Multibump solutions for a class of nonlinear elliptic problems

The paper is concerned with a class of semilinear elliptic Dirichlet problems approximating degenerate equations. By using variational methods, it is proved that, if the degeneration set consists of connected components, then there exist at least multibump positive solutions. Received March 25, 1996 / Accepted May 26, 1997     

A class of nonlinear elliptic problems

We obtain a strict coercivity estimate, (generalizing that of T. I. Seidman [J. Differential Equations 19 (1975), 242–257] in considering spatial variation) for second order elliptic operators A: u ? ?▽ · γ(·, ▽u) with γ “radial in the gradient” ?γ(·, ξ) = a(·, |ξ|)ξ for ξ ? $\text{R}$^m. The estimate is then applied to obtain existence of solutions of boundary value problems: $\text{?▽ ·}\text{a}\text{?}\text{(·, u, |▽u|) ▽u = f(·, u, ▽u)}$ with Dirichlet conditions.     

Flux-splitting schemes for parabolic problems

Splitting with respect to space variables can be used in solving boundary value problems for second-order parabolic equations. Classical alternating direction methods and locally one-dimensional schemes could be examples of this approach. For problems with rapidly varying coefficients, a convenient tool is the use of fluxes (directional derivatives) as independent variables. The original equation is written as a system in which not only the desired solution but also directional derivatives (fluxes) are unknowns. In this paper, locally one-dimensional additional schemes (splitting schemes) for second-order parabolic equations are examined. By writing the original equation in flux variables, certain two-level locally one-dimensional schemes are derived. The unconditional stability of locally one-dimensional flux schemes of the first and second approximation order with respect to time is proved.     

Positive solutions of a class of nonlinear elliptic eigenvalue problems

This paper is mainly concerned with the natural order relationship between positive solutions of the elliptic eigenvalue Dirichlet problem: in and u=0 on . Under suitable conditions, we prove that there are 2m-1 positive solutions satisfying . It seems that standard arguments do not provide such a result. Several authors, including P. Hess, proved the existence of equal number of positive solutions without such a relationship between them. We also prove that in Hess's result as well as in ours some sufficient condition is also necessary if the domain possesses a particular shape. At last, as an illustrative example, we study the diagram of positive solutions when with and d being both parameters. Received: 13 July 2000; in final form: 22 August 2001 / Published online: 1 February 2002     

Monotone schemes for semilinear elliptic systems related to ecology

Semilinear elliptic systems of partial differential equations related to ecology are studied, with Dirichlet boundary conditions. Monotone sequences of functions which satisfy scalar equations are constructed so that they will converge to upper and lower bounds for the solutions of the systems. In case a related system has a unique positive solution, then these sequences will converge to the solution of the original system. Applications of the monotone sequences to uniqueness and stability are also given.     

Attractivity properties of nonnegative solutions for a class of nonlinear degenerate parabolic problems

Summary Monotonicity methods are developped to investigate attractivity properties of non-negative stationary solutions for a class of nonlinear degenerate parabolic problems in any space dimension. Applications to specific problems suggested by population dynamics are also discussed.     

Existence, uniqueness and behavior of solutions for a class of nonlinear parabolic problems

We prove existence, uniqueness, regularity results and estimates describing the behavior (both for large and small times) of a solution u of some nonlinear parabolic equations of Leray-Lions type including the p-Laplacian. In particular we show how the summability of the initial datum u₀ and the value of p influence the behavior of the solution u, producing ultracontractive or supercontractive estimates or extinction in finite time or different kinds of decay estimates.     

Monotone numerical schemes for a Dirichlet problem for elliptic operators in divergence form

We consider a second‐order differential operator A( x )=?∑?_ia_ij( x )?_j+ ∑?_j(b_j( x )·)+c( x ) on ?^d, on a bounded domain D with Dirichlet boundary conditions on ?D, under mild assumptions on the coefficients of the diffusion tensor a_ij. The object is to construct monotone numerical schemes to approximate the solution of the problem A( x )u( x )=µ( x ), x ∈D, where µ is a positive Radon measure. We start by briefly mentioning questions of existence and uniqueness introducing function spaces needed to prove convergence results. Then, we define non‐standard stencils on grid‐knots that lead to extended discretization schemes by matrices possessing compartmental structure. We proceed to discretization of elliptic operators, starting with constant diffusion tensor and ending with operators in divergence form. Finally, we discuss W‐convergence in detail, and mention convergence in C and L₁ spaces. We conclude by a numerical example illustrating the schemes and convergence results. Copyright © 2008 John Wiley & Sons, Ltd.     

High‐order difference schemes for 2D elliptic and parabolic problems with mixed derivatives

We propose a 9‐point fourth‐order finite difference scheme for 2D elliptic problems with a mixed derivative and variable coefficients. The same approach is extended to derive a class of two‐level high‐order compact schemes with weighted time discretization for solving 2D parabolic problems with a mixed derivative. The schemes are fourth‐order accurate in space and second‐ or lower‐order accurate in time depending on the choice of a weighted average parameter μ. Unconditional stability is proved for 0.5 ≤ μ ≤ 1, and numerical experiments supporting our theoretical analysis and confirming the high‐order accuracy of the schemes are presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 366–378, 2007     

Finite element Galerkin approximations to a class of nonlinear and nonlocal parabolic problems

In this article, a finite element Galerkin method is applied to a general class of nonlinear and nonlocal parabolic problems. Based on an exponential weight function, new a priori bounds which are valid for uniform in time are derived. As a result, existence of an attractor is proved for the problem with nonhomogeneous right hand side which is independent of time. In particular, when the forcing function is zero or decays exponentially, it is shown that solution has exponential decay property which improves even earlier results in one dimensional problems. For the semidiscrete method, global existence of a unique discrete solution is derived and it is shown that the discrete problem has an attractor. Moreover, optimal error estimates are derived in both and ‐norms with later estimate is a new result in this context. For completely discrete scheme, backward Euler method with its linearized version is discussed and existence of a unique discrete solution is established. Further, optimal estimates in ‐norm are proved for fully discrete schemes. Finally, several numerical experiments are conducted to confirm our theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1232–1264, 2016     

A general class of free boundary problems for fully nonlinear parabolic equations

Annali di Matematica Pura ed Applicata (1923 -) - In this paper, we consider the fully nonlinear parabolic free boundary problem $$\begin{aligned} \left\{ \begin{array}{ll} F(D^2u) -\partial _{t}...     

Finite-difference schemes for parabolic problems on graphs

We consider a reaction–diffusion parabolic problem on branched structures. The Hodgkin–Huxley reaction–diffusion equations are formulated on each edge of the graph. The problems are coupled by some conjugation conditions at branch points. It is important to note that two different types of the flux conservation equations are considered. The first one describes a conservation of the axial currents at branch points, and the second equation defines the conservation of the current flowing at the soma in neuron models. We study three different types of finite-difference schemes. The fully implicit scheme is based on the backward Euler algorithm. The stability and convergence of the discrete solution is proved in the maximum norm, and the analysis is done by using the maximum principle method. In order to decouple computations at each edge of the graph, we consider two modified schemes. In the predictor algorithm, the values of the solution at branch points are computed by using an explicit approximation of the conservation equations. The stability analysis is done using the maximum principle method. In the predictor–corrector method, in addition to the previous algorithm, the values of the solution at the branch points are recomputed by an implicit algorithm, when the discrete solution is obtained on each subdomain. The stability of this algorithm is investigated numerically. The results of computational experiments are presented.