Blow-up phenomena in nonlocal eigenvalue problems: when theories of $L^1$ and $L^2$ meet
Hardy Chan, David Gómez-Castro, Juan Luis Vázquez
备注：
The explicit constant in equation (2.3) (which is deduced in Lemma B.1) has been corrected from $Γ(s+1)^2$ to $Γ(s+1) Γ(s)$. The proof of Lemma B.1 has been slightly modified accordingly. It does not affect any of the other results in the text
We develop a linear theory of very weak solutions for nonlocal eigenvalue problems $\mathcal L u = λu + f$ involving integro-differential operators posed in bounded domains with homogeneous Dirichlet exterior condition, with and without singular boundary data. We consider mild hypotheses on the Green's function and the standard eigenbasis of the operator. The main examples in mind are the fractional Laplacian operators.
Without singular boundary datum and when $λ$ is not an eigenvalue of the operator, we construct an $L^2$-projected theory of solutions, which we extend to the optimal space of data for the operator $\mathcal L$. We present a Fredholm alternative as $λ$ tends to the eigenspace and characterise the possible blow-up limit. The main new ingredient is the transfer of orthogonality to the test function.
We then extend the results to singular boundary data and study the so-called large solutions, which blow up at the boundary. For that problem we show that, for any regular value $λ$, there exist "large eigenfunctions" that are singular on the boundary and regular inside. We are also able to present a Fredholm alternative in this setting, as $λ$ approaches the values of the spectrum.
We also obtain a maximum principle for weighted $L^1$ solutions when the operator is $L^2$-positive. It yields a global blow-up phenomenon as the first eigenvalue is approached from below.
Finally, we recover the classical Dirichlet problem as the fractional exponent approaches one under mild assumptions on the Green's functions. Thus "large eigenfunctions" represent a purely nonlocal phenomenon.
In this paper we consider a singular nonlocal viscoelastic problem with a nonlinear source term and a possible damping term. We proved that if the initial data enter into the stable set, the solution exists globally and decays to zero with a more general rate, and if the initial data enter into the unstable set, the solution with non-positive initial energy as well as positive initial energy blows up in finite time. These are achieved by using the potential well theory, the modified convexity method and the perturbed energy method.
In this paper we consider initial boundary value problem for nonlinear nonlocal parabolic equation with absorption under nonlinear nonlocal boundary condition and nonnegative initial datum. We prove comparison principle, global existence and blow-up of solutions.
In this article, we study the local existence of solutions for a wave equation with a nonlocal in time nonlinearity. Moreover, a blow-up results are proved under some conditions on the dimensional space, the initial data and the nonlinear forcing term.
We consider a family of non-local problems that model the effects of transport and vortex stretching in the incompressible Euler equations. Using modulation techniques, we establish stable self-similar blow-up near a family of known self-similar blow-up solutions.
In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence results. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.
In this paper, we consider the solution of a nonlocal parabolic equation. Focusing on the solutions with initial data at high energy level, we find the criteria for global existence and finite time blow up for the corresponding solution respectively. Moreover, we prove that there always exists blow up solution with negative Nehari functional no matter how large the energy is.
In this paper we consider a semilinear parabolic equation with nonlinear and nonlocal boundary condition and nonnegative initial datum. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given.
This paper is devoted to the analysis of non-negative solutions for the chemotaxis model with nonlocal source in bounded domain. The qualitative behavior of solutions is determined by the nonlinearity from the aggregation and the reaction. The nonlocal nonlinear Fisher-KPP reaction helps preventing blow-up phenomena in chemotaxis system. For appropriately chosen exponents, the global existence of classical solutions is proved with arbitrary initial data, where a modified Moser-Alikakos iteration method plays a key role in the a priori estimates.
A time-space fractional reaction-diffusion equation in a bounded domain is considered. Under some conditions on the initial data, we show that solutions may experience blow-up in a finite time. However, for realistic initial conditions, solutions are global in time. Moreover, the asymptotic behavior of bounded solutions is analysed.
