在本文中，我们首先研究对偶分数阶抛物型方程 \partial^α_t u(x,t)+(-Δ)^s u(x,t) = f(u(x,t)) in B_1(0)×R，该方程受限于外部消失条件。我们证明，对于每个 t∈R，正有界解 u(·,t) 在 R^n 的单位球内必须关于原点径向对称且严格递减。为了克服由算子 ∂^α_t+(-Δ)^s 的双重非局部性带来的挑战，引入了一些新颖的技术。然后，我们在全空间中建立齐次方程 ∂^α_t u(x,t)+(-Δ)^s u(x,t) = 0 in R^n×R 的 Liouville 定理。我们首先证明无界域中反对称函数的极值原理，从而推导出 u(x,t) 关于 x 必须是常数。接着，我们只需建立 Marchaud 分数阶方程 ∂^α_t u(t) = 0 in R 的 Liouville 定理。为了规避由算子 ∂^α_t 的非局部和单边性质引起的困难，我们引入了一些新思路和更简单的方法。我们不再扰动反对称函数，而是直接在解 u(t) 本身采用扰动技术。该方法为建立单边算子 ∂^α_t（包括更一般的 Marchaud 时间导数）的 Liouville 定理提供了一条更简洁直观的途径。
英文摘要：
In this paper, we first study the dual fractional parabolic equation \begin{equation*} \partial^α_t u(x,t)+(-Δ)^s u(x,t) = f(u(x,t))\ \ \mbox{in}\ \ B_1(0)\times\R , \end{equation*} subject to the vanishing exterior condition. We show that for each $t\in\R$, the positive bounded solution $u(\cdot,t)$ must be radially symmetric and strictly decreasing about the origin in the unit ball in $\R^n$.
To overcome the challenges caused by the dual non-locality of the operator $\partial^α_t+(-Δ)^s$, some novel techniques were introduced.
Then we establish the Liouville theorem for the homogeneous equation in the whole space \begin{equation*}\label{B} \partial^α_t u(x,t)+(-Δ)^s u(x,t) = 0\ \ \mbox{in}\ \ \R^n\times\R . \end{equation*}
We first prove a maximum principle in unbounded domains for anti-symmetric functions to deduce that $u(x,t)$ must be constant with respect to $x.$
Then it suffices for us to establish the Liouville theorem for the Marchaud fractional equation
\begin{equation*} \partial^α_t u(t) = 0\ \ \mbox{in}\ \ \R . \end{equation*} To circumvent the difficulties arising from the nonlocal and one-sided nature of the operator $\partial_t^α$, we bring in some new ideas and simpler approaches. Instead of disturbing the anti-symmetric function, we employ a perturbation technique directly on the solution $u(t)$ itself. This method provides a more concise and intuitive route to establish the Liouville theorem for one-sided operators $\partial_t^α$, including even more general Marchaud time derivatives.
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 consider Pogorelov type estimates and Liouville type theorems to parabolic $k$-Hessian equations of the form $-u_t σ_k (D^2u) =1$ in $\mathbb{R}^n\times (-\infty, 0]$. We derive that any \textbf{$k+1$-convex-monotone} solution to $-u_t σ_k (D^2u) =1$ when $u(x,0)$ satisfies a quadratic growth and $0<m_1\le -u_t\le m_2$ must be a linear function of $t$ plus a quadratic polynomial of $x$.
In this paper, we establish the gradient and Pogorelov estimates for $k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form $-u_tσ_k(λ(D^2u))=ψ(x,t,u)$. We also apply such estimates to obtain a Liouville type result, which states that any $k$-convex-monotone and $C^{4,2}$ solution $u$ to $-u_tσ_k(λ(D^2u))=1$ in $\mathbb{R}^n\times(-\infty,0]$ must be a linear function of $t$ plus a quadratic polynomial of $x$, under some growth assumptions on $u$.
