Motivated by a nonlocal free boundary problem, we study uniform properties of solutions to a singular perturbation problem for a boundary-reaction-diffusion equation, where the reaction term is of combustion type. This boundary problem is related to the fractional Laplacian. After an optimal uniform Hölder regularity is shown, we pass to the limit to study the free boundary problem it leads to.
Slow-fast system in Rosales-Majda combustion model with fractional order kinetics
Institut de Mathématiques de Bordeaux UMR CNRS 5251, université de Bordeaux（波尔多大学数学研究所）；School of Mathematics and Statistics, Yunnan University（云南大学数学与统计学院）；IMB UMR 5584, Université de Bourgogne Europe, CNRS（勃艮第欧洲大学数学与图像实验室）
We prove a regularity result for the unstable elliptic free boundary problem $Δu = -χ_{\{u>0\}}$ related to traveling waves in a problem arising in solid combustion. The maximal solution and every local minimizer of the energy are regular, that is, $\{u=0\}$ is locally an analytic surface and $u|_{\bar{\{u>0\}}}, u|_{\bar{\{u<0\}}}$ are locally analytic functions. Moreover we prove a partial regularity result for solutions that are non-degenerate of second order: here $\{u=0\}$ is analytic up to a closed set of Hausdorff dimension $n-2$. We discuss possible singularities.
We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation \[ (-Δ)^s u = χ_{\{u>c\}} \] for $0<s<1$ and an arbitrary constant $c$. Our main object of study is the free boundary $\partial\{u>c\}$. We study the behavior of the free boundary and prove an upper bound for the Hausdorff dimension of the singular set. We also show that when $s\leq 1/2$ certain symmetric solutions are stable; however, when $s>1/2$ these solutions are not stable and therefore not minimizers of the corresponding functional.
We prove that an a priori BMO gradient estimate for the two phase singular perturbation problem implies Lipschitz regularity for the limits. This problem arises in the mathematical theory of combustion where the reaction-diffusion is modelled by the $p$-Laplacian. A key tool in our approach is the weak energy identity. Our method proves a natural and intrinsic characterization of the free boundary points and can be applied to more general classes of solutions.
This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential.
A numerical scheme for solving fractional initial value problems involving the Atangana-Baleanu fractional derivative is presented. Some examples for the proposed method are included, both for equations and systems of fractional initial value problems.
In this paper we consider Sobolev inequalities associated with singular problems for the fractional $p$-Laplacian operator in a bounded domain of $\mathbb{R}^{N}$, $N\geq 2$.
In this paper, we prove the exact asymptotic behavior of singular positive solutions of fractional semi-linear equations $$(-Δ)^σu = u^p~~~~~~~~in ~~ B_1\backslash \{0\}$$ with an isolated singularity, where $σ\in (0, 1)$ and $\frac{n}{n-2σ} < p < \frac{n+2σ}{n-2σ}$.
This article deals with time-fractional diffusion equations with time-dependent singular source term. Whenever the order of the time-fractional derivative is either multi-term, distributed or space-dependent, we prove that the system admits a unique weak solution enjoying a Duhamel representation, provided that the time-dependence of the source term is a distribution.
In this paper we consider a system of two reaction-diffusion equations that models diffusional-thermal combustion with stepwise ignition-temperature kinetics and fractional reaction order 0 < $α$ < 1. We turn the free interface problem into a scalar free boundary problem coupled with an integral equation. The main intermediary step is to reduce the scalar problem to the study of a non-C 1 vector field in dimension 2. The latter is treated by qualitative topo-logical methods based on the Poincar{é}-Bendixson Theorem. The phase portrait is determined and the existence of a stable manifold at the…
We study a class of fractional $p$-Laplacian problems with weights which are possibly singular on the boundary of the domain. We provide existence and multiplicity results as well as characterizations of critical groups and related applications.
We consider parabolic PDEs associated with fractional type operators drifted by non-linear singular first order terms. When the drift enjoys some boundedness properties in appropriate Lebesgue and Besov spaces, we establish by exploiting a priori Besov-type estimates, the H{ö}lder continuity of the solutions. In particular, we handle the almost critical case in whole generality.
We provide fundamental properties of the first eigenpair for fractional $p$-Laplacian eigenvalue problems under singular weights, which is related to Hardy type inequality, and also show that the second eigenvalue is well-defined. We obtain a-priori bounds and the continuity of solutions to problems with such singular weights with some additional assumptions. Moreover, applying the above results, we show a global bifurcation emenating from the first eigenvalue, the Fredholm alternative for non-resonant problems, and obtain the existence of infinitely many solutions for some nonlinear problems …
We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, \ $\min\bigl\{(-Δ)^su,\,u-φ\bigr\}=0$ in $\mathbb R^n$, for general obstacles $φ$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in \cite{GP} to all $s\in(0,1)$.
