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.
We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$ to the following fractional Hardy-Hénon system $$ \left\{\begin{aligned} &(-Δ)^{s_1}u(x)=|x|^αv^p(x),\ \ x\in\mathbb{R}_+^n, \\&(-Δ)^{s_2}v(x)=|x|^βu^q(x),\ \ x\in\mathbb{R}_+^n, \\&u(x)\geq 0,\ \ v(x)\geq 0,\ \ x\in\mathbb{R}_+^n, \end{aligned}\right. $$ where $0<s_1,s_2<1$, $n>2\max\{s_1,s_2\}$. Nonexistence of anti-symmetric solutions are obtained in some appropriate domains of $(p,q)$ under some corresponding assumptions of $α,β$ via the methods of moving spheres and moving planes. Particularly, for the case $s_1=s_2$, one …
We are concerned with nonexistence results for a class of quasilinear parabolic differential problems with a potential in $Ω\times(0,+\infty)$, where $Ω$ is a bounded domain. In particular, we investigate how the behavior of the potential near the boundary of the domain and the power nonlinearity affect the nonexistence of solutions. Particular attention is devoted to the special case of the semilinear parabolic problem, for which we show that the critical rate of growth of the potential near the boundary ensuring nonexistence is sharp.
We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.
In this paper, we present some suitable distributional identities of the solutions for nonhomogeneous elliptic equations involving the Hardy-Leray potentials and study qualitative properties of the solutions to the corresponding nonhomogeneous problems by the distributional identities. We address some applications on the nonexistence of some nonhomogeneous problems with the Hardy-Leray potentials and the nonexistence principle eigenvalue with some indefinite potentials.
In this paper we prove a nonexistence result for nonlinear parabolic problems with zero lower order term whose model is $$ \begin{cases} u_{t}- Δ_p u+|u|^{q-1}u=λ& \text{in}\ (0,T)\timesΩ u(0,x)=0 & \text{in}\ Ω,\\ u(t,x)=0 & \text{on}\ (0,T)\times\partialΩ, \end{cases} $$ where $Δ_p ={\rm div }(|\nabla u|^{p-2}\nabla u)$ is the usual $p$-laplace operator, $λ$ is measure concentrated on a set of zero parabolic $r$-capacity ($1<p<r$), and $q$ is large enough.
We are concerned with the nonexistence of sign-changing global weak solutions for a class of semilinear parabolic differential inequalities with convection terms in exterior domains. A weight function of the form $t^α|x|^σ$ is considered in front of the power nonlinearity. Two types of non-homogeneous boundary conditions are investigated: Neumann-type and Dirichlet-type boundary conditions. Using a unified approach, for each case, we establish sufficient criteria for the nonexistence of global weak solutions. When $α=0$, the critical exponent in the sense of Fujita is obtained. This exponent i…
We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-Δ)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first prove that all positive bounded solutions are monotone increasing along the $x_1$ direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains. To this end, we introduce seve…
In this paper, we give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains. Our methods are self-contained and new. The main ideas and key ingredients will be discussed in the next section after Theorem 1.1 for punctured domains, and after Theorem 1.4 for exterior domains. We will also explain why all the previous methods and techniques do not apply to our general setting. Let us give here a foretaste of our line of attack: Based on the imbalance between the Hard…