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 …
Liouville theorem and sharp solvability for solutions of the parabolic Monge-Ampère equation with periodic data
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education（北京师范大学数学科学学院，教育部数学与复杂系统重点实验室）
In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm Liouville problem in a general star graph with mixed Dirichlet and Ne…
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In this paper, we first establish Hopf's lemmas for parabolic fractional equations and parabolic fractional $p$-equations. Then we derive an asymptotic Hopf's lemma for antisymmetric solutions to parabolic fractional equations. We believe that these Hopf's lemmas will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations.
In this paper, we are concerned with stable solutions to the fractional elliptic equation $$ (-Δ)^s u=e^u\mbox{ in }\mathbb R^{N}, $$ where $(-Δ)^s$ is the fractional Laplacian with $0<s<1$. We establish the nonexistence of stable solutions provided that $N<10s$. This result is optimal when $s\uparrow 1$.
We consider positive solutions of cooperative parabolic Lotka-Volterra systems with equal diffusion coefficients, in bounded and unbounded domains. The systems are complemented by the Dirichlet or Neumann boundary conditions. Under suitable assumptions on the coefficients of the reaction terms, these problems possess both global solutions and solutions which blow up in finite time. We show that any solution $(u,v)$ defined on the time interval $(0,T)$ satisfies a universal estimate of the form $$u(x,t)+v(x,t)\leq C(1+t^{-1}+(T-t)^{-1}),$$ where $C$ does not depend on $x,t,u,v,T$. In particular…
We deal with the higher-order fractional Laplacians by two methods: the integral method and the system method. The former depends on the integral equation equivalent to the differential equation. The latter works directly on the differential equations. We first derive nonexistence of positive solutions, often known as the Liouville type theorem, for the integral and differential equations. Then through an delicate iteration, we show symmetry for positive solutions.
We prove a Liouville-type theorem for semilinear parabolic systems of the form $${\partial_t u_i}-Δu_i =\sum_{j=1}^{m}β_{ij} u_i^ru_j^{r+1}, \quad i=1,2,...,m$$ in the whole space ${\mathbb R}^N\times {\mathbb R}$. Very recently, Quittner [{\em Math. Ann.}, DOI 10.1007/s00208-015-1219-7 (2015)] has established an optimal result for $m=2$ in dimension $N\leq 2$, and partial results in higher dimensions in the range $p< N/(N-2)$. By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron, we partially improve the results of Quittner in dimensions $N\geq 3$. In particul…
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Let $0<α,β<2$ be any real number. In this paper, we investigate the following semilinear system involving the fractional Laplacian \begin{equation*} \left\{\begin{array}{lll} (-\lap)^{α/2} u(x)=f(v(x)), & (-\lap)^{β/2} v(x)=g(u(x)), & \qquad x\in\mathbb{R}^n_+, u,v\geq0, & \qquad x\in\mathbb{R}^n\setminus\mathbb{R}^n_+. \end{array}\right. \end{equation*} Applying a direct method of moving planes for the fractional Laplacian, without any decay assumption on the solutions at infinity, we prove Liouville theorems of nonnegative solutions under some natural conditions on $f$ and $g$.
We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, or on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.
We study different maximum principles for non-local non-linear operators with non-standard growth that arise naturally in the context of fractional Orlicz-Sobolev spaces and whose most notable representative is the fractional $g-$Laplacian: \[ (-Δ_g)^su(x):=\textrm{p.v.}\int_{\mathbb{R}^n}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{dy}{|x-y|^{n+s}}, \] being $g$ the derivative of a Young function. We further derive qualitative properties of solutions such as a Liouville type theorem and symmetry results and present several possible extensions and some interesting open questions. These are the…
This paper is devoted to the fractional Laplacian system with critical exponents. We use the method of moving sphere to derive a Liouville Theorem, and then prove the solutions in R^n\{0} are radially symmetric and monotonically decreasing radially. Together with blow up analysis and the Pohozaev integral, we get the upper and lower bound of the local solutions in B_1\{0}. Our results is an extension of the classical work by Caffarelli et al [6, 7], Chen et al[16]
In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \begin{equation*} u(x)=\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-α}}dy, \end{equation*} where $u: \mathbb{R}^{n} \to \mathbb{R}^{k}$ with $k \geq 1$ and $1<α<n/2$. We prove that $u \in L^2(\mathbb{R}^n) \Rightarrow u \equiv 0$ on $\mathbb{R}^n$, as long as $u$ is a bounded and differentiable solution.
