In this paper we study the asymptotic behavior of solutions to systems of strongly coupled integral equations with oscillatory coefficients. The system of equations is motivated by a peridynamic model of the deformation of heterogeneous media that additionally accounts for short-range forces. We consider the vanishing nonlocality limit on the same length scale as the heterogeneity and show that the system's effective behavior is characterized by a coupled system of local equations that are elliptic in the sense of Legendre-Hadamard. This effective system is characterized by a fourth-order tensor that shares properties with Cauchy elasticity tensors that appear in the classical equilibrium equations for linearized elasticity.
This paper is concerned with the global stability of non-critical/critical traveling waves with oscillations for time-delayed nonlocal dispersion equations. We first theoretically prove that all traveling waves, especially the critical oscillatory traveling waves, are globally stable in a certain weighted space, where the convergence rates to the non-critical oscillatory traveling waves are time-exponential, and the convergence to the critical oscillatory traveling waves are time-algebraic. Both of the rates are optimal. The approach adopted is the weighted energy method with the fundamental s…
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 $α\in (0,2)$.
We investigate asymptotic behavior of solutions for nonlocal elliptic boundary value problems in plane angles and in ${\mathbb R}^2\backslash\{0\}$. Such problems arise as model ones when studying asymptotics of solutions for nonlocal elliptic problems in bounded domains. We obtain explicit formulas for the asymptotic coefficients in terms of eigenvectors and associated vectors of both adjoint nonlocal operators (acting in spaces of distributions) and formally adjoint (with respect to the Green formula) nonlocal problems.
The paper is devoted to the study of asymptotic behavior of solutions for nonlocal elliptic problems in weighted spaces. We deal with the most difficult case where the support of nonlocal terms intersects with the boundary of a plane bounded domain. In this situation, a general form of the asymptotics is investigated and coefficients in the asymptotics are calculated.
In this paper we consider a nonlocal evolution problem and obtain by a scaling method the first term in the asymptotic behavior of the solutions. The method employed treats in different way the smooth and the rough part of the solution.
We consider a space-inhomogeneous Kolmogorov-Petrovskii-Piskunov (KPP) equation with a nonlocal diffusion and an almost-periodic nonlinearity. By employing and adapting the theory of homogenization, we show that solutions of this equation asymptotically converge to its stationary states in regions of space separated by a front that is determined by a Hamilton-Jacobi variational inequality.
We consider a space-inhomogeneous Kolmogorov-Petrovskii-Piskunov (KPP) equation with a nonlocal diffusion and a stationary ergodic nonlinearity. By employing and adapting the theory of stochastic homogenization, we show that solutions of this equation asymptotically converge to its stationary states in regions of space separated by a front that is determined by a Hamilton-Jacobi variational inequality.
In this paper we study the asymptotic behaviour of a nonlocal nonlinear parabolic equation governed by a parameter. After giving the existence of unique branch of solutions composed by stable solutions in stationary case, we gives for the parabolic problem $L^\infty $ estimates of solution based on using the Moser iterations and existence of global attractor. We finish our study by the issue of asymptotic behaviour in some cases when $t\to \infty$.
In this paper we study the asymptotic behaviour of a nonlocal nonlinear parabolic equation governed by a parameter. After giving the existence of unique branch of solutions composed by stable solutions in stationary case, we gives for the parabolic problem $L^\infty $ estimates of solution based on using the Moser iterations and existence of global attractor. We finish our study by the issue of asymptotic behaviour in some cases when $t\to \infty$.
The fractional heat operator $(\partial_t-Δ_x)^s$ and Continuous Time Random Walks (CTRWs) are interesting and sophisticated mathematical models that can describe complex anomalous systems. In this paper, we prove asymptotic mean value representation formulas for functions with respect to $(\partial_t-Δ_x)^s$ and we introduce new nonlocal, nonlinear parabolic operators related to a tug-of-war which accounts for waiting times and space-time couplings. These nonlocal, nonlinear parabolic operators and equations can be seen as nonlocal versions of the evolutionary infinity Laplace operator.
In this paper the study of a nonlocal second order Cahn-Hilliard-type singularly perturbed family of functions is undertaken. The kernels considered include those leading to Gagliardo fractional seminorms for gradients. Using Gamma convergence the integral representation of the limit energy is characterized leading to an anisotropic surface energy on interfaces separating different phases.
We prove existence of solutions and study the nonlocal-to-local asymptotics for nonlocal, convective, Cahn-Hilliard equations in the case of a W^{1,1} convolution kernel and under homogeneous Neumann conditions. Any type of potential, possibly also of double-obstacle or logarithmic type, is included. Additionally, we highlight variants and extensions to the setting of periodic boundary conditions and viscosity contributions, as well as connections with the general theory of evolutionary convergence of gradient flows.
