Nonlocal Equation Research Articles

Existence and uniqueness of solutions to the Peierls–Nabarro model in anisotropic media

We study the existence and uniqueness of solutions to the vector field Peierls–Nabarro (PN) model for curved dislocations in a transversely isotropic medium. Under suitable assumptions for the misfit potential on the slip plane, we reduce the 3D PN model to a nonlocal scalar Ginzburg–Landau equation. For a particular range of elastic coefficients, the nonlocal scalar equation with explicit nonlocal positive kernel is derived. We prove that any stable steady solution has a one-dimensional profile. As a result, we obtain that solutions to the scalar equation, as well as the original 3D system, are characterized as a one-parameter family of straight dislocations. This paper generalizes results found previously for the full isotropic case to an anisotropic setting.

An application of Sobolev's inequality to one-dimensional Kirchhoff equations

We consider nonlocal differential equations with convolution coefficients of the form−M((a⁎(g∘|u′|))(1))u″(t)=λf(t,u(t)), t∈(0,1), in which the coefficient function M imparts a nonlocal structure to the problem. The function g satisfies p-q growth. A model case occurs when g(t):=tp, where p>1, and a(t)≡1 – i.e., the problem−M(‖u′‖Lpp)u″(t)=λf(t,u(t)), t∈(0,1). By an application of the one-dimensional Sobolev inequality, together with a specially constructed order cone, we are able to demonstrate existence of at least one positive solution to this equation when subjected to Dirichlet boundary conditions. Our methodology utilises a more recently developed topological fixed point theory, which allows for the use of sets that are unbounded in the ambient norm.

The role of antisymmetric functions in nonlocal equations

We use a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms, in combination with the method of moving planes, to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that non-negative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set Ω ⊂ R n \Omega \subset \mathbb {R}^n must be radially symmetric if one of their level surfaces is parallel to the boundary of Ω \Omega ; in turn, Ω \Omega must be a ball. Furthermore, we discuss maximum principles and the Harnack inequality for antisymmetric functions in the fractional setting and provide counter-examples to these theorems when only ‘local’ assumptions are imposed on the solutions. The construction of these counter-examples relies on an approximation result that states that ‘all antisymmetric functions are locally antisymmetric and s s -harmonic up to a small error’.

Traveling Waves for a Sign-Changing Nonlocal Evolution Equation with Delayed Nonlocal Response

Traveling Waves for a Sign-Changing Nonlocal Evolution Equation with Delayed Nonlocal Response

On regularity and asymptotic stability for semilinear nonlocal pseudo-parabolic equations

We deal with a class of nonlocal pseudo-parabolic equations involving strong nonlinearities. The questions on existence, regularity and stability of solutions are addressed by using local estimates, fixed point arguments, and the relation between the Hilbert scales and fractional Sobolev spaces.

Viscosity Solutions for Nonlocal Equations with Space-Dependent Operators

Viscosity Solutions for Nonlocal Equations with Space-Dependent Operators

Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D

In this contribution, we study an optimal control problem for the celebrated nonlocal Cahn-Hilliard equation endowed with the singular Flory-Huggins potential in the three-dimensional setting. The control enters the governing state system in a nonlinear fashion in the form of a prescribed solenoidal, that is a divergence-free, vector field, whereas the cost functional to be minimized is of tracking-type. The novelties of the present paper are twofold: in addition to the control application, the intrinsic difficulties of the optimization problem forced us to first establish new regularity results on the nonlocal Cahn-Hilliard equation that were unknown even without the coupling with a velocity field and are therefore of independent interest. This happens to be shown using the recently proved separation property along with ad hoc Hölder regularities and a bootstrap method. For the control problem, the existence of an optimal strategy as well as first-order necessary conditions are then established.

Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional

<abstract><p>In this paper, the asymptotic behavior of solutions to a fractional stochastic nonlocal reaction-diffusion equation with polynomial drift terms of arbitrary order in an unbounded domain was analysed. First, the stochastic equation was transformed into a random one by using a stationary change of variable. Then, we proved the existence and uniqueness of solutions for the random problem based on pathwise uniform estimates as well as the energy method. Finally, the existence of a unique pullback attractor for the random dynamical system generated by the transformed equation is shown.</p></abstract>

Stackelberg exact controllability of a class of nonlocal parabolic equations

This paper deals with a multi-objective control problem for a class of nonlocal parabolic equations, where the non-locality is expressed through an integral kernel. We present the Stackelberg strategy that combines the concepts of controllability to trajectories with optimal control. The strategy involves two controls: a main control (the leader) and a secondary control (the follower). The leader solves a controllability to trajectories problem which consists to drive the state of the system to a prescribed target at a final time while the follower solves an optimal control problem which consists to minimize a given cost functional. The paper considers two cases: in the first case, both the leader and the follower act in the interior of the domain, and in the second case, the leader acts in the interior of the domain and the follower acts on a small part of the boundary. These results are applied to both linear and nonlinear nonlocal parabolic systems.