In this paper we consider the space-fractional Schrödinger equation with a singular potential for a wide class of fractional hypoelliptic operators. Such analysis can be conveniently realised in the setting of graded Lie groups. The paper is a continuation and extension of a previous one where the classical Schrödinger equation on $\mathbb R^n$ with singular potentials was considered.
We study a nonlinear, nonlocal eigenvalue problem driven by the fractional p-Laplacian with an indefinite, singular weight chosen in an optimal class. We prove the existence of an unbounded sequence of positive variational eigenvalues and alternative characterizations of the first and second eigenvalues. Then, by means of such characterizations, we prove strict decreasing monotonicity of such eigenvalues with respect to the weight function.
In this paper, the space-fractional Schrödinger equations with singular potentials are studied. Delta-like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of 'weakened' solutions, calling them very weak solutions. The existence, uniqueness and consistency results are proved in an appropriate sense. Numerical simulations are done, and a particle accumulating effect is observed in the singular cases. From the mathematical point of view, a "splitting of the strong singularity" phenomena is also observed.
We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem. The main step is an infinite-dimensional Lyapunov-Schmidt reduction method, that reduces the problem to an (infinite dimensional) Toda type system.
In this paper, we study the existence of infinitely many weak solutions to a fractional Kirchhoff-Schrödinger-Poisson system involving the weak singularity, i.e. when $0<γ<1$. Further, we obtain the existence of a solution with the strong singularity, i.e. when $γ>1$. We employ variational techniques to prove the existence and multiplicity results. Moreover, a $L^{\infty}$ estimate is obtained by using the Moser iteration method.
In this paper, we study singular solutions of linear problems with fractional Laplacian. First, we establish Bôcher type theorems on a punctured ball via distributional approach. Then, we develop a few interesting maximum principles on a punctured ball. Our distributional approach only requires the basic local L-1 integrability. We also introduce several simple and useful lemmas, which enable us to unify the treatments for both Laplacian and fractional Laplacian. These theorems, lemmas and the methods introduced here can be adapted and applied in other situations.
In this paper we study the existence of a least energy sign-changing solution to a nonlocal elliptic PDE involving singularity by using the Nehari manifold method, the constraint variational method and Brouwer degree theory.
We study the formation of singularities for the Euler-Alignment system with influence function $ψ=\frac{k_α}{|x|^α}$ in 1D. As in [20] the problem is reduced to the analysis of a nonlocal 1D equation. We show the existence of singularities in finite time for any $α$ in the range $0<α<2$ in both the real line and the periodic case.
We are interested in nonlinear fractional Schrödinger equations with singular potential of form \begin{equation*} (-Δ)^su=\fracλ{|x|^α}u+|u|^{p-1}u,\quad \mathbf R^n\setminus\{0\}, \end{equation*} where $s\in (0,1)$, $α>0$, $p\ge1$ and $λ\in \mathbf R$. Via Caffarelli-Silvestre extension method, we obtain existence, nonexistence, regularity and symmetry properties of solutions to this equation for various $α$, $p$ and $λ$.
We consider a space-fractional wave equation with a singular mass term depending on the position and prove that it is very weak well-posed. The uniqueness is proved in some appropriate sense. Moreover, we prove the consistency of the very weak solution with classical solutions when they exist. In order to study the behaviour of the very weak solution near the singularities of the coefficient, some numerical experiments are conducted where the appearance of a wall effect for the singular masses of the strength of $δ^2$ is observed.
This paper is concerned with a Neumann type problem for singularly perturbed fractional nonlinear Schrödinger equations with subcritical exponent. For some smooth bounded domain $Ω\subset \mathbf R^n$, our boundary condition is given by \begin{equation*} \int_Ω\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy=0\quad\mbox{for }x\in \mathbf R^n\setminus\barΩ. \end{equation*} We establish existence of nonnegative small energy solutions, and also investigate the integrability of the solutions on $\mathbf R^n$.
The aim of this paper is to study the singular solutions to fractional elliptic equations with absorption $$ \left\{\arraycolsep=1pt \begin{array}{lll} (-Δ)^αu+|u|^{p-1}u=0,\quad & \rm{in}\quadΩ\setminus\{0\},\\[2mm] u=0,\quad & \rm{in}\quad \R^N\setminusΩ,\\[2mm] \lim_{x\to 0}u(x)=+\infty, \end{array} \right. $$ where $p>0$, $Ω$ is an open, bounded and smooth domain of $\R^N\ (N\ge2)$ with $0\inΩ$. We analyze the existence, nonexistence, uniqueness and asymptotic behavior of the solutions.