In this paper, we consider a nonlocal parabolic equation associated with initial and Dirichlet boundary conditions. Firstly, we discuss the vacuum isolating behavior of solutions with the help of a family of potential wells. Then we obtain a threshold of global existence and blow up for solutions with critical initial energy. Furthermore, for those solutions satisfy $J(u_0)\leq d$ and $I(u_0)\neq 0$, we show that global solutions decay to zero exponentially as time tends to infinity and the norm of blow-up solutions increase exponentially.
In this article we study global existence and blow-up of solutions for a general class of nonlocal nonlinear wave equations with power-type nonlinearities, $u_{tt}-Lu_{xx}=B(- |u|^{p-1}u)_{xx}, ~(p>1)$, where the nonlocality enters through two pseudo-differential operators $L$ and $B$. We establish thresholds for global existence versus blow-up using the potential well method which relies essentially on the ideas suggested by Payne and Sattinger. Our results improve the global existence and blow-up results given in the literature for the present class of nonlocal nonlinear wave equations and c…
Blow-up analysis and extremal functions for nonlocal interaction functionals in dimension $N$
Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro（巴里阿尔多·莫罗大学数学系）；Institut für Mathematik, Goethe-Universität Frankfurt（法兰克福歌德大学数学研究所）
This paper studies the non-negative solutions of the Keller-Segel model with a nonlocal nonlinear source in a bounded domain. The competition between the aggregation and the nonlocal reaction term is highlighted: when the growth factor is stronger than the dampening effect, with the help of the nonlocal term, the model admits a classical solution which is uniformly bounded. Moreover, when the growth factor is of the same order compared to the dampening effect, the nonlocal nonlinear exponents can prevent the chemotactic collapse. Global existence of classical solutions is shown for an appropri…
We study the initial-value problem for a general class of nonlinear nonlocal coupled wave equations. The problem involves convolution operators with kernel functions whose Fourier transforms are nonnegative. Some well-known examples of nonlinear wave equations, such as coupled Boussinesq-type equations arising in elasticity and in quasi-continuum approximation of dense lattices, follow from the present model for suitable choices of the kernel functions. We establish local existence and sufficient conditions for finite time blow-up and as well as global existence of solutions of the problem.
In this paper, we develop a direct {\em blowing-up and rescaling} argument for a nonlinear equation involving the fractional Laplacian operator. Instead of using the conventional extension method introduced by Caffarelli and Silvestre, we work directly on the nonlocal operator. Using the integral defining the nonlocal elliptic operator, by an elementary approach, we carry on a {\em blowing-up and rescaling} argument directly on nonlocal equations and thus obtain a priori estimates on the positive solutions for a semi-linear equation involving the fractional Laplacian. We believe that the ideas…
This study deals with the analysis of the Cauchy problem of a general class of nonlocal nonlinear equations modeling the bi-directional propagation of dispersive waves in various contexts. The nonlocal nature of the problem is reflected by two different elliptic pseudodifferential operators acting on linear and nonlinear functions of the dependent variable, respectively. The well-known doubly dispersive nonlinear wave equation that incorporates two types of dispersive effects originated from two different dispersion operators falls into the category studied here. The class of nonlocal nonlinea…
We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \I\partial_t q(x,t)+\partial_{x}^2q(x,t)+2σq^{2}(x,t)\overline{q(-x,t)}=0 $ with initial data $q(x,0)\in H^{1,1}(\mathbb{R})$. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $H^{1,1}$ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main to…
We investigate a system of nonlocal transport equations in one spatial dimension. The system can be regarded as a model for the 3D Euler equations in the hyperbolic flow scenario. We construct blowup solutions with control up to the blowup time.
