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Using a modified version of Weinstein's argument for constrained minimization in nonlinear dispersive equations, we prove the existence of solitary waves in fully nonlocally nonlinear equations, as long as the linear multiplier is of positive and slightly higher order than the Coifman–Meyer nonlinear multiplier. It is therefore the relative order of the linear term over the nonlinear one that determines the method and existence for these types of equations. In analogy to Korteweg–De Vries-type equations and water waves in the capillary regime, smooth solutions of all amplitudes can be found. We consider two structural types of symmetric Coifman–Meyer symbols $ n(\xi-\eta,\eta) $, and show that cyclical symmetry is necessary for the existence of a functional formulation. Estimates for the solution and wave speed are given as the solutions tend to the bifurcation point of solitary waves. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for convolution-type nonlocal linear and nonlinear Schrödinger equations (NSEs) is studied. The equations include the general differential operators. The existence, uniqueness, Lp-regularity properties of linear problem and the existence, uniqueness, and blow-up at finite time of the nonlinear problem are obtained. By choosing differential operators including in equations, the regularity properties of a different type of nonlocal Schrödinger equation are studied. [ABSTRACT FROM AUTHOR]

We study nonlocal elliptic and parabolic equations on C1,τ open sets in weighted Sobolev spaces, where τ ∈ (0, 1). The operators we consider are infinitesimal generators of symmetric stable Lévy processes, whose Lévy measures are allowed to be very singular. Additionally, for parabolic equations, the measures are assumed to be merely measurable in the time variable. [ABSTRACT FROM AUTHOR]

In this paper we consider the homogenization of the evolution problem associated with a jump process that involves three different smooth kernels that govern the jumps to/from different parts of the domain. We assume that the spacial domain is divided into a sequence of two subdomains A n ∪ B n and we have three different smooth kernels, one that controls the jumps from A n to A n , a second one that controls the jumps from B n to B n and the third one that governs the interactions between A n and B n . Assuming that χ A n (x) → X (x) weakly in L ∞ (and then χ B n (x) → 1 - X (x) weakly in L ∞ ) as n → ∞ and that the initial condition is given by a density u 0 in L 2 we show that there is an homogenized limit system in which the three kernels and the limit function X appear. When the initial condition is a delta at one point, δ x ¯ (this corresponds to the process that starts at x ¯ ) we show that there is convergence along subsequences such that x ¯ ∈ A n j or x ¯ ∈ B n j for every n j large enough. We also provide a probabilistic interpretation of this evolution equation in terms of a stochastic process that describes the movement of a particle that jumps in Ω according to the three different kernels and show that the underlying process converges in distribution to a limit process associated with the limit equation. We focus our analysis in Neumann type boundary conditions and briefly describe at the end how to deal with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

We present a systematic study on a class of nonlocal integral functionals for functions defined on a bounded domain and the naturally induced function spaces. The function spaces are equipped with a seminorm depending on finite differences weighted by a position-dependent function, which leads to heterogeneous localization on the domain boundary. We show the existence of minimizers for nonlocal variational problems with classically defined, local boundary constraints, together with the variational convergence of these functionals to classical counterparts in the localization limit. This program necessitates a thorough study of the nonlocal space; we demonstrate properties such as a Meyers--Serrin theorem, trace inequalities, and compact embeddings, which are facilitated by new studies of boundary-localized convolution operators. [ABSTRACT FROM AUTHOR]

In this paper we prove some results on the boundary behavior of solutions to fractional elliptic problems. Firstly, we establish a Hopf Lemma for solutions to some integro-differential equations. The main novelty of our result is that we do not assume any global condition on the sign of the solutions. Secondly, we show that non-trivial radial solutions cannot have infinitely many zeros accumulating at the boundary. We provide concrete examples to show that the results obtained are sharp. [ABSTRACT FROM AUTHOR]

Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $ L^{p} $ vector fields defined on a domain $ \Omega $ that is either a bounded domain in $ \mathbb{R}^{d} $ or $ \mathbb{R}^{d} $ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $ L^{p} $ compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals.