This paper is devoted to the study of the well-posedness of a singular nonlinear fractional pseudo-hyperbolic system. The fractional derivative is described in Caputo sense. The equations are supplemented by classical and nonlocal boundary conditions. Upon some a priori estimates and density arguments, we establish the existence and uniqueness of the strongly generalized solution for the associated linear fractional system in some Sobolev fractional spaces. On the basis of the obtained results for the linear fractional system, we apply an iterative process in order to establish the well-posedn…
In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the setting of graded Lie groups. The uniqueness of the very weak solution, and the consistency with the classical solution are also proved, under suitable considerations. This extends and improves the results obtained in the first part of this work which was devoted to the classical Euclidean Klein-Gordon equation.
We provide an asymptotic analysis of a non-local Fisher-KPP type equation in periodic media and with a non-local stable operator of order $α$ $\in$ (0, 2). We perform a long time-long range scaling in order to prove that the stable state invades the unstable state with a speed which is exponential in time.
We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in a previous paper of ours, and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem. We also investigate the quadratic behaviour of solutions near the zero level set, at least for small val…
We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the 'gradient' of the solution, convoluted with a singular term b. Our first result is the well-posedness for this problem: We show existence and uniqueness of a (local in time) mild solution. The main result is about blow-up of said solution, and in particular we find sufficient conditions on the initial datum and on the term b to ensure blow-up of the solution in finite time.
The standard problem for the classical heat equation posed in a bounded domain $Ω$ of $\mathbb R^n$ is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial and boundary value problem can still be solved on the condition that the non-zero boundary data must be singular, i.e., the solution $u(t,x)$ blows up as $x$ approaches $\partial Ω$ in a definite way. In this paper we construct a theory of existence and uniqueness of solutions of the parabolic problem with singular data taken in a very precise sense, and also admit…
In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^α, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om. \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > sp, s \in (0,1), \la >0, 0 < q \leq 1 $ and $α\le p^*_s-1$. We use variational methods to show the existence and multiplicity of positive solutions of above problem with respect to parameter $\la$.
In this work, we consider an initial problem for second order partial differential equations with Caputo fractional derivatives in the time-variable and Bessel operator in the space-variable. For non-local boundary conditions, we present a solution of this problem in an explicit form representing it by the Fourier-Bessel series. The obtained solution is written in terms of multinomial Mittag-Leffler functions and first kind Bessel functions.
In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation $$ (-Δ)^σu = |x|^αu^p ~~~~~~~~~~~ in ~~ B_1 \backslash \{0\} $$ with an isolated singularity at the origin, where $σ\in (0, 1)$ and the punctured unit ball $B_1 \backslash \{0\} \subset \mathbb{R}^n$ with $n \geq 2$. When $-2σ< α< 2σ$ and $\frac{n+α}{n-2σ} < p < \frac{n+2σ}{n-2σ}$, we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of…
In this article, we investigate the existence, uniqueness, nonexistence, and regularity of weak solutions to the nonlinear fractional elliptic problem of type $(P)$ (see below) involving singular nonlinearity and singular weights in smooth bounded domain. We prove the existence of weak solution in $W_{loc}^{s,p}(Ω)$ via approximation method. Establishing a new comparison principle of independent interest, we prove the uniqueness of weak solution for $0 \leq δ< 1+s- \frac{1}{p}$ and furthermore the nonexistence of weak solution for $δ\geq sp.$ Moreover, by virtue of barrier arguments we study t…
In this paper, we are concerned with the following type of fractional problems: $$ \begin{cases}\dis (-Δ)^{s} u-μ\frac{u}{|x|^{2s}}-λu=|u|^{2^*_{s}-2}u+f(x,u), &\text{in} Ω,\ \ \, u=0\,&\text{in} \R^N\backslashΩ \end{cases} \eqno {(*)} $$ where $s\in (0,1)$, $2^*_{s}=2N/(N-2s)$ is the critical Sobolev exponent, $f(x,u)$ is a lower order perturbation of critical Sobolev nonlinearity. We obtain the existence of the solution for (*) through variational methods. In particular we derive a Brézis-Nirenberg type result when $f(x,u)=0$.
In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler- Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on t…
We prove the existence of a solution of (--$Δ$) s u + f (u) = 0 in a smooth bounded domain $Ω$ with a prescribed boundary value $μ$ in the class of positive Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f (u) = u p and $μ$ is a Dirac mass, we prove the existence of several critical exponents p.
In this work we investigate positivity properties of nonlocal Schrödinger type operators, driven by the fractional Laplacian, with multipolar, critical, and locally homogeneous potentials. On one hand, we develop a criterion that links the positivity of the spectrum of such operators with the existence of certain positive supersolutions, while, on the other hand, we study the localization of binding for this kind of potentials. Combining these two tools and performing an inductive procedure on the number of poles, we establish necessary and sufficient conditions for the existence of a configur…