We consider the nonlinear eigenvalue problem $[D(u(t))u(t)']' + λg(u(t)) = 0$, $u(t) > 0$, $t \in I := (0,1)$, $u(0) = u(1) = 0$, which comes from the porous media type equation. Here, $D(u) = pu^{2n} + \sin u$ ($n \in \mathbb{N}$, $p > 0$: given constants), $g(u) = u$ or $g(u) = u + \sin u$. $λ> 0$ is a bifurcation parameter which is a continuous function of $α= \Vert u_λ\Vert_\infty$ of the solution $u_λ$ corresponding to $λ$, and is expressed as $λ= λ(α)$. Since our equation contains oscillatory term in diffusion term, it seems significant to study how this oscillatory term gives effect to …
In this paper we study the asymptotic behavior as time goes to infinity of the solution to a nonlocal diffusion equation with absorption modeled by a powerlike reaction $-u^p$, $p>1$ and set in $\R^N$. We consider a bounded, nonnegative initial datum $u_0$ that behaves like a negative power at infinity. That is, $|x|^αu_0(x)\to A>0$ as $|x|\to\infty$ with $0<α\le N$. We prove that, in the supercritical case $p>1+2/α$, the solution behaves asymptotically as that of the heat equation --with diffusivity $\a$ related to the nonlocal operator-- with the same initial datum.
This paper concerns with existence, uniqueness and asymptotic behavior of the solutions for a nonlocal coupled system of reaction-diffusion. We prove the existence and uniqueness of weak solutions by the Faedo-Galerkin method and exponential decay of solutions by the classic energy method. We improve the results obtained by Chipot-Lovato and Menezes for coupled systems. A numerical scheme is presented.
We study a nonlocal evolution equation that involves a pseudo-parabolic third-order term. The equation models almost uni-directional two-phase flow in Brinkman regimes. We prove the existence of weak solutions for this equation. We also give a series of numerical examples that demonstrate the ability of the equation to support overshooting and explore the behavior of solutions in various limit regimes.
Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well known equivalence between harmonic functions and mean value properties. In the nonlocal setting of fractional harmonic functions, such an equivalence still holds, and many applications are now-days available. The nonlinear case, corresponding to the $p$-Laplace operator, has also been recently investigated, whereas the validity of a nonlocal, nonlinear, counterpart remains an open problem. In this paper, we propose a formula for the \emph{…
In this paper, we study the asymptotic behavior of the solutions of nonlocal bistable reaction-diffusion equations starting from compactly supported initial conditions. Depending on the relationship between the nonlinearity, the interaction kernel and the diffusion coefficient, we show that the solutions can either: propagate, go extinct or remain pinned. We especially focus on the latter regime where solutions are pinned by thoroughly studying discontinuous ground state solutions of the problem for a specific interaction kernel serving as a case study. We also present a detailed numerical ana…
Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the …
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 investigate the existence of nonnegative solutions for the problem $$ -\mathcal{L}_{K}u+V(x)u=f(u) $$ in $\mathbb R^n$, where $-\mathcal{L}_{K}$ is a integro-differential operator with measurable kernel $K$ and $V$ is a continuous potential. Under apropriate hypothesis, we prove, using variational methods, that the above equation has solution.
In this paper, we are going to investigate Cauchy problem for nonlocal nonlinear Schrödinger equation with the initial potential $q_0(x)$ in weighted sobolev space $H^{1,1}(\mathbb{R})$, \begin{align*} iq_t(x,t)&+q_{xx}(x,t)+2σq^2(x,t)\bar q(-x,t)=0,\quadσ=\pm1,\\ q(x,0)&=q_0(x). \end{align*} We show that the solution can be represented by the solution of a Riemann-Hilbert problem (RH problem), and assuming no discrete spectrum, we majorly apply $\bar\partial$-steepest cescent descent method on analyzing the long-time asymptotic behavior of it.
We study the curl-div-system with variable coefficients and a nonlocal homogenisation problem associated with it. Using, in part refining, techniques from nonlocal $H$-convergence for closed Hilbert complexes, we define the appropriate topology for possibly nonlocal and non-periodic coefficients in curl-div systems to model highly oscillatory behaviour of the coefficients on small scales. We address curl-div systems under various boundary conditions and analyse the limit of the ratio of small scale over large scale tending to zero. Already for standard Dirichlet boundary conditions and local c…
In this paper, we consider a nonlocal advection model for two populations on a bounded domain. The first part of the paper is devoted to the existence and uniqueness of solutions and the associated semi-flow properties. Here we use the notion of solution integrated along the characteristics. Next, by proving segregation property, we construct an energy functional to investigate the asymptotic behavior of the solution. In order to get some compactness of the positive orbit, we use the narrow convergence in the space of Young measures. By using this idea, we get a description of the asymptotic b…
In this paper, we furnish van der Corput types estimates for oscillatory integrals with respect to a large parameter, where the phase is allowed to have a stationary point of real order and the amplitude to have an integrable singularity. The resulting estimates show explicitly the influence of these two particular points on the decay. These results are then applied to the solutions of a family of dispersive equations whose generators are Fourier multipliers. We explore the effect of a limitation to compact frequency bands and of singular frequencies of the initial condition on the decay. Unif…
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.
