhlenbeck type operators. We state the optimality of the constant and, then, the nonexistence of positive exponentially bounded solutions to the parabolic problem.
In the present paper we define quasimeromorphic mappings on homogeneous groups and study their properties. We prove an analogue of results of L. Ahlfors, R. Nevanlinna and S. Rickman, concerning the value distribution for quasimeromorphic mappings on $\mathbb H$-type Carnot groups for parabolic and hyperbolic type domains.
This paper deals with the Cauchy problem for the Hardy-Hénon equation (and its fractional analogue). Local well-posedness for initial data in the class of continuous functions with slow decay at infinity is investigated. Small data (in critical weak-Lebesgue space) global well-posedness is obtained in $C_{b}([0,\infty);L^{q_c,\infty}(\RR^{n}))$. As a direct consequence, global existence for data in strong critical Lebesgue $L^{q_c}(\RR^{n})$ follows under a smallness condition while uniqueness is unconditional. Besides, we prove the existence of self-similar solutions and examine the long time…
The aim goal of this paper is to treat the following problem \begin{equation*} \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u &=&\dyle ł\dfrac{u^{p-1}}{|x|^{ps}} & \text{ in } Ø_{T}=Ω\times (0,T), \\ u&\ge & 0 & \text{ in }\ren \times (0,T), \\ u &=& 0 & \text{ in }(\ren\setminusØ) \times (0,T), \\ u(x,0)&=& u_0(x)& \mbox{ in }Ø, \end{array}% \right. \end{equation*} where $Ω$ is a bounded domain containing the origin, $$ (-\D^s_{p})\, u(x,t):=P.V\int_{\ren} \,\dfrac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}} \,dy$$ with $1<p<N, s\in (0,1)$ and $f, u_0$ are non negative functions. The m…
In this manuscript, we prove existence of viscosity solutions to singular parabolic equations in Carnot groups. We develop the analysis by constructing appropriate deterministic games adapted to the algebraic and differential structures of Carnot groups. We point out that the proof of existence does not require comparison and it is based on an Arzela-Ascoli-type theorem.
We give a new proof of the $L^2$ version of Hardy's uncertainty principle based on calculus and on its dynamical version for the heat equation. The reasonings rely on new log-convexity properties and the derivation of optimal Gaussian decay bounds for solutions to the heat equation with Gaussian decay at a future time. We extend the result to heat equations with lower order variable coefficient.
An analogy of nonexistence result by Baras and Goldstein (1984), for the heat equation with inverse singular potential, is proved for 2mth-order linear parabolic equations with Hardy-supercritical singular potentials. Extensions to other linear and nonlinear singular PDEs are discussed.
In this paper we develop a Fefferman-Stein theorem, a Hardy-Littlewood theorem and sharp function estimations in weighted Sobolev spaces. We also provide uniqueness and existence results for second-order elliptic and parabolic partial differential systems in weighed Sobolev spaces.
In this article, we prove a weighted Hardy inequality for $1<p<+\infty$ and dimension $d\ge1$. If $p>d$, then we can deduce from our weighted Hardy inequality a Poincaré inequality. The proof of the weighted Hardy inequality is based on the method of vector fields firstly introduced by Mitidieri \cite{MR1769903}. By the same method, we show for $1<p<+\infty$ and $d\ge1$ that weighted Caffarelli-Kohn-Nirenberg inequalities hold true. As an application of our weighted Hardy inequality, we prove a non-existence result for a p-Kolmogorov parabolic equation.
In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality \[ |Δu - u_t| \leq \frac{M}{|x|^2} |u|, \] with the critical inverse square potential. Our main result sharpens a previous one of Vessella concerned with the subcritical case.
In this paper, we study both elliptic and parabolic equations in non-divergence form with singular degenerate coefficients. Weighted and mixed-norm $L_p$-estimates and solvability are established under some suitable partially weighted BMO regularity conditions on the coefficients. When the coefficients are constants, the operators are reduced to extensional operators which arise in the study of fractional heat equations and fractional Laplace equations. Our results are new even in this setting and in the unmixed case. For the proof, we establish both interior and boundary Lipschitz estimates f…
This paper is devoted mainly to the global existence problem for the two-dimensional parabolic-parabolic Keller-Segel in the full space. We derive a critical mass threshold below which global existence is ensured. Using carefully energy methods and ad hoc functional inequalities we improve and extend previous results in this direction. The given threshold is supposed to be the optimal criterion, but this question is still open. This global existence result is accompanied by a detailed discussion on the duality between the Onofri and the logarithmic Hardy-Littlewood-Sobolev inequalities that un…
Sharp Lifespan Estimates and Fujita Phenomena for Fractional Hardy-Hénon Type Parabolic Equations
Imam Abdulrahman Bin Faisal University（伊玛目阿卜杜勒拉赫曼·本·费萨尔大学）；Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University（伊玛目阿卜杜勒拉赫曼·本·费萨尔大学基础与应用科学研究中心）；Institute of Mathematics and Mathematical Modeling（数学与数学建模研究所）
In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant in the left-hand side of the inequality is optimal. As applications, we establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic problem involving the Ornstein-Uhlenbeck operator perturbed by a critical singular potential in two dimensional case, according to the size of the coefficient of the critical potential. These results can be considered as counterparts in the l…
We give general conditions to state the weighted Hardy inequality \[ c\int_{\mathbb{R}^N}\frac{φ^2} {|x|^2}dμ\leq\int_{\mathbb{R}^N}|\nabla φ|^2 dμ+C\int_{\mathbb{R}^N} φ^2dμ,\quad φ\in C_c^{\infty}(\mathbb{R}^N),\,c\leq c_{0,μ}, \] with respect to a probability measure $dμ$. Moreover, the optimality of the constant $c_{0,μ}$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential \[ Lu+Vu=\left(Δu+\frac{\nabla μ}μ\cdot \nabla u\right)+\frac{c}{|x|^2}u \] for which the existence of positive solutions to the corresponding parabolic problem can …
We consider the weighted parabolic problem of the type \begin{equation*} \begin{split} \left\{\begin{array}{ll} u_t-\mathrm{div}(ω_2(x)|\nabla u|^{p-2} \nabla u )= λω_1(x) |u|^{p-2}u,& x\inΩ, u(x,0)=f(x),& x\inΩ, u(x,t)=0,& x\in\partialΩ,\ t>0, \end{array}\right. \end{split} \end{equation*} for quite a general class of possibly unbounded weights $ ω_1,ω_2$ satisfying the Hardy-type inequality. We prove existence of a global weak solution in the weighted Sobolev spaces provided that $λ$ is smaller than the optimal constant in the inequality.
Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the singularity of solutions to linear and subcritical semilinear parabolic equations with Hardy type potentials. As a remarkable byproduct, a unique continuation property is obtained.