We show that any viscosity solution to a general fully nonlinear nonlocal elliptic equation can be approximated by smooth (C^\infty) solutions. [ABSTRACT FROM AUTHOR]

Hölder estimates and Harnack inequalities are studied for fully nonlinear integro-differential equations under some mild assumptions. We allow the kernels of variable order and critically close to 2. [ABSTRACT FROM AUTHOR]

Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of L p vector fields defined on a domain Ω that is either a bounded domain in R d or R d itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The L p compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals. Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of vector fields defined on a domain that is either a bounded domain in or itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals. [ABSTRACT FROM AUTHOR]

Some equations are provided for the Variance Gamma process using the definition other than that based on a time-changed Brownian motion. A new nonlocal equation is obtained involving generalized Weyl derivatives, which is true even in the drifted case. The connection to special functions is in focus, and a space equation for the process is studied. In conclusion, the convergence in distribution of a compound Poisson process to the Variance Gamma process is observed.

Partial differential equations are a classical and very active field of research. One of its salient features is to break the rigid distinction between the evolution of the theory and the applications to real world phenomena, since the two are intimately intertwined in the harmonious development of such a fascinating and multifaceted topic of investigation.

This paper is concerned with some properties of solutions for a two-dimensional linear nonlocal peridynamic model involving a smooth truncated integrable tensor kernel. For the stationary Dirichlet-type problem with zero boundary data, the higher integrability and H 1 regularity of its weak solution are established. On the other hand, for the nonstationary problem with a local damping term 2 r u t (r > 0) , the existence results in L p , L β p and Hölder spaces are obtained via successive approximation methods, Bessel potential theory and standard ODE arguments, respectively. Further, under certain suitable assumptions on kernels, we can recover the corresponding local problem in the limit of δ → 0 and prove that the limit of nonlocal solutions is a weak solution of the local counterparts exactly. [ABSTRACT FROM AUTHOR]

We consider a class of generalized nonlocal $p$ -Laplacian equations. We find some proper structural conditions to establish a version of nonlocal Harnack inequalities of weak solutions to such nonlocal problems by using the expansion of positivity and energy estimates. [ABSTRACT FROM AUTHOR]

We study a natural alternating method of Schwarz type (domain decomposition) for a certain class of couplings between local and nonlocal operators. We show that our method fits into Lions's framework and prove, as a consequence, convergence in both the continuous and the discrete settings. [ABSTRACT FROM AUTHOR]

Some equations are provided for the Variance Gamma process using the definition other than that based on a time-changed Brownian motion. A new nonlocal equation is obtained involving generalized Weyl derivatives, which is true even in the drifted case. The connection to special functions is in focus, and a space equation for the process is studied. In conclusion, the convergence in distribution of a compound Poisson process to the Variance Gamma process is observed. [ABSTRACT FROM AUTHOR]

We study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations: P${\mathcal {P}}$(−Δ)su+u=a(x)|u|p−1u+f(x)inRNu∈Hs(RN),$$\begin{equation} \hspace*{5pc} {\left\lbrace \begin{aligned} (-\Delta)^s u + u &= a(x) |u|^{p-1}u+f(x)\;\;\text{in}\;\mathbb {R}^{N}\\ u &\in H^{s}{\big(\mathbb {R}^{N}\big)}, \end{aligned} \right.} \end{equation}$$where s∈(0,1)$s \in (0,1)$, N>2s$N>2s$, 1

We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schrödinger equation (− Δ) s u + V (x) u = u 2 s ∗ − 1 − 휖 in ℝ N , where 휖 > 0, s ∈ (0,1), 2 s ∗ : = 2 N N − 2 s and N > 4s, as we deal with finite energy solutions. We show that the ground state u휖 blows up and precisely with the following rate ∥ u 휖 ∥ L ∞ (ℝ N) ∼ 휖 − N − 2 s 4 s , as 휖 → 0 + . We also localize the concentration points and, in the case of radial potentials V, we prove local uniqueness of sequences of ground states which exhibit a concentrating behavior. [ABSTRACT FROM AUTHOR]