We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positive smooth solutions to many special cases of the nonlinear equation. In particular, we apply gradient estimates to discuss some Yamabe-type problems of complete Riemannian manifolds and smooth metric measure spaces.
Two fractional two-phase Stefan-like problems are considered by using Riemann-Liouville and Caputo derivatives of order $α\in (0, 1)$ verifying that they coincide with the same classical Stefan problem at the limit case when $α=1$. For both problems, explicit solutions in terms of the Wright functions are presented. Even though the similarity of the two solutions, a proof that they are different is also given. The convergence when $α\nearrow 1$ of the one and the other solutions to the same classical solution is given. Numerical examples for the dimensionless version of the problem are also pr…
We obtain a new Liouville comparison principle for entire weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$ u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*) $$ in the half-space ${\mathbb S} = {\mathbb R}^1_+ \times \mathbb R^n$. Here $n\geq 1$, $q>0$ and $$ {\mathcal L}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i} [ a_{ij}(t, x) \frac{\partial}{{\partial}x_j}],$$ where $a_{ij}(t,x)$, $i,j=1,...,n$, are functions defined, measurable and locally bounded in $\mathbb S$, and such that $a_{ij}(t,x)=a_{ji}(t,x)$ …
We consider non-negative solutions to the semilinear space-fractional diffusion problem $(\partial_t+(-Δ)^{α/2})u=ρ(x)u^p$ on whole space $R^n$ with nonnegative initial data and with $(-Δ)^{α/2}$ being the $α$-Laplacian operator, $α\in (0,2)$. Here $p>0$ and $ρ(x)$ is a non-negative locally integrable function. For $ρ(x)=1$ we show that the fujita exponent is $p_F=1+\fracα{n}$ and the Liouville type result for the stationary equation is true for $0<p\leq 1+\fracα{n-α}$. When $p=1/2$ and $ρ(x)$ satisfies an integrable condition, there is at least one positive solution. This existence result is …
We establish a Liouville type theorem for the fractional Lane-Emden system: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-Δ)^αu=v^q&{\rm in}\,\,\R^N,\\ (-Δ)^αv=u^p&{\rm in}\,\,\R^N, \end{array} \right. \end{eqnarray*} where $ α\in(0,1) $, $ N>2α$ and $ p,q $ are positive real numbers and in an appropriate new range. To prove our result we will use the local realization of fractional Laplacian, which can be constructed as Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre \cite{CS}. Our proof is based on a monotonicity argument for …
In this paper, we prove Liouville type theorems for stable solutions to the weighted fractional Lane-Emden system \begin{align*} (-Δ)^s u = h(x)v^p,\quad (-Δ)^s v= h(x)u^q, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N, \end{align*} where $1<q\leq p$ and $h$ is a positive continuous function in $\mathbb{R}^N$ satisfying $\displaystyle{\liminf_{|x|\to \infty}}\frac{h(x)}{|x|^\ell} > 0$ with $\ell > 0.$ Our results generalize the results established in \cite{HHM16} for the Laplacian case (correspond to $s=1$) and improve the previous work \cite{TuanHoang21}. As a consequence, we prove classification…
In this paper, we show the nonexistence results for the Kirchhoff elliptic, parabolic, and hyperbolic type equations on the Heisenberg groups. Also, the pseudo-parabolic and pseudo-hyperbolic equations of the Kirchhoff-type are under consideration. To prove these results we use the test function method. In addition, the analogous results are transferred to the cases of systems. Also, we give some examples of non-local nonlinearities.
