A nonlocal nonlinear diffusion equation with blowing up boundary conditions

We deal with boundary value problems (prescribing Dirichlet or Neumann boundary conditions) for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation. First, we prove existence, uniqueness and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models

Let be a nonnegative, smooth function with , supported in , symmetric, , and strictly increasing in . We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation

We prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as : they converge to the mean value of the initial data. Next, we consider a discrete version of the above problem. Under suitable hypotheses we prove that the discrete model has properties analogous to the continuous one. Moreover, solutions of the discrete problem converge to the continuous ones when the mesh parameter goes to zero. Finally, we perform some numerical experiments.

     

Positive solutions for the boundary value problems of one-dimensional p-Laplacian with delay

This paper is concerned with the existence of positive solutions for the boundary value problem of one-dimensional p-Laplacian with delay. The proof is based on the Guo–Krasnoselskii fixed-point theorem in cones.     

Positive solutions of four-point boundary value problems for higher-order p-Laplacian operator

In this paper, we study the existence of positive solutions for the nonlinear four-point singular boundary value problem for higher-order with p-Laplacian operator. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear singular boundary value problem with p-Laplacian operator are obtained.     

Multiple positive solutions for some p-Laplacian boundary value problems

This paper deals with the existence of multiple positive solutions for the one-dimensional p-Laplacian subject to one of the following boundary conditions: or where φ_p(s)=|s|^p−2s, p>1. By means of a fixed point theorem due to Avery and Peterson, sufficient conditions are obtained that guarantee the existence of at least three positive solutions. The interesting point is the nonlinear term f is involved with the first-order derivative explicitly.     

A nonlocal nonlinear diffusion equation in higher space dimensions

We study the initial-value problem for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation, in the whole RN, N?1, or in a bounded smooth domain with Neumann or Dirichlet boundary conditions. First, we prove the existence, uniqueness and the validity of a comparison principle for solutions of these problems. In RN we show that if initial data is bounded and compactly supported, then the solutions is compactly supported for all positive time t, this implies the existence of a free boundary. Concerning the Neumann problem, we prove that the asymptotic behavior of the solutions as t→∞, they converge to the mean value of the initial data. For the Dirichlet problem we prove that the asymptotic behavior of the solutions as t→∞, they converge to zero.     

New results of periodic solutions for a kind of Duffing type p-Laplacian equation

By using Mawhin–Manásevich continuation theorem, some new sufficient conditions for the existence and uniqueness of periodic solutions of Duffing type p-Laplacian differential equation are established, which are complement of previously known results.     

Multiple nontrivial solutions for nonlinear periodic problems with the p-Laplacian

We consider a nonlinear periodic problem driven by the scalar p-Laplacian with a nonsmooth potential (hemivariational inequality). Using the degree theory for multivalued perturbations of (S)₊-operators and the spectrum of a class of weighted eigenvalue problems for the scalar p-Laplacian, we prove the existence of at least three distinct nontrivial solutions, two of which have constant sign.     

Global existence and blow-up of solutions to a nonlocal evolution p-laplace system with nonlinear boundary conditions

In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution.     

Small-amplitude solutions of the sine-Gordon equation on an interval under dirichlet or Neumann boundary conditions

Summary We give a complete classification of the small-amplitude finite-gap solutions of the sine-Gordon (SG) equation on an interval under Dirichlet or Neumann boundary conditions. Our classification is based on an analysis of the finite-gap solutions of the boundary problems for the SG equation by means of the Schottky uniformization approach.On leave from IPPI, Moscow, Russia     

On a degenerate variational inequality with Neumann boundary conditions

A stopping time problem for degenerate reflected diffusions is studied in this paper. We give a characterization of the optimal cost as the maximum solution of a degenerate elliptic variational inequality with Neumann boundary conditions.The author would like to thank Professor L. C. Evans for very helpful suggestions on this topic.     

Small sets for Neumann boundary conditions

For an open set D ? ?ⁿ and a relatively closed subset E ? D of Lebesgue measure zero, we investigate conditions for the property that Brownian motion with reflexion at the boundary on D and D \ E are the same. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a class of parabolic equations with nonlocal boundary conditions

In this paper we study a class of parabolic equations subject to a nonlocal boundary condition. The problem is a generalized model for a theory of ion-diffusion in channels. By using energy method, we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution. Moreover, under some assumptions on the nonlinear boundary condition, it is shown that the solution blows up in finite time. Finally, the long-time behavior of solution to a linear problem is also studied in the paper.     

On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations

Inspired by the penalization of the domain approach of Lions and Sznitman, we give a sense to Neumann and oblique derivatives boundary value problems for nonlocal, possibly degenerate elliptic equations. Two different cases are considered: (i) homogeneous Neumann boundary conditions in convex, possibly non-smooth and unbounded domains, and (ii) general oblique derivatives boundary conditions in smooth, bounded, and possibly non-convex domains. In each case we give appropriate definitions of viscosity solutions and prove uniqueness of solutions of the corresponding boundary value problems. We prove that these boundary value problems arise in the penalization of the domain limit from whole space problems and obtain as a corollary the existence of solutions of these problems.     

Variational-hemivariational inequalities with small perturbations of nonhomogeneous Neumann boundary conditions

In this paper variational-hemivariational inequalities with nonhomogeneous Neumann boundary conditions are investigated. Under an appropriate oscillating behavior of the nonlinear term, the existence of infinitely many solutions to this type of problems, even under small perturbations of nonhomogeneous Neumann boundary conditions, is established.     

Asymptotic behavior of the solutions of the p -Laplacian equation

The asymptotic behavior of the solutions for p-Laplacian equations as p → ∞ is studied.     

Nonlinear eigenvalue Neumann problems with discontinuities

In this paper we study nonlinear eigenvalue problems with Neumann boundary conditions and discontinuous terms. First we consider a nonlinear problem involving the p-Laplacian and we prove the existence of a solution for the multivalued approximation of it, then we pass to semilinear problems and we prove the existence of multiple solutions. The approach is based on the critical point theory for nonsmooth locally Lipschitz functionals.     

A fourth-order method of the convection-diffusion equations with Neumann boundary conditions

In this paper, we have developed a fourth-order compact finite difference scheme for solving the convection-diffusion equation with Neumann boundary conditions. Firstly, we apply the compact finite difference scheme of fourth-order to discrete spatial derivatives at the interior points. Then, we present a new compact finite difference scheme for the boundary points, which is also fourth-order accurate. Finally, we use a Padé approximation method for the resulting linear system of ordinary differential equations. The presented scheme has fifth-order accuracy in the time direction and fourth-order accuracy in the space direction. It is shown through analysis that the scheme is unconditionally stable. Numerical results show that the compact finite difference scheme gives an efficient method for solving the convection-diffusion equations with Neumann boundary conditions.