This work investigates a nonlocal sinh-Gordon equation with a singularly perturbed parameter in a ball. Under the Robin boundary condition, the solution asymptotically forms a quite steep boundary layer in a thin annular region, and rapidly becomes a flat curve outside this region. {Focusing more particularly on the structure of the thin annular layer in this region, the pointwise asymptotic expansion involving the domain-size is evaluated more sharply, where the domain-size exactly appears in the second term of the asymptotic expansion.} It should be stressed that the standard argument of mat…
In this paper we analyze the asymptotic behavior of several fractional eigenvalue problems by means of Gamma-convergence methods. This method allows us to treat different eigenvalue problems under a unified framework. We are able to recover some known results for the behavior of the eigenvalues of the $p-$fractional laplacian when the fractional parameter $s$ goes to 1, and to extend some known results for the behavior of the same eigenvalue problem when $p$ goes to $\infty$. Finally we analyze other eigenvalue problems not previously covered in the literature.
We derive damping estimates and asymptotics of $L^p$ operator norms for oscillatory integral operators with finite type singularities. The methods are based on incorporating finite type conditions into $L^2$ almost orthogonality technique of Cotlar-Stein.
In this paper, we study the asymptotic behavior as $\varepsilon\to0^+$ of solutions $u\_\varepsilon$ to the nonlocal stationary Fisher-KPP type equation$$ \frac{1}{\varepsilon^m}\int\_{\mathbb{R}^N}J\_\varepsilon(x-y)(u\_\varepsilon(y)-u\_\varepsilon(x))\mathrm{d}y+u\_\varepsilon(x)(a(x)-u\_\varepsilon(x))=0\text{ in }\mathbb{R}^N, $$where $\varepsilon>0$ and $0\leq m<2$. Under rather mild assumptions and using very little technology, we prove that there exists one and only one positive solution $u\_\varepsilon$ and that $u\_\varepsilon\to a^+$ as $\varepsilon\to0^+$ where $a^+=\max\{0,a\}$. T…
The aim of this paper is to study a concentration-compactness principle for inhomogeneous fractional Sobolev space $H^s (\mathbb{R}^N)$ for $0<s\leq N/2.$ As an application we establish Palais-Smale compactness for the Lagrangian associated to the fractional Schrödinger equation $(-Δ)^{s} u + a(x)u= f(x,u)$ for $0<s<1.$ Moreover, we prove the existence of nontrivial nonnegative solutions to this class of elliptic equations for a wide class of possible singular potentials $a(x)$; not necessarily bounded away from zero. We consider possible oscillatory nonlinearities and that may not satisfy the…
We consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting the family of fractional interaction kernels $|z|^{-n-s}$, with $s\in(0,1)$ and $n$ the dimension of the ambient space. The fractional Young's law (contact angle condition) predicted by these models coincides, in the limit as $s\to 1^-$, with the classical Young's law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for $s$ close to $1$, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion coefficient $σ$ is…
We study the initial value problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with symmetric boundary conditions: $q(x,t)\to Ae^{2iA^2t}$ as $x\to\pm\infty$, where $A>0$ is an arbitrary constant. We describe the asymptotic stage of modulation instability for the NNLS equation by computing the large-time asymptotics of the solution $q(x,t)$ of this initial value problem. We shown that it exhibits a non-universal, in a sense, behavior: the asymptotics of $|q(x,t)|$ depends on details of the initial data $q(x,0)$. Th…
We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K, which might be singular. In the first part of the paper, we show that these functionals are indeed perimeters in a generalised sense and we establish existence of minimisers for the corresponding Plateau problem. Also, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe flat boundary conditions. A Gamma-convergence result is discussed in the second …
This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen-Cahn equation in a bounded domain $Ω\subset\mathbb{R}^n$, with or without a source term in the right hand side of the equation (commonly called chemical potential). Compare to the usual Allen-Cahn equation, the Laplace operator is here replaced by the fractional Laplacian $(-Δ)^s$ with $s\in(0,1/2)$, as defined in Fourier space. In the singular limit $\varepsilon\to 0$, we show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to surf…
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 …
We study the nonlocal scalar field equation with a vanishing parameter \[ \left\{\begin{array}{lll} (-Δ)^s u+εu &=|u|^{p-2}u -|u|^{q-2}u \quad\text{in}\quad\mathbb{R}^N \\ u >0, & u \in H^s(\mathbb{R}^N), \end{array} \right. \] where $s\in(0,1)$, $N>2s$, $q>p>2$ are fixed parameters and $ε>0$ is a vanishing parameter. For $ε>0$ small, we prove the existence of a ground state solution and show that any positive solution of above problem is a classical solution and radially symmetric and symmetric decreasing. We also obtain the decay rate of solution at infinity. Next, we study the asymptotic be…
We obtain an asymptotic representation formula for harmonic functions with respect to a linear anisotropic nonlocal operator. Furthermore we get a Bourgain-Brezis-Mironescu type limit formula for a related class of anisotropic nonlocal norms.
We study local (the heat equation) and nonlocal (convolution type problems with an integrable kernel) evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions to both local and nonlocal problems is given by the solution of the heat equation, but on a star shaped graph in which there is only one node and as many infinite edges as in the original graph. In this way we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single …
