On a Model for the Evolution of Morphogens in a Growing Tissue

Abstract

We analyze a recently proposed model [I. Averbukh et al., Development, 141 (2014), pp. 2150-2156 for the regulation of growth and patterning in developing tissues by diffusing morphogens. We show that solutions of the underlying coupled systems of nonlinear PDEs exist, are unique, and are stable in a suitable sense. The key tool in the analysis is the transformation of the underlying system to a porous medium equation. Numerical experiments illustrating the model are also presented.

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],…”

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

“…Several mathematical models have been proposed to answer the previous questions, see [1, 5, 9, 13, 15-18, 20, 21]; a complete and critical description of them can be found in [6][7][8].…”

“…The well-posedness (i.e., existence, uniqueness and stability) for (1.1) in the cases h [ logð2Þ; h ¼ logð2Þ; h\ logð2Þ has been proved in [6][7][8], respectively. This paper is devoted to the long time analysis of the solution (M(t, x), u(t, x), L(t)) of (1.1) under the assumptions h !…”

“…Such equivalence is based on a suitable change of variables inspired by the characteristics method, all the details can be found in [6,Section 2] and [7,Section 2]. The equivalence holds for all the values of h, but when h\ logð2Þ the solution of (1.1) and (1.3) are less regular than the ones in the case h !…”

“…The equivalence holds for all the values of h, but when h\ logð2Þ the solution of (1.1) and (1.3) are less regular than the ones in the case h ! logð2Þ: Indeed, the arguments of [8] are more subtle than the ones in [6,7]. Due to this loss of regularity in this paper we consider only h !…”