In the present paper, we study an inhomogeneous pseudo-parabolic equation with nonlocal nonlinearity $$u_t-kΔu_t-Δu=I^γ_{0+}(|u|^{p})+ω(x),\,\ (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ where $p>1,\,k\geq 0$, $ω(x)\neq0$ and $I^γ_{0+}$ is the left Riemann-Liouville fractional integral of order $γ\in(0,1).$ Based on the test function method, we have proved the blow-up result for the critical case $γ=0,\,p=p_c$ for $N\geq3$, which answers an {\bf open question} posed in \cite{Zhou}, and in particular when $k=0$ it improves the result obtained in \cite{Bandle}. An interesting fact is that in the cas…
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 study the following elliptic system with critical exponent: \begin{displaymath} \begin{cases}-Δu_j-\frac{λ_j}{|x|^2}u_j=u_j^{2^*-1}+\sum\limits_{k\neq j}β_{jk}α_{jk}u_j^{α_{jk}-1}u_k^{α_{kj}},\;\;x\in\R^N, u_j\in D^{1,2}(\R^N),\quad u_j>0 \;\; \hbox{in} \quad \R^N\setminus \{0\},\quad j=1,...,r.\end{cases}\end{displaymath} Here $N\geq 3, r\geq2, 2^*=\frac{2N}{N-2}, λ_j\in (0, \frac{(N-2)^2}{4})$ for all $ j=1,...,r $; $β_{jk}=β_{kj}$; \; $α_{jk}>1, α_{kj}>1,$ satisfying $α_{jk}+α_{kj}=2^* $ for all $k\neq j$. Note that the nonlinearities $u_j^{2^*-1}$ and the coupling terms all are critical…
In this paper, we primarily consider the following semilinear elliptic equation \begin{eqnarray*} \arraycolsep=1pt\left\{ \begin{array}{lll} \displaystyle -Δu= h(x,u)\quad \ &{\rm in}\ Ω,\\[1.5mm] \phantom{ -Δ} \displaystyle u\ge 0\qquad &{\rm on}\ \partialΩ, \end{array}\right. \end{eqnarray*} where $Ω$ is an exterior domain in $R^N$ with $N\ge 3$, and derive optimal nonexistence results of positive supersolution. Our argument is based on a nonexistence result of positive supersolution of a linear elliptic problem with Hardy potential. We also establish sharp nonexistence results of positive s…
In this paper, we are interested to analyze a nonlocal nonlinear parabolic equation with fractional Laplacian. We show that there are no nontrivial global weak solutions using the test function method.
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）
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 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.
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 \[W(x)\leq \min\{m,ω_1(x)\},\qquad m\in\mathbb{R}_+.\]
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 Choquard equation with a local perturbation \begin{equation*} -Δu=λu+(I_α\ast|u|^p)|u|^{p-2}u+μ|u|^{q-2}u,\ x\in \mathbb{R}^{N} \end{equation*} having prescribed mass \begin{equation*} \int_{\mathbb{R}^N}|u|^2dx=a^2. \end{equation*} For a $L^2$-critical or $L^2$-supercritical perturbation $μ|u|^{q-2}u$, we prove nonexistence, existence and symmetry of normalized ground states, by using the mountain pass lemma, the Pohožaev constraint method, the Schwartz symmetrization rearrangements and some theories of polarizations. In particular, our results cover the Hardy-Littlewood-Sobolev …
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 $q$.
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…
Inhomogeneous parabolic equations with Hardy potential and memory on the Heisenberg group
Indian Institute of Technology (BHU)（印度理工学院（贝拿勒斯 Hindu 大学））；Birla Institute of Technology and Science Pilani, Dubai Campus（比拉理工科学研究所 pilani 迪拜校区）；Nazarbayev University（纳扎尔巴耶夫大学）
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…
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.
\begin{abstract} We state the following weighted Hardy inequality \begin{equation*} c_{o, μ}\int_{{\R}^N}\frac{φ^2 }{|x|^2}\, dμ\le \int_{{\R}^N} |\nablaφ|^2 \, dμ+ K \int_{\R^N}φ^2 \, dμ\quad \forall\, φ\in H_μ^1 %\qquad c\le c_μ, \end{equation*} in the context of the study of the Kolmogorov operators \begin{equation*} Lu=Δu+\frac{\nabla μ}μ\cdot\nabla u \end{equation*} perturbed by inverse square potentials and of the related evolution problems. The function $μ$ in the drift term is a probability density on $\R^N$. We prove the optimality of the constant $c_{o, μ}$ and state existence and no…