Consider a nonlocal conservation where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (a…
For a competition-diffusion blow-up system involving the fractional Laplacian of the form \begin{equation*}\label{syst1} -(-Δ)^su=uv^2,\quad-(-Δ)^sv=vu^2,\quad u,v>0 \ \mathrm{in} \ \mathbb{R}^N, \end{equation*} whith $s\in(0,1)$, we prove that the maximal asymptotic growth rate for its entire solutions is $2s$. Moreover, since we are able to construct symmetric solutions to the problem, when $N=2$ with prescribed growth arbitrarily close to the critical one, we can conclude that the asymptotic bound found is optimal. Finally, we prove existence of genuinely higher dimensional solutions, when …
In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in $S^1$,} \end{equation} where $(-Δ)^\frac{1}{2}$ stands for the fractional Laplacian and $κ$ is a bounded function. We interpret the above equation as the prescribed curvature equation to a curve in conformal parametrization. We also establish a relation between this equation and the analogous equation in $\mathbb{R}$ \begin{equation} (-Δ)^\frac{1}{2} u =Ke^u \quad \text{in }\mathbb{R}, \end{equation} with $K$ bounded on $\mathbb{R}$.
We prove global existence and blow-up of solutions of initial-boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for large values of time.
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特应用数学研究所，代尔夫特理工大学）
In this note we discuss the global dynamics of an integrable nonlocal NLS on $\mathbb{R}$, which has been the object of recent investigation by integrable systems methods. We prove two results which are in striking contrast with the case of the local cubic focusing NLS on $\mathbb{R}$. First, finite time blow-up solutions exist with arbitrarily small initial data in $H^s(\mathbb{R})$, for any $s\geqslant0$. On the other hand, the solitons of the local NLS, which are also solutions of the nonlocal equation, are unstable by blow-up for the latter.
In this paper we study a class of nonlinearities for which a nonlocal parabolic equation with Neumann-Robin boundary conditions, for $p$-Laplacian, has finite time blow-up solutions.
Long-time behaviour of a nonlocal stochastic fractional reaction--diffusion equation arising in tumour dynamics
Karlstad University（卡尔斯特德大学）；Indian Institute of Technology Bhubaneswar（布巴内斯瓦尔印度理工学院）；Indian Institute of Technology Roorkee（罗尔基印度理工学院）；University of Aegean（爱琴海大学）；Periyar University（佩里亚尔大学）
We consider a nonlocal parabolic PDE, which may be regarded as the standard semilinear heat equation with power nonlinearity, where the nonlinear term is divided by some Sobolev norm of the solution. In this paper, we are interested in constructing blowup solutions under some critical regimes. We had a complete new phenomenon of the blowup speed with a log correction term.
We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey space). Then, asymptotics of subcritical solutions is determined. These results are compared with conditions on the initial data leading to a finite time blowup.
In this paper, we consider the asymptotic behavior of the nonlocal parabolic problem \[ u_{t}=Δu+\displaystyle\frac{λf(u)}{\big(\int_Ωf(u)dx\big)^{p}}, x\in Ω, t>0, \] with homogeneous Dirichlet boundary condition, where $λ>0, p>0$, $f$ is nonincreasing. It is found that: (a) For $0<p\leq1$, $u(x,t)$ is globally bounded and the unique stationary solution is globally asymptotically stable for any $λ>0$; (b) For $1<p<2$, $u(x,t)$ is globally bounded for any $λ>0$; (c) For $p=2$, if $0<λ<2|\partialΩ|^2$, then $u(x,t)$ is globally bounded, if $λ=2|\partialΩ|^2$, there is no stationary solution and…
We study the global existence and space-time asymptotics of solutions for a class of nonlocal parabolic semilinear equations. Our models include the Nernst-Planck and the Debye-Hukel drift-diffusion systems as well as parabolic-elliptic systems of chemotaxis. In the case of a model of self-gravitating particles, we also give a result on the finite time blow up of solutions with localized and oscillating complex-valued initial data, using a method by S. Montgomery-Smith.
We investigate the global existence and blow-up of solutions to the Keller-Segel model with nonlocal reaction term $u\left(M_0-\int_{\R^2} u dx\right)$ in dimension two. By introducing a transformation in terms of the total mass of the populations to deal with the lack of mass conservation, we exhibit that the qualitative behavior of solutions is decided by a critical value $8π$ for the growth parameter $M_0$ and the initial mass $m_0$. For general solutions, if both $m_0$ and $M_0$ are less than $8π$, solutions exist globally in time using the energy inequality, whereas there are finite time …
We study a multi-dimensional nonlocal active scalar equation of the form $u_t+v\cdot \nabla u=0$ in $\mathbb R^+\times \mathbb R^d$, where $v=Λ^{-2+α}\nabla u$ with $Λ=(-Δ)^{1/2}$. We show that when $α\in (0,2]$ certain radial solutions develop gradient blowup in finite time. In the case when $α=0$, the equations are globally well-posed with arbitrary initial data in suitable Sobolev spaces.
