EXISTENCE OF SOLUTIONS FOR NONLINEAR EQUATIONS WITH MIXED LOCAL AND NONLOCAL OPERATORS

Small perturbations in the type of boundary conditions for an elliptic operator

We are concerned with the following elliptic equation with a general nonlocal integrodifferential operator $${\mathcal {L}}_K$$ $$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} -{\mathcal {L}}_Ku=\lambda u+f(x,u), &{}\quad \text {in}\quad \Omega ,\\ u=0, &{} \quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}$$ where $$\Omega $$ be an open-bounded set of $${\mathbb {R}}^n$$ with continuous boundary, $$\lambda \in {\mathbb {R}}$$ is a real parameter, and f is a nonlinear term with subcritical growth. We show the existence of a ground state and infinitely many pairs of solutions. The proof is based on the method of Nehari manifold for the equation with $$\lambda <\lambda _1$$ , where $$\lambda _1$$ is the first eigenvalue of the nonlocal operator $$-{\mathcal {L}}_K$$ with homogeneous Dirichlet boundary condition, and the method of generalized Nehari manifold for the equation with $$\lambda \ge \lambda _1$$ . As a concrete example, we derive the existence and multiplicity of solutions for the equation driven by fractional Laplacian $$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^\alpha u=\lambda u+f(x,u),&{}\quad \text {in}\quad \Omega ,\\ u=0, &{}\quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}$$ where $$0<\alpha <1$$ . The results presented here may be viewed as the extension of some classical results for the Laplacian to nonlocal fractional setting.

The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey

A nonlocal observation operator has been developed to assimilate GPS radio occultation (RO) refractivity with WRF 3DVAR. For simplicity, in the past GPS RO refractivity was often assimilated using a local observation operator with the assumption that the GPS RO observation was representative of a model local point. Such an operator did not take into account the effects of horizontal inhomogeneity on the derived GPS RO refractivity. In order to more accurately model the observables, Sokolovskiy et al. (2005a) developed a nonlocal observation operator; which would take into account the effects of horizontal inhomogeneity on GPS RO measurements. This nonlocal observation operator calculates the integrated amount of the model refractivity along the ray paths centered at the perigee points. For comparative purposes, the nonlocal observation operator can be simplified by limiting the length of integration near the RO point. This is called the “local operator variant”, which is equivalent to the original local operator except that the original one is performed with fixed tangent points at observation levels. For computational efficiency, assimilation using both the nonlocal operator and local operator variant now is performed with smear tangent points at the mean height of each model vertical level. In this study, the statistics of observation errors using both local and nonlocal operators were estimated based on WRF simulations. The observation errors produced by the nonlocal operator are about two times smaller than those generated by the local operator and in agreement with Sokolovskiy et al. (2005b). Each of the three operators is used to assimilate GPS RO refractivity soundings from the FORMOSAT-3/COSMIC mission using the WRF 3DVAR system. The WRF model then is applied to simulate Typhoons Kaemi (July 2006) which struck Taiwan with significant torrential rainfall. The analysis increments produced by the nonlocal operator and local operator variant are quite similar in horizontal and vertical distributions; whereas, the former is slightly stretched along the ray's direction, as a result of the longer integration length. The simulated typhoon tracks prior to landfall are quite similar for the three operators. Both the nonlocal operator and local operator variant improve the detoured track after landfall as predicted by the local operator. The nonlocal operator outperforms the two local operators in rainfall prediction at later times. The performances of the nonlocal operator in general are promising and can replace the local operator at no marked cost of computational efficiency.

Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem

Establishing variational formulation is an effective way to study the existence and uniqueness of the solution of certain elliptic partial differential equation with boundary condition. For the solution of certain elliptic partial differential equation with boundary condition, we know that the numerical solution obtained by the finite element method approximates the solution of this equation. Moreover, to avoid gridding overly complex domains, we can use the Chimera method to decompose the domain into several overlapping sub-domains. In this paper, we study Poisson’s equation with the homogeneous Dirichlet boundary condition. By analyzing the existence and uniqueness of the solution of the corresponding variational formulation, we know the existence and uniqueness of the solution of Poisson’s equation with the homogeneous Dirichlet boundary condition. We use the Chimera method and the finite element method to deal with Poisson’s equation with the homogeneous Dirichlet boundary condition by constructing two iterative sequences and analyzing their properties.

Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian

Generic bifurcation of steady-state solutions

We study the large time behavior of the solution of a homogenous string equation with a homogenous Dirichlet boundary condition at the left end and a homogenous Dirichlet or Neumann boundary condition at the right end. A pointwise interior actuator gives a linear viscous damping term.

A reaction–diffusion predator–prey system with homogeneous Dirichlet boundary conditions describes the lethal risk of predator and prey species on the boundary. The spatial pattern formations with the homogeneous Dirichlet boundary conditions are characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Compared with homogeneous Neumann boundary conditions, we see that the homogeneous Dirichlet boundary conditions may depress the spatial patterns produced through the diffusion-induced instability. In addition, the existence of semi-trivial steady states and the global stability of the trivial steady state are characterized by the comparison technique.

In recent years, the fractional Laplacian has attracted the attention of many researchers, the corresponding fractional obstacle problems have been widely applied in mathematical finance, particle systems, and elastic theory. Furthermore, the monotonicity of the numerical scheme is beneficial for numerical stability. The purpose of this work is to introduce a monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. Through successful monotone discretization of the fractional Laplacian, the monotonicity is preserved for the fractional obstacle problem and the uniform boundedness, existence, and uniqueness of the numerical solutions of the fractional obstacle problems are proved. A policy iteration is adopted to solve the discrete nonlinear problems, and the convergence after finite iterations can be proved through the monotonicity of the scheme. Our improved policy iteration, adapted to solution regularity, demonstrates superior performance by modifying discretization across different regions. Numerical examples underscore the efficacy of the proposed method.

A version of fractional diffusion on bounded domains, subject to 'homogeneous Dirichlet boundary conditions' is derived from a kinetic transport model with homogeneous inflow boundary conditions. For nonconvex domains, the result differs from standard formulations. It can be interpreted as the forward Kolmogorow equation of a stochastic process with jumps along straight lines, remaining inside the domain.

The nonlinear Schrödinger equation in the finite line

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set \(\Omega \subset {\mathbb {R}}^N\). Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.

We study the following equation \begin{equation*}\frac{\partial u(t,x)}{\partial t}=\Delta u(t,x)+b\bigl(u(t,x)\bigr)+\sigma \dot{W}(t,x),\quad t>0,\end{equation*} where $\sigma $ is a positive constant and $\dot{W}$ is a space–time white noise. The initial condition $u(0,x)=u_{0}(x)$ is assumed to be a nonnegative and continuous function. We first study the problem on $[0,1]$ with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together with a theorem of Bonder and Groisman in (Phys. D 238 (2009) 209–215), our first result shows that the solution blows up in finite time if and only if for some $a>0$, \begin{equation*}\int _{a}^{\infty }\frac{1}{b(s)}\,\mathrm{d}s<\infty,\end{equation*} which is the well-known Osgood condition. We also consider the same equation on the whole line and show that the above condition is sufficient for the nonexistence of global solutions. Various other extensions are provided; we look at equations with fractional Laplacian and spatial colored noise in $\mathbf{R}^{d}$.