scholarly journals The spreading fronts in a mutualistic model with delay

2016 ◽  
Vol 09 (06) ◽  
pp. 1650080
Author(s):  
Mei Li
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Ecological Model ◽  
Local Existence ◽  
Free Boundaries ◽  
Local Existence And Uniqueness ◽  
Fast Solution ◽  
Global Fast Solution ◽  
Semilinear Parabolic

This paper is concerned with a system of semilinear parabolic equations with two free boundaries describing the spreading fronts of the invasive species in a mutualistic ecological model. The local existence and uniqueness of a classical solution are obtained and the asymptotic behavior of the free boundary problem is studied. Our results indicate that two free boundaries tend monotonically to finite or infinite limits at the same time, and the free boundary problem admits a global slow solution with unbounded free boundaries if the intra-specific competitions are strong, while if the intra-specific competitions are weak, there exist the blowup solution and global fast solution.

An evolutional free-boundary problem of a reaction–diffusion–advection system

2017 ◽  
Vol 147 (3) ◽  
pp. 615-648 ◽  
Author(s):  
Ling Zhou ◽  
Shan Zhang ◽  
Zuhan Liu
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Ecological Model ◽  
Reaction Diffusion ◽  
Spreading Speed ◽  
Heterogeneous Environment ◽  
Long Run ◽  
Strong Competition ◽  
Asymptotic Spreading

In this paper we consider a system of reaction–diffusion–advection equations with a free boundary, which arises in a competition ecological model in heterogeneous environment. The evolution of the free-boundary problem is discussed, which is an extension of the results of Du and Lin (Discrete Contin. Dynam. Syst. B19 (2014), 3105–3132). Precisely, when u is an inferior competitor, we prove that (u, v) → (0, V) as t→∞. When u is a superior competitor, we prove that a spreading–vanishing dichotomy holds, namely, as t→∞, either h(t)→∞ and (u, v) → (U, 0), or limt→∞h(t) < ∞ and (u, v) → (0, V). Moreover, in a weak competition case, we prove that two competing species coexist in the long run, while in a strong competition case, two species spatially segregate as the competition rates become large. Furthermore, when spreading occurs, we obtain some rough estimates of the asymptotic spreading speed.

Approximation of minimal surfaces with free boundaries: convergence results

2018 ◽  
Vol 39 (3) ◽  
pp. 1391-1420
Author(s):  
Tristan Jenschke
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Minimal Surfaces ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Free Boundaries ◽  
Convergence Estimate ◽  
Convergence Results ◽  
Discrete Finite Element

Abstract In a previous paper we developed a penalty method to approximate solutions of the free boundary problem for minimal surfaces by solutions of certain variational problems depending on a parameter $\lambda $. There we showed existence and $C^2$-regularity of these solutions as well as convergence to the solution of the free boundary problem for $\lambda \to \infty $. In this paper we develop a fully discrete finite element procedure for approximating solutions of these variational problems and prove a convergence estimate, which includes an order of convergence with respect to the grid size.

A free boundary problem arising in the modelling of internal oxidation of binary alloys

1995 ◽  
Vol 6 (3) ◽  
pp. 225-245
Author(s):  
Bei Hu ◽  
Jianhua Zhang
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Internal Oxidation ◽  
Free Boundary Problem ◽  
Binary Alloys ◽  
Local Existence ◽  
Discontinuous Coefficients ◽  
Parabolic Partial Differential Equation ◽  
One Dimensional ◽  
A Free Boundary Problem

A one-dimensional free boundary problem arising in the modelling of internal oxidation of binary alloys is studied in this paper. The free boundary of this problem is determined by the equation u = 0, where u is the solution of a parabolic partial differential equation with discontinuous coefficients across the free boundary. Local existence, uniqueness and the regularity of the free boundary are established. Global existence is also studied.

Sign in / Sign up

Export Citation Format

Share Document