We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated to fully nonlinear elliptic equations of order $2s$, with $s\in (1/2,1)$, and a coercive gradient term with subcritical power $0<p<2s$. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0<p<2s$, involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry-Lions for the case subquadratic case $1<p<2$.
This paper is concerned with the compactness of metrics of the disk with prescribed Gaussian and geodesic curvatures. We consider a blowing-up sequence of metrics and give a precise description of its asymptotic behavior. In particular, the metrics blow-up at a unique point on the boundary and we are able to give necessary conditions on its location. It turns out that such conditions depend locally on the Gaussian curvatures but they depend on the geodesic curvatures in a nonlocal way. This is a novelty with respect to the classical Nirenberg problem where the blow-up conditions are local, and…
In this paper, we will consider the $L^2$-critical fractional Schrödinger equation $iu_t-|D|^βu+|u|^{2β}u=0$ with initial data $u_0\in H^{β/2}(\mathbb{R})$ and $β$ close to $2$. We will show that the solution blows up in finite time if the initial data has negative energy and slightly supercritical mass. We will also give a specific description for the blow-up dynamics. This is an extension of the work of F. Merle and P. Raphaël for $L^2$-critical Schrödinger equations but the nonlocal structure of this equation and the lack of some symmetries make the analysis more complicated, hence some new…
The nonlocal nonlinear aggregation equation in one space dimension is investigated. In the so-called attractive case smooth solutions blow up in finite time, so that weak measure solutions are introduced. The velocity involved in the equation becomes discontinuous, and a particular care has to be paid to its definition as well as the formulation of the corresponding flux. When this is done, the notion of duality solutions allows to obtain global in time existence and uniqueness for measure solutions. An upwind finite volume scheme is also analyzed, and the convergence towards the unique soluti…
This paper is concerned with the analysis of the Cauchy problem of a general class of two-dimensional nonlinear nonlocal wave equations governing anti-plane shear motions in nonlocal elasticity. The nonlocal nature of the problem is reflected by a convolution integral in the space variables. The Fourier transform of the convolution kernel is nonnegative and satisfies a certain growth condition at infinity. For initial data in $L^{2}$ Sobolev spaces, conditions for global existence or finite time blow-up of the solutions of the Cauchy problem are established.
We study the existence and behaviour of blowing-up solutions to the fully fractional heat equation $$ \mathcal{M} u=u^p,\qquad x\in\mathbb{R}^N,\;0<t<T $$ with $p>0$, where $\mathcal{M}$ is a nonlocal operator given by a space-time kernel $M(x,t)=c_{N,σ}t^{-\frac N2-1-σ}e^{-\frac{|x|^2}{4t}}{1}_{\{t>0\}}$, $0<σ<1$. This operator coincides with the fractional power of the heat operator, $\mathcal{M}=(\partial_t-Δ)^σ$ defined through semigroup theory. We characterize the global existence exponent $p_0=1$ and the Fujita exponent $p_*=1+\frac{2σ}{N+2(1-σ)}$, and study the rate at which the blowing…
This paper is concerned with a class of nonlocal dispersive models -- the $θ$-equation proposed by H. Liu [ On discreteness of the Hopf equation, {\it Acta Math. Appl. Sin.} Engl. Ser. {\bf 24}(3)(2008)423--440]: $$ (1-\partial_x^2)u_t+(1-θ\partial_x^2)(\frac{u^2}{2})_x =(1-4θ)(\frac{u_x^2}{2})_x, $$ including integrable equations such as the Camassa-Holm equation, $θ=1/3$, and the Degasperis-Procesi equation, $θ=1/4$, as special models. We investigate both global regularity of solutions and wave breaking phenomena for $θ\in \mathbb{R}$. It is shown that as $θ$ increases regularity of solution…
