We study the existence and uniqueness of mild and strong solutions of nonlocal nonlinear diffusion problems of $p$-Laplacian type with nonlinear boundary conditions posed in metric random walk spaces. These spaces include, among others, weighted discrete graphs and $\mathbb{R}^N$ with a random walk induced by a nonsingular kernel. We also study the case of nonlinear dynamical boundary conditions. The generality of the nonlinearities considered allow us to cover the nonlocal counterparts of a large scope of local diffusion problems: Stefan problems, Hele-Shaw problems, diffusion in porous media problems, obstacle problems, and more. Nonlinear semigroup theory is the basis for this study.
We establish Harnack's estimates for positive weak solutions to a mixed local and nonlocal doubly nonlinear parabolic equation. All results presented in this paper are provided together with quantitative estimates.
We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Hölder inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions.
The theory of elliptic equations involving singular nonlinearities is well studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $(P_λ)$(See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of non-smooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.
We study the existence, bifurcations, and stability of stationary solutions for the doubly-nonlocal Fisher-KPP equation. We prove using Lyapunov-Schmidt reduction that under suitable conditions on the parameters, a bifurcation from the non-trivial homogeneous state can occur. The kernel of the linearized operator at the bifurcation is two-dimensional and periodic stationary patterns are generated. Then we prove that these patterns are, again under suitable conditions, locally asymptotically stable. We also compare our results to previous work on the nonlocal Fisher-KPP equation containing a lo…
The purpose of this article is twofold. First, an issue of regularity of weak solution to the problem $(P)$ (See below) is addressed. Secondly, we investigate the question of $H^s$ versus $C^0$- weighted minimizers of the functional associated to problem $(P)$ and then give applications to existence and multiplicity results.
In this paper, we investigate the asymptotic dynamics of Fisher-KPP equations with nonlocal dispersal operator and nonlocal reaction term in time periodic and space heterogeneous media. We first show the global existence and boundedness of nonnegative solutions, and then obtain some sufficient conditions ensuring the uniform persistence. In particular, we study the existence, uniqueness and global stability of positive time periodic solutions under several different conditions.
We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional $p$-Laplace equations which includes the fractional parabolic $p$-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results …
This work is devoted to the study of the existence of solutions to nonlocal equations involving the fractional Laplacian. These equations have a variational structure and we find a nontrivial solution for them using the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. In addition, we require rather general assumptions on the local operator. As far as we know, this result is new and represent a fractional version of a classical theorem obtained working with Laplacian equations.
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 prove existence of variational solutions for a class of doubly nonlinear nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u^m &+ \text{P.V.}\int_{\mathbb{R}^N} \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\&+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qr}} \,dy = 0,\,m>0,\,p>1,\,s,r\in (0,1). \end{align*} We make use of the approach of minimizing movements pioneered by DeGiorgi and Ambrosio and refined by Bögelein, Duzaar, Marcellini, and co-authors to study nonlinear parabolic equations with non-standard growth.
The nonlinear and nonlocal PDE $$ |v_t|^{p-2}v_t+(-Δ_p)^sv=0 \, , $$ where $$ (-Δ_p)^s v\, (x,t)=2 \,\text{PV} \int_{\mathbb{R}^n}\frac{|v(x,t)-v(x+y,t)|^{p-2}(v(x,t)-v(x+y,t))}{|y|^{n+sp}}\, dy, $$ has the interesting feature that an associated Rayleigh quotient is non-increasing in time along solutions. We prove the existence of a weak solution of the corresponding initial value problem which is also unique as a viscosity solution. Moreover, we provide Hölder estimates for viscosity solutions and relate the asymptotic behavior of solutions to the eigenvalue problem for the fractional $p$-Lap…
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…
Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations
This article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{ \begin{split} (-Δ)^su &= λ|u|^{q-2}u + \left(\int_Ω\frac{|v(y)|^{2^*_μ}}{|x-y|^μ}~\mathrm{d}y\right) |u|^{2^*_μ-2}u\; \text{in}\; Ω (-Δ)^sv &= δ|v|^{q-2}v + \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}~\mathrm{d}y \right) |v|^{2^*_μ-2}v \; \text{in}\; Ω u &=v=0\; \text{in}\; \mb R^n\setminusΩ, \end{split} \right. \end{equation*} where $Ω$ is a smooth bounded dom…
We study traveling waves for a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions. We describe relations between speeds and asymptotic of profiles of traveling waves, and prove the uniqueness of the profiles up to shifts.
We consider a Fisher-KPP-type equation, where both diffusion and nonlinear part are nonlocal, with anisotropic probability kernels. Under minimal conditions on the coefficients, we prove existence, uniqueness, and uniform space-time boundedness of the positive solution. We investigate existence, uniqueness, and asymptotic behavior of monotone traveling waves for the equation. We also describe the existence and main properties of the front of propagation.
