100,999 research outputs found
Modeling contact inhibition of growth: traveling waves
No full textWe consider a simplified 1-dimensional PDE-model describing the effect of contact inhibition in growth processes of normal and abnormal cells. Varying the value of a significant parameter, numerical tests suggest two different types of contact inhibition between the cell populations: the two populations move with constant velocity and exhibit spatial segregation, or they stop to move and regions of coexistence are formed. In order to understand the different mechanisms, we prove that there exists a segregated traveling wave solution for a unique wave speed, and we present numerical results on the "stability" of the segregated waves. We conjecture the existence of a non-segregated standing wave for certain parameter values
Partially overlapping travelling waves in a parabolic-hyperbolic system
No full textWe study the existence of travelling wave solutions of a one-dimensional parabolic-hyperbolic system for u (x, t) and v (x, t), which arises as a model for contact inhibition of cell growth. Compared to the scalar Fisher-KPP equation, the structure of the travelling wave solutions is surprisingly rich and strongly parameter-dependent. In the present paper we consider a parameter regime where the minimal wave speed is positive. We show that there exists a branch of travelling wave solutions for wave speeds which are larger than the minimal one. But the main result is more surprising: for certain values of the parameters the travelling wave with minimal wave speed is not segregated (a solution is called segregated if the product uv vanishes almost everywhere) and in that case there exists a second branch of "partially overlapping" travelling wave solutions for speeds between the minimal one and that of the (unique) segregated travelling wave
Travelling wave solutions of a parabolic-hyperbolic system for contact inhibition of cell-growth
No full textA density dependent diffusion equation in population dynamics: stabilization to equilibrium
No full textWe study an evolution problem corresponding to the nonlinear diffusion equation with no flux boundary conditions. This problem has a continuum of stationary solutions. We prove the existence and uniqueness of the solution of the evolution problem and construct a Lyapunov functional in order to show that the solution stabilizes as
Pseudoparabolic regularization of forward-backward parabolic equations: a logarithmic nonlinearity
No full textWe study the initial-boundary value problem
u_t = Δφ(u) + εΔ[ψ(u)]_t in Q := Ω×(0, T],
φ(u) + ε[ψ(u)]_t = 0 in ∂Ω×(0, T],
u = u_0 ≥0 in Ω×{0},
with measure-valued initial data, assuming that the regularizing term ψ has logarithmic growth (the case of power-type ψ was dealt with in an earlier work). We prove that this case is intermediate between the case of power-type ψ and that of bounded ψ, to be addressed in a forthcoming paper. Specifically, the support of the singular part of the solution with respect to the Lebesgue measure remains constant in time (as in the case of power-type ψ), although the singular part itself need not be constant (as in the case of bounded ψ, where the support of the singular part can also increase). However, it turns out that the concentrated part of the solution with respect to the Newtonian capacity remains constant
Letter, [Author unclear] to Paulina T. Merritt
No full textHandwritten letter to Paulina Merritt from an unknown author, October 1, 1876.
On a class of forward–backward parabolic equations: Existence of solutions
No full textWe study the initial-boundary value problemu(t) = [phi(u)](xx) + is an element of[psi(u)](txx) in Omega x (0, T) phi(u) + is an element of[psi(u)] t = 0 in partial derivative Omega x (0, T) u = u(0) in Omega x 0,where Omega is an interval and u0 is a nonnegative Radon measure on Omega. The map phi is increasing in (0, alpha) and decreasing in (alpha, infinity) for some alpha > 0, and satisfies phi(0) = phi(infinity) = 0. The regularizing map psi is increasