In this article, we consider the semilinear elliptic problem −ε∆u = h(|x|)(u− a(|x|))(1− u) in B1(0) with the Neumann boundary condition. The function a is a C1 function satisfying |a(x)| < 1 for x ∈ [0, 1] and a′(0) = 0. In particular we consider the case a(r) = 0 on some interval I ⊂ [0, 1]. The function h is a positive C1 function satisfying h′(0) = 0. We investigate an asymptotic profile of...

The precise dynamics of a reaction-diffusion model of autocatalytic chemical reaction is described. It is shown that exactly either one, two, or three steady states exists, and the solution of dynamical problem always approaches to one of steady states in the long run. Moreover it is shown that a global codimension one manifold separates the basins of attraction of the two stable steady states....

The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class optimization is strongly utilized engineering, biology and finance. In this paper, a stochastic gradient method proposed for the resolution nonconvex problem on Hilbert space. We show that, suitable assumptions, strong or weak accumulation points iterates produced by c...

In [14], a new local minimax method that characterizes a saddle point as a solution to a local minimax problem is established. Based on the local characterization, a numerical minimax algorithm is designed for finding multiple saddle points. Numerical computations of many examples in semilinear elliptic PDE have been successfully carried out to solve for multiple solutions. One of the important...

Let be a bounded open subset of , function in Stummel classes where and&#x0D; semilinear monotone elliptic equation, is symmetric matrix, elliptic, bounded, and non decreasing Lipschitz. By proving weighted estimation for class with its weight which allows us to use Stampacchia’s lemma, we obtained the existence uniqueness solution this equation.

This paper is concerned with the forward and inverse problems for fractional semilinear elliptic equation (??)su+a(x,u)=0 0<s<1. For problem, we proved problem well-posed has a unique solution small exterior data. The consider here consists of two cases. First demonstrate that an unknown coefficient a(x,u) can be uniquely determined from knowledge measurements, known as Dirichlet-to-Neumann map...