Solutions to an advanced functional partial differential equation of the pantograph type

Abstract

A model for cells structured by size undergoing growth and division leads to an initial boundary value problem that involves a first-order linear partial differential equation with a functional term. Here, size can be interpreted as DNA content or mass. It has been observed experimentally and shown analytically that solutions for arbitrary initial cell distributions are asymptotic as time goes to infinity to a certain solution called the steady size distribution. The full solution to the problem for arbitrary initial distributions, however, is elusive owing to the presence of the functional term and the paucity of solution techniques for such problems. In this paper, we derive a solution to the problem for arbitrary initial cell distributions. The method employed exploits the hyperbolic character of the underlying differential operator, and the advanced nature of the functional argument to reduce the problem to a sequence of simple Cauchy problems. The existence of solutions for arbitrary initial distributions is established along with uniqueness. The asymptotic relationship with the steady size distribution is established, and because the solution is known explicitly, higher-order terms in the asymptotics can be readily obtained.

Asymmetric cell division with stochastic growth rate. Dedicated to the memory of the late Spartak Agamirzayev

“…The cell growth model to is based on a model proposed by Sinko and Streifer for planarian worms. The functional PDE was studied, among others, by Hall and Wake, Hall et al, Begg et al, Metz and Diekmann, and Zaidi et al Perthame and Ryzhik established the existence of a unique eigenvalue λ and the corresponding positive eigenfunction y ( x ) towards which all solutions to converge exponentially for large time, ie, as t → ∞ . Here, is a normalization constant.…”

“…The PDE (7) is a special case of a more general coagulation-fragmentation equation for which there is a dearth of general solution techniques. Here, we solve (7) analytically and establish directly the long time asymptotic behaviour of solutions.…”

“…The PDE (7) can be reduced to a simplified form by using a sequence of transformations. The transformation n(x, t) = e −(b+ )tñ (x, t) reduces (7) to…”

A functional partial differential equation arising in a cell growth model with dispersion

“…There are no general techniques for solving such problems even for a restricted class of growth and division rates. The full problem was solved, however, for the simplest case of constant rates G and B …”

“…The full problem was solved, however, for the simplest case of constant rates G and B. 4 Most of the research on this model focussed on the long time asymptotic solution to the problem. Hall and Wake 1 referred to such solutions as steady size distributions (SSDs), and for the case of constant G and B they proposed an SSD solution based on the separable solution n(x, t) = N(t)y(x) to the pde (see also Hall and Wake 5 ).…”