We study propagation over $\mathbb{R}^d$ of the solution to a nonlocal nonlinear equation with anisotropic kernels, which can be interpretted as a doubly nonlocal reaction-diffusion equation of the Fisher--KPP-type. We prove that if the kernel of the nonlocal diffusion is exponentially integrable in a direction and if the initial condition decays in this direction faster than any exponential function, then the solution propagates at most linearly in time in that direction. Moreover, if both the kernel and the initial condition have the above properties in any direction (being, in general, anis…
In this paper we study the Cauchy problem for doubly dissipative elastic waves in two space dimensions, where the damping terms consist of two different friction or structural damping. We derive energy estimates and diffusion phenomena with different assumptions on initial data. Particularly, we find the dominant influence on diffusion phenomena by introducing a new threshold of diffusion structure.
We consider a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions in a space of bounded functions on $\mathbb{R}^d$. Using the properties of the corresponding semiflow, we prove the existence of monotone traveling waves along those directions where the diffusion kernel is exponentially integrable. Among other properties, we prove continuity, strict monotonicity and exponential integrability of the traveling wave profiles.
We study a class of nonlocal, energy-driven dynamical models that govern the motion of closed, embedded curves from both an energetic and dynamical perspective. Our energetic results provide a variety of ways to understand physically motivated energetic models in terms of more classical, combinatorial measures of complexity for embedded curves. This line of investigation culminates in a family of complexity bounds that relate a rather broad class of models to a generalized, or weighted, variant of the crossing number. Our dynamic results include global well-posedness of the associated partial …
This paper is concerned with the critical sharp traveling wave for doubly nonlinear diffusion equation with time delay, where the doubly nonlinear degenerate diffusion is defined by $\Big(\big|(u^m)_x\big|^{p-2}(u^m)_x\Big)_x$ with $m>0$ and $p>1$. The doubly nonlinear diffusion equation is proved to admit a unique sharp type traveling wave for the degenerate case $m(p-1)>1$, the so-called slow-diffusion case. This sharp traveling wave associated with the minimal wave speed $c^*(m,p,r)$ is monotonically increasing, where the minimal wave speed satisfies $c^*(m,p,r)<c^*(m,p,0)$ for any time del…
We advance a variational method to prove qualitative properties such as symmetries, monotonicity, upper and lower bounds, sign properties, and comparison principles for a large class of doubly-nonlinear evolutionary problems including gradient flows, some nonlocal problems, and systems of nonlinear parabolic equations. Our method is based on the so-called Weighted-Energy-Dissipation (WED) variational approach. This consists in defining a global parameter-dependent functional over entire trajectories and proving that its minimizers converge to solutions to the target problem as the parameter go…
In this paper, we consider the inverse reaction coefficient problems (IRCPs) for nonlocal diffusion equation and multi-term time-fractional nonlocal diffusion equation from the average nonlocal flux data in external reaction region. Based on the nonlocal maximum principle we established, the uniqueness theorem for IRCPs are proved.
We study front propagation phenomena for a large class of nonlocal KPP-type reaction-diffusion equations in oscillatory environments, which model various forms of population growth with periodic dependence. The nonlocal diffusion is an anisotropic integro-differential operator of order $α\in (0,2)$.
In this paper, extending previous results of \cite{J1}, we obtain pointwise nonlinear stability of periodic traveling reaction-diffusion waves, assuming spectral linearized stability, under nonlocalized perturbations. More precisely, we establish pointwise estimate of nonlocalized modulational perturbation under a small initial perturbation consisting of a nonlocalized modulation plus a localized perturbation decaying algebraically.
We give an application of the Crandall-Rabinowitz theorem on local bifurcation to a system of nonlinear parabolic equations with nonlocal reaction and cross-diffusion terms as well as nonlocal initial conditions. The system arises as steady-state equations of two interacting age-structured populations.
This work addresses the regularity of solutions for a nonlocal diffusion equation over the space of periodic distributions. The spatial operator for the nonlocal diffusion equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular as well as distributional solutions of the nonlocal diffusion equation, integrable as well as singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kern…
The famous Fisher-KPP reaction diffusion model combines linear diffusion with the typical Fisher-KPP reaction term, and appears in a number of relevant applications. It is remarkable as a mathematical model since, in the case of linear diffusion, it possesses a family of travelling waves that describe the asymptotic behaviour of a wide class solutions $0\leq u(x,t)\leq 1$ of the problem posed in the real line. The existence of propagation wave with finite speed has been confirmed in the cases of "slow" and "pseudo-linear" doubly nonlinear diffusion too, see arXiv:1601.05718. We investigate her…
We investigate the differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion at either supercritical or critical type cases. Under the suitable conditions on the drift velocity and the forcing term in terms of the spatial Hölder regularity, we prove that the vanishing viscosity solution is differentiable with some Hölder continuous derivatives for any positive time.