We introduce a mathematically rigorous formulation for a nonlocal interface problem with jumps and propose an asymptotically compatible finite element discretization for the weak form of the interface problem. After proving the well-posedness of the weak form, we demonstrate that solutions to the nonlocal interface problem converge to the corresponding local counterpart when the nonlocal data are appropriately prescribed. Several numerical tests in one and two dimensions show the applicability of our technique, its numerical convergence to exact nonlocal solutions, its convergence to the local limit when the horizons vanish, and its robustness with respect to the patch test. [ABSTRACT FROM AUTHOR]

Partial differential equations are a classical and very active field of research. One of its salient features is to break the rigid distinction between the evolution of the theory and the applications to real world phenomena, since the two are intimately intertwined in the harmonious development of such a fascinating and multifaceted topic of investigation. [ABSTRACT FROM AUTHOR]

In this paper we provide a rate of convergence for periodic homogenization of Hamilton–Jacobi–Bellman equations with nonlocal diffusion. The result is based on the regularity of the associated effective problem, where convexity plays a crucial role. The necessary regularity estimates are made possible by a representation formula we obtain for the effective Hamiltonian, a result that has an independent interest. [ABSTRACT FROM AUTHOR]

In this paper we study two different ways of coupling a local operator with a nonlocal one so that the resulting equation is related to an energy functional. In the first strategy the coupling is given via source terms in the equation, and in the second one a flux condition in the local part appears. For both models we prove existence and uniqueness of a solution that is obtained via direct minimization of the related energy functional. In the second part of this paper we extend these ideas to local/nonlocal elasticity models in which we couple classical local elasticity with nonlocal peridynamics. [ABSTRACT FROM AUTHOR]

We propose a method to couple local and nonlocal diffusion models. By inheriting desirable properties such as patch tests, asymptotic compatibility and unintrusiveness from related splice and optimization-based coupling schemes, it enables the use of weak (or variational) formulations, is computationally efficient and straightforward to implement. We prove well-posedness of the coupling scheme and demonstrate its properties and effectiveness in a variety of numerical examples. [ABSTRACT FROM AUTHOR]

Nonlocal reaction–diffusion equations describe various biological and biomedical applications. Their mathematical properties are essentially different in comparison with the local equations, and this difference can lead to important biological implications. This review will present the state of the art in the investigation of nonlocal reaction–diffusion models in biomedical applications. We will consider various models arising in mathematical immunology, neuroscience, cancer modelling, and we will discuss their mathematical properties, nonlinear dynamics, resulting spatiotemporal patterns and biological significance. [ABSTRACT FROM AUTHOR]

This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity:

In this paper we study a nonlocal reaction–diffusion equation in which the diffusion depends on the gradient of the solution. Firstly, we prove the existence and uniqueness of regular and strong solutions. Secondly, we obtain the existence of global attractors in both situations under rather weak assumptions by defining a multivalued semiflow (which is a semigroup in the particular situation when uniqueness of the Cauchy problem is satisfied). Thirdly, we characterize the attractor either as the unstable manifold of the set of stationary points or as the stable one when we consider solutions only in the set of bounded complete trajectories. [ABSTRACT FROM AUTHOR]

We study an initial value problem with fractional Laplacian and a singular integral drift term. This equation quantifies fractal interfaces in statistical mechanics. The singularity of the drift term is a generalization of existing results. Making use of some important boundedness properties of Calderón-Zygmund operator in Lp and Lipschitz spaces, we obtain local and global existence theorems. [ABSTRACT FROM AUTHOR]

