Long time behavior of a model for the evolution of morphogens in a growing tissue

Abstract

The long time behavior of a model for the regulation of growth and patterning in developing tissues by diffusing morphogens is analyzed. Such model is expressed in terms of a system of nonlinear PDEs. The key tool in the analysis is the transformation of such system to an equation with singular diffusion. Keywords Singular diffusion Á Neumann boundary conditions Á Long time behavior Á Morphogen evolution Mathematics Subject Classification 35K55 Á 35K65 Á 35Q92 Á 34B15 Á 35B40 • How can a diffusing morphogen concentration scale with growing tissue? This article is part of the section ''Applications of PDEs'' edited by Hyeonbae Kang.

On a singular limit as $$\theta \rightarrow 0$$ for a model for the evolution of morphogens in a growing tissue

“…from now on we assume that β > 1 ( 3 ). Let us also briefly recall the results on the asymptotic behavior for t → ∞ of N β (t, y) (see [6, Theorem 2.1]) and of (M θ (t, x), u θ (t, x), L θ (t)) (see [6,Theorem1.1]). If the assumptions (1.2) and (1.3) or (1.2) and (1.4) hold, then the function N β (•, y) is monotone and its limit…”

“…Key tool for the analysis of the well-posedness (see [3,4,7]) and of the asymptotic behavior as t → ∞ (see [5,6]) is the definition of a suitable family of "characteristic" curves which start at the points of [0, L 0 ] and "cover" {(t, x) | t ≥ 0, 0 ≤ x ≤ L θ (t)}. Let us briefly recall them because they are also useful in this paper.…”

“…n(t, z) ed N β (t, y) share the same regularity (see[4, Theorem 2.1] and[6, Theorem2.3]) 5. The differentiation in the weak sense of the function v m ∂zω and the role of the test function ω − can be justified by the following observations…”

Free boundary problem for the role of planktonic cells in biofilm formation and development

“…The biofilm thickness L(t) represents the free boundary of the mathematical problem. Its evolution is governed by the following ordinary differential equation [7,16,[19][20][21][22][23],…”

Regularity results for a class of doubly nonlinear very singular parabolic equations

“…when 3−p < m+p < 2 and m⟨1, p⟩1. Besides their intrinsic mathematical interest, such kind of equations appear in recent models of morphogenesis (see [1] and [10]); in [7,8] and [9] one can find results regarding the wellposedness of the model as well as qualitative properties of the equations. Here we prove suitable L r − L ∞ estimates, a result of expansion of positivity and Harnack estimates.…”