We introduce a spatially extended transport kinetic FitzHugh-Nagumo model with forced local interactions and prove that its hydrodynamic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. Our approach is based on a relative entropy method, where the macroscopic quantities of the kinetic model are compared with the solution to the nonlocal reaction-diffusion system. This approach allows to make the rigorous link between kinetic and reaction-diffusion models.
We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions.
In this paper we consider a model that involves nonlocal diffusion and a classical convective term. Using a scaling argument and a new compactness argument we obtain the first term in the asymptotic behavior of the solutions.
We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [4,8]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form $J_ε(x)=\fra…
We establish the logarithmic Bramson correction to the position of solutions to the Fisher--KPP equation with nonlocal diffusion. Solutions with step-like initial data typically resemble a front at position $c_{*} t - \frac{3}{2 λ_{*}} \log t + \mathcal{O}(1)$ for explicit constants $c_{*}$ and $λ_{*}$. However, certain singular diffusions exhibit more exotic behavior.
We construct an example of a one-dimensional parabolic integro-differential equation with nonlocal diffusion which does not have asymptotically finite-dimensional dynamics in the corresponding state space. This example is more natural in the class of evolutionary equations of parabolic type than those known earlier.
In this paper, we study the global dynamics of a general $2\times 2$ competition models with nonsymmetric nonlocal dispersal operators. Our results indicate that local stability implies global stability provided that one of the diffusion rates is sufficiently small. This paper continues the work in \cite{BaiLi2017}, where competition models with symmetric nonlocal operators are considered.
We establish a priori Lipschitz estimates for equations with mixed local and nonlocal diffusion, coercive gradient terms and unbounded right-hand side in Lebesgue spaces through an integral refinement of the Bernstein method. This relies on a nonlinear, nonlocal and variational version of the Bochner identity that involves the adjoint equation of the linearization of the initial problem.
The aim of this paper is to provide a comprehensive study of some linear nonlocal diffusion problems in metric measure spaces. These include, for example, open subsets in $\mathbb{R}^N$, graphs, manifolds, multi-structures or some fractal sets. For this, we study regularity, compactness, positiveness and the spectrum of the stationary nonlocal operator. Then we study the solutions of linear evolution nonlocal diffusion problems, with emphasis in similarities and differences with the standard heat equation in smooth domains. In particular prove weak and strong maximum principles and describe th…
We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincaré-type inequality and classification results for stable solutions, and we apply them to the study of an associated nonlocal problem. We also establish a counterexample in the corresponding framework for the fractional Laplacian.
In this paper we consider a subcritical model that involves nonlocal diffusion and a classical convective term. In spite of the nonlocal diffusion, we obtain an Oleinik type estimate similar to the case when the diffusion is local. First we prove that the entropy solution can be obtained by adding a small viscous term $μu_{xx}$ and letting $μ\to 0$. Then, by using uniform Oleinik estimates for the viscous approximation we are able to prove the well-posedness of the entropy solutions with $L^1$-initial data. Using a scaling argument and hyperbolic estimates given by Oleinik's inequality, we obt…
In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type Lévy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the sup…
We prove existence, uniqueness and regularity of solutions of nonlocal heat equations associated to anisotropic stable diffusion operators. The main features are that the right-hand side has very few regularity and that the spectral measure can be singular in some directions. The proofs require having good enough estimates for the corresponding heat kernels and their derivatives.
In this paper, we first consider two scalar nonlocal diffusion problems with a free boundary and a fixed boundary. We obtain the global existence, uniqueness and longtime behaviour of solution of these two problems. The spreading-vanishing dichotomy and sharp criteria for spreading and vanishing are established. We also prove that accelerated spreading could happen if and only if a threshold condition is violated by kernel function. Then we discuss a classical Lotka-Volterra predator-prey model with nonlocal diffusions and a free boundary which can be seen as nonlocal diffusion counterpart of …
We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by "nonlocal diffusion" instead of "local diffusion". We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria f…
We develop general heterogeneous nonlocal diffusion models and investigate their connection to local diffusion models by taking a singular limit of focusing kernels. We reveal the link between the two groups of diffusion equations which include both spatial heterogeneity and anisotropy. In particular, we introduce the notion of deciding factors which single out a nonlocal diffusion model and typically consist of the total jump rate and the average jump length. In this framework, we also discuss the dependence of the profile of the steady state solutions on these deciding factors, thus shedding…
A mutualist model with nonlocal diffusions and a free boundary is first considered. We prove that this problem has a unique solution defined $t\ge0$, and its dynamics are governed by a spreading-vanishing dichotomy. Some criteria for spreading and vanishing are also given. Of particular importance is that we find that the solution of this problem has quite rich longtime behaviors, which vary with the conditions satisfied by kernel functions and are much different from those of the counterpart with local diffusion and free boundary. At last, we extend these results to the model with nonlocal di…
