Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions

Abstract

In this paper, we consider a particular type of nonlinear McKendrick-von Foerster equation with a diffusion term and Robin boundary condition. We prove the existence of a global solution to this equation. The steady state solutions to the equations that we consider have a very important role to play in the study of long time behavior of the solution. Therefore we address the issues pertaining to the existence of solution to the corresponding state equation. Furthermore, we establish that the solution of McKendrick-von Foerster equation with diffusion converges pointwise to the solution of its steady state equations as time tends to infinity.

A numerical scheme for a diffusion equation with nonlocal nonlinear boundary condition

“…In particular, the McKendrick-von Foerster equation is one amongst the important models whenever age structure is a vital feature in the modeling (see [5,6,7]). In the recent years, the McKendrick-Von Foerster equation with diffusion (M-V-D) has attracted interest of many engineers as well as mathematicians due to its applications in the modeling of thermoelasticity, neuronal networks etc, (see [1,2,15,8,9,14]). The main difficulty in the study of M-V-D is due to the nonlocal nature of the partial differential equation (PDE) and the boundary condition.…”

“…In [8], the authors considered the M-V-D with nonliner nonlocal Robin boundary condition and studied the existence and uniqueness of the solution. The authors of [10] proposed a convergent numerical scheme to the M-V-D. On the other hand, the existence of a global solution to the M-V-D in a bounded domain with nonliner nonlocal Robin boundary condition was proved when d = d(x) in [9]. Recently in [4], an implicit finite difference scheme has been introduced to approximate the solution to the M-V-D in a bounded domain with nonliner nonlocal Robin boundary condition at both the boundary points.…”

Global existence of solutions of initial boundary value problem for nonlocal parabolic equation with nonlocal boundary condition

“…For the global existence and blow-up of solutions for parabolic equations with nonlocal boundary conditions, we refer to previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein. Initial-boundary value problems for nonlocal parabolic equations with nonlocal boundary conditions were considered in many papers also (see, for example, previous works [18][19][20][21][22][23][24] ).…”

“…Throughout this paper we suppose that a(x, t), b(x, t), k(x, y, t) and u 0 (x) satisfy the following conditions: a(x, t), b(x, t) ∈ C α loc (Ω × [0, ∞)), 0 < α < 1, a(x, t) ≥ 0, b(x, t) ≥ 0; k(x, y, t) ∈ C(∂Ω × Ω × [0, ∞)), k(x, y, t) ≥ 0; u 0 (x) ∈ C(Ω), u 0 (x) ≥ 0, x ∈ Ω, u 0 (x) = Ω k(x, y, 0)u l 0 (y) dy, x ∈ ∂Ω. For global existence and blow-up of solutions for parabolic equations with nonlocal boundary conditions we refer to [1,2,6], [11]- [18], [21,23,24,30,32,33] and the references therein. Initial-boundary value problems for nonlocal parabolic equations with nonlocal boundary conditions were considered in many papers also (see, for example, [4,7,9,10,26,27,34]).…”