The present work is devoted to the study of the solvability questions for a nonlocal problem with an integro-differential conjugation condition for a nonlocal fourth-order mixed type equation with degeneration. The case of a — order Gerasimov–Caputo type fractional operator was considered. The solution of the nonlocal mixed type differential equation was studied in the class of regular functions. The Fourier series method was used and a countable system of ordinary differential equations was obtained. When conditions of smoothness are fulfilled, then using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series was proved. If these conditions are violated, then the problem will have an infinite set of non-trivial solutions. The stability of unknown function of the considering problem with respect to the nonlocal condition was studied. [ABSTRACT FROM AUTHOR]

Here, the Cauchy problem for linear and nonlinear nonlocal Schrödinger equations are studied. The equation involves a convolution integral operators with a general kernel operator functions whose Fourier transform are operator functions defined in a Hilbert space H together with some growth conditions. By assuming enough smoothness on the initial data and the operator functions, the local and global existence and uniqueness of solutions are established. We can obtain a different classes of nonlocal Schr ödinger equations by choosing the space H and linear operators, which occur in a wide variety of physical systems [ABSTRACT FROM AUTHOR]

Fourier multiplier analysis is developed for nonlocal peridynamic-type Laplace operators, which are defined for scalar fields in R n . The Fourier multipliers are given through an integral representation. We show that the integral representation of the Fourier multipliers is recognized explicitly through a unified and general formula in terms of the hypergeometric function 2 F 3 in any spatial dimension n. Asymptotic analysis of 2 F 3 is utilized to identify the asymptotic behavior of the Fourier multipliers m (ν) as ∥ ν ∥ → ∞. We show that the multipliers are bounded when the peridynamic Laplacian has an integrable kernel and diverge when the kernel is singular. The bounds and decay rates are presented explicitly in terms of the dimension n, the integral kernel and the peridynamic Laplacian nonlocality. The asymptotic analysis is applied in the periodic setting to prove a regularity result for the peridynamic Poisson equation and, moreover, show that its solution converges to the solution of the classical Poisson equation. [ABSTRACT FROM AUTHOR]

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup ⁎ inf type Harnack inequality of Schoen for integral equations. [ABSTRACT FROM AUTHOR]

In this paper we consider the homogenization problem for a nonlocal equation that involve different smooth kernels. We assume that the spacial domain is divided into a sequence of two subdomains An ∪ Bn and we have three different smooth kernels, one that controls the jumps from An to An, a second one that controls the jumps from Bn to Bn and the third one that governs the interactions between An and Bn. Assuming that χAn (x) → X (x) weakly-* in L∞ (and then χBn (x) → (1−X) (x) weakly-* in L∞) as n → ∞ we show that there is an homogenized limit system in which the three kernels and the limit function X appear. We deal with both Neumann and Dirichlet boundary conditions. Moreover, we also provide a probabilistic interpretation of our results. [ABSTRACT FROM AUTHOR]

We study integral operators Lu(χ)=∫ℝℕψ(u(x)−u(y))J(x−y)dy$\mathcal{L}u\left( \chi \right)=\int{_{_{\mathbb{R}}\mathbb{N}}\psi \left( u\left( x \right)-u\left( y \right) \right)J\left( x-y \right)dy}$of the type of the fractional p-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincaré inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel J considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem Lu=f$\mathcal{L}u=f$in a bounded domain Ω,$\Omega ,$and boundary condition u ≡ 0 on Ωc;${{\Omega }^{c}};$both cases f = f(x) and f = f(u) are considred, including the generalized eigenvalue problem f(u)=λψ(u).$f\left( u \right)=\lambda \psi \left( u \right).$

We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an infinite-dimensional dynamical system for the fast variables is coupled to a finite-dimensional dynamical system for slow variables. We prove the existence of center-manifolds for generic models of this type, and study the reduced, finite-dimensional dynamics near bifurcations of (possibly) patterned steady states in the layer problem. Theoretical results are complemented with detailed examples and numerical simulations covering systems of local and nonlocal reaction-diffusion equations, neural field models, and delay-differential equations. We provide analytical foundations for numerical observations recently reported in the literature, such as spatio-temporal canards and slow passages through Hopf bifurcations in spatially extended systems subject to slow parameter variations. We also provide a theoretical analysis of slow passage through a Turing bifurcation in local and nonlocal models. [ABSTRACT FROM AUTHOR]