In this paper, we establish the following Liouville theorem for fractional \emph{p}-harmonic functions. {\em Assume that $u$ is a bounded solution of $$(-\lap)^s_p u(x) = 0, \;\; x \in \mathbb{R}^n,$$ with $0<s<1$ and $p \geq 2$. Then $u$ must be constant.} A new idea is employed to prove this result, which is completely different from the previous ones in deriving Liouville theorems. For any given hyper-plane in $\mathbb{R}^n$, we show that $u$ is symmetric about the plane. To this end, we established a {\em maximum principle} for anti-symmetric functions on any half space. We believe that th…
In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$ (-Δ)^s u= h(x_N)f(u) \quad \hbox{in }\mathbb{R}^{N} $$ where $(-Δ)^s$ stands for the fractional Laplacian with $s\in (0,1)$, and the functions $h$ and $f$ are nondecreasing. The main feature is that the function $h$ changes sign in $\mathbb{R}$, therefore the problem is sometimes termed as indefinite. As an application we obtain a priori bounds for positive solutions of some boundary value problems, which give existence of such solutions by means of bifurcation methods.
We revisit rescaling methods for nonlinear elliptic and parabolic problems and show that, by suitable modifications, they may be used for nonlinearities that are not scale invariant even asymptotically and whose behavior can be quite far from power like. In this enlarged framework, by adapting the doubling-rescaling method from [37, 38], we show that the equivalence found there between universal estimates and Liouville theorems remains valid. In the parabolic case we also prove a Liouville type theorem for a rather large class of non scale invariant nonlinearities. This leads to a number of ne…
We investigate existence, Liouville type theorems and regularity results for the 3D stationary and incompressible fractional Navier-Stokes equations: in this setting the usual Laplacian is replaced by its fractional power $(-Δ)^{\fracα{2}}$ with $0<α<2$. By applying a fixed point argument, weak solutions can be obtained in the Sobolev space $\dot{H}^{\fracα{2}}(\mathbb{R})$ and if we add an extra integrability condition, stated in terms of Lebesgue spaces, then we can prove for some values of $α$ that the zero function is the unique smooth solution. The additional integrability condition is al…
We obtain a new Liouville comparison principle for weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*)$$ in the whole space ${\mathbb E} = {\mathbb R} \times \mathbb R^n$. Here, $n\geq 1$, $q>0$ and $$ {\mathcal L}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i}\left [ a_{ij}(t, x) \frac{\partial}{{\partial}x_j}\right],$$ where $a_{ij}(t,x)$, $i,j=1,\dots,n$, are functions that are defined, measurable and locally bounded in $\mathbb E$, and such that $a_{ij}(t,x)=a_…
We introduce new lower bounds for the fractional Fisher information. Equipped with these bounds we study a hyperbolic-parabolic model of chemotaxis and prove the global existence of solutions in certain dissipation regimes.
This note is a synthesis of my reflexions on some questions that have emerged during the MATRIX event "Recent Trends on Nonlinear PDEs of Elliptic and Parabolic Type" concerning the qualitative properties of solutions to some non local reaction-diffusion equations of the form L[u](x) + f (u(x)) = 0, for x $\in$ R n \ K, where K $\subset$ R N is a bounded smooth compact "obstacle", L is non local operator and f is a bistable nonlinearity. When K is convex and the nonlocal operator L is a continuous operator of convolution type then some Liouville-type results for solutions satisfying some asymp…
We study a nonlocal nonlinear parabolic problem with a fractional time derivative. We prove a Krylov-Safonov type result; mainly, we prove Holder regularity of solutions. Our estimates remain uniform as the order of the fractional time derivative approaches 1.
We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi-Nash-Moser Holder regularity theorem for solutions in a divergence form equation. We also prove results regarding existence, uniqueness, and higher regularity in time.