Approximate controllability of neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space

In this paper, our main purpose is to establish the controllability results for nonlocal neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space as well as to generalize the results that existed in the literature on this topic. We present three types of conditions on the nonlocal initial condition's function to prove the existence of a mild solution for nonlocal neutral Hilfer fractional differential equations of Sobolev-type, and we then derive the approximate controllability results for the system. With help of an approximate technique, we establish the existence and controllability results under the weaker hypothesis (continuous only) on the nonlocal initial condition's function. The main tools applied in our analysis are semigroup theory, fractional calculus, resolvent operator theory, the theory of fractional powers of operators, Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, and approximating technique. Finally, we provide two examples as applications to illustrate our main results.

General soliton solutions for the complex reverse space-time nonlocal mKdV equation on a finite background

Three kinds of Darboux transformations are constructed by means of the loop group method for the complex reverse space-time (RST) nonlocal modified Korteweg–de Vries equation, which are different from that for the PT symmetric (reverse space) and reverse time nonlocal models. The N-periodic, the N-soliton, and the N-breather-like solutions, which are, respectively, associated with real, pure imaginary, and general complex eigenvalues on a finite background are presented in compact determinant forms. Some typical localized wave patterns such as the doubly periodic lattice-like wave, the asymmetric double-peak breather-like wave, and the solitons on singly or doubly periodic waves are graphically shown. The essential differences and links between the complex RST nonlocal equations and their local or PT symmetric nonlocal counterparts are revealed through these explicit solutions and the solving process.

The Schrödinger equation and the two-slit experiment of quantum mechanics

The one-dimensional time-dependent complex-valued Schrödinger equation is used to describe the interference diffraction patterns of two-slit experiments in quantum mechanics. The properties of the diffraction patterns are related to the initial data of the Schrödinger equation. The Schrödinger equation solutions are compared to Fraunhofer diffraction formulas in geometric optics, and to the solutions of nonlocal advection diffusion equations.

On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution

<abstract><p>In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved.</p></abstract>

Existence and uniqueness of random nonlocal differential equations with colored noise

Existence and uniqueness of random nonlocal differential equations with colored noise

Growth of solutions for a system of nonlocal singular viscoelastic equation with sources and distributed delay terms

Growth of solutions for a system of nonlocal singular viscoelastic equation with sources and distributed delay terms

Soliton solutions and a bi-Hamiltonian structure of the fifth-order nonlocal reverse-spacetime Sasa-Satsuma-type hierarchy via the Riemann-Hilbert approach

<p>Our objective is to explore the intricacies of a nonlinear nonlocal fifth-order scalar Sasa-Satsuma equation in reverse spacetime which is rooted in a nonlocal $ 5 \times 5 $ matrix AKNS spectral problem. Starting with this spectral problem, we derive both local and nonlocal symmetry relations through rotations within a defined group. We then formulate a specific type of Riemann-Hilbert problem, facilitating the generation of soliton solutions. These solutions are generated by utilizing vectors that reside in the kernel of the matrix Jost solutions. Under the condition where reflection coefficients are null, the jump matrix reduces to the identity, leading to soliton solutions via the corresponding Riemann-Hilbert problem. The explicit formulas of these soliton solutions enable a comprehensive exploration of their dynamics.</p>

Exact controllability for nonlocal wave equations with nonlocal boundary conditions

Exact controllability for nonlocal wave equations with nonlocal boundary conditions

Dynamical behavior of a temporally discrete non-local reaction-diffusion equation on bounded domain

This paper focuses on the study of global dynamics of a class of temporally discrete non-local reaction-diffusion equations on bounded domains. Similar to classical reaction diffusion equations and integro-difference equations, temporally discrete reaction-diffusion equations can also be used to describe the dispersal phenomena in population dynamics. In this paper, we first derived a temporally discrete reaction diffusion equation model with time delay and nonlocal effects to model the evolution of a single species population with age-structured located in a bounded domain. By establishing a new maximum principle and applying the monotone iteration method, the global stabilities of the trivial solution and the positive steady state solution are obtained respectively under some appropriate assumptions.

Asymptotic behavior and blow-up of solutions for a nonlocal parabolic equation with a special diffusion process

Asymptotic behavior and blow-up of solutions for a nonlocal parabolic equation with a special diffusion process

Sharp asymptotic of solutions to some nonlocal parabolic equations

Sharp asymptotic of solutions to some nonlocal parabolic equations