In this note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form Lu = 0 in Ω, u = g in RN \ Ω, in non-smooth domains Ω. When g is smooth enough, then it is easy to transform this problem into an homogeneous Dirichlet problem with a bounded right-hand side for which the boundary regularity is well understood. Here, we study the case in which g ∈ C0,α, and establish the optimal Hölder regularity of u up to the boundary. Our results extend previous results of Grubb for C∞ domains Ω. [ABSTRACT FROM AUTHOR]

We consider the fractional mean-field equation on the interval I = (− 1 , 1) (− Δ) 1 2 u = ρ e u ∫ I e u d x , subject to Dirichlet boundary conditions, and prove that existence holds if and only if ρ < 2 π. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of nonlocal type. [ABSTRACT FROM AUTHOR]

We study the following system of equations L i (u i) = H i (u 1 , ⋯ , u m) in R n , when m ≥ 1 , u i : R n → R and H = (H i) i = 1 m is a sequence of general nonlinearities. The nonlocal operator L i is given by L i (f (x)) : = lim ϵ → 0 ⁡ ∫ R n ∖ B ϵ (x) [ f (x) − f (z) ] J i (z − x) d z , for a sequence of even, nonnegative and measurable jump kernels J i. We prove a Poincaré inequality for stable solutions of the above system for a general jump kernel J i. In particular, for the case of scalar equations, that is when m = 1 , it reads ∬ R 2 n A y (∇ x u) [ η 2 (x) + η 2 (x + y) ] J (y) d x d y ≤ ∬ R 2 n B y (∇ x u) [ η (x) − η (x + y) ] 2 J (y) d x d y , for any η ∈ C c 1 (R n) and for some nonnegative A y (∇ x u) and B y (∇ x u). This is a counterpart of the celebrated inequality derived by Sternberg and Zumbrun in [46] for semilinear elliptic equations that is used extensively in the literature to establish De Giorgi type results, to study phase transitions and to prove regularity properties. We then apply this inequality to finite range jump processes and to jump processes with decays to prove De Giorgi type results in two dimensions. In addition, we show that whenever H i (u) ≥ 0 or ∑ i = 1 m u i H i (u) ≤ 0 then Liouville theorems hold for each u i in one and two dimensions. Lastly, we provide certain energy estimates under various assumptions on the jump kernel J i and a Liouville theorem for the quotient of partial derivatives of u. [ABSTRACT FROM AUTHOR]

In this paper, we analyse nonlocal equations in perforated domains. We consider nonlocal problems of the form $f(x) = \int \nolimits _{B} J(x-y) (u(y) - u(x)) {\rm d}y$ with x in a perforated domain $\Omega ^\epsilon \subset \Omega $. Here J is a nonsingular kernel. We think about $\Omega ^\epsilon $ as a fixed set Ω from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside Ω. In the latter case we impose that u vanishes in the holes but integrate in the whole ℝ N (B = ℝ N ) and in the former we just consider integrals in ℝ N minus the holes ($B={\open R} ^N \setminus (\Omega \setminus \Omega ^\epsilon)$). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of $\Omega ^\epsilon $ has a weak limit, $\chi _{\epsilon } \rightharpoonup {\cal X}$ weakly* in L ∞(Ω), we analyse the limit as ε → 0 of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls, we obtain that the critical radius is of the order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behaviour of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems. [ABSTRACT FROM AUTHOR]

We construct heteroclinic orbits for a strongly nonlocal integro-differential equation. Since the energy associated to the equation is infinite in such strongly nonlocal regime, the proof, based on variational methods, relies on a renormalized energy functional, exploits a perturbation method of viscosity type, combined with an auxiliary penalization method, and develops a free boundary theory for a double obstacle problem of mixed local and nonlocal type. The description of the stationary positions for the atom dislocation function in a perturbed crystal, as given by the Peierls–Nabarro model, is a particular case of the result presented. [ABSTRACT FROM AUTHOR]

We prove new comparison principles for viscosity solutions of nonlinear integro-differential equations. The operators to which the method applies include but are not limited to those of Lévy-Ito type. The main idea is to use an optimal transport map to couple two different Lévy measures and use the resulting coupling in a doubling of variables argument. [ABSTRACT FROM AUTHOR]

We present a series of recent results on some new classes of free boundary problems.Differently from the classical literature, the problems considered have either a “nonlocal” feature (e.g., the interaction or/and the interfacial energy may depend on global quantities) or a “nonlinear” flavor (namely, the total energy is the nonlinear superposition of energy components, thus losing the standard additivity and scale invariances of the problem).

In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee the uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is a dynamic gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.

We present a new approach for studying the large time asymptotic behavior to the nonlocal equation ∂ t u = J u − χ 0 u in the whole R n , where J u = ∫ R n J (⋅ , y) u (y , t) d y. We obtain a condition so that the solutions decay in time at the rate of a regular varying function R , even for non-symmetric J , for a wide class of R. This condition involves a sharp bound for J k u 0 (u 0 = u | t = 0) or the kernels of J k , and the decay rate is proved by analyzing an exponential series with regular varying coefficients. Classical cases, including the fractional Laplacian, the compactly support regular kernels, and algebraic decaying kernels are shown to satisfy our condition by employing the sharp Young's inequality and the Fourier transform technique. In particular, we have generalized the pioneer work by Chasseigne et al. (2006) [11]. Finally, we exhibit nonlocal equation that has solutions with a prescribed decay rate, for a wide class of regular varying function. [ABSTRACT FROM AUTHOR]

We show bifurcation of localized spike solutions from spatially constant states in systems of nonlocally coupled equations in the whole space. The main assumptions are a generic bifurcation of saddle-node or transcritical type for spatially constant profiles, and a symmetry and second moment condition on the convolution kernel. The results extend well known results for spots, spikes, and fronts, in locally coupled systems on the real line, and for radially symmetric profiles in higher space dimensions. Rather than relying on center manifolds, we pursue a more direct approach, deriving leading order asymptotics and Newton corrections for error terms. The key ingredient is smoothness of Fourier multipliers arising from discrepancies between nonlocal operators and their local long-wavelength approximations. [ABSTRACT FROM AUTHOR]

We investigate a nonlocal equation ∂ t u = ∫ R n J (x − y) u (y , t) d y − ‖ J ‖ L 1 u (x , t) + a (x , t) u p in R n , where a is unbounded and J belongs to a weighted space. Crucial weighted L p and interpolation estimates for the Green operator are established by a new method based on the sharp Young's inequality, the asymptotic behavior of a regular varying coefficients exponential series, and the properties of auxiliary functions Γ = (1 + | x | 2 ∕ η) b ∕ 2 that − Γ ∕ η ≲ J ∗ Γ − Γ ≲ Γ ∕ η and η − b + ∕ 2 ≲ Γ ∕ x b ≲ η − b − ∕ 2. Blow-up behaviors are investigated by employing test functions ϕ R = Γ (η = R) instead of principal eigenfunctions. Global well-posedness in weighted L p spaces for the Cauchy problem is proved. When a ∼ x σ the Fujita exponent is shown to be 1 + (σ + 2) ∕ n. Our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlocal evolution equation posed in perforated domains. We consider problems of the form ut(t,x)=∫RN∖AϵJ(x−y)(u(t,y)−u(t,x))dy+f(t,x) with x in a perturbed domain Ωϵ⊂Ω. We think about Ωε as a fixed set Ω from where we have removed the subset Aε that we call the holes. Moreover, we take J as a nonsingular kernel. Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of Ωε has a weak limit, χϵ⇀X weakly∗ in L∞(Ω) as ε→0, we analyze the limit of the solutions proving a nonlocal homogenized evolution equation. [ABSTRACT FROM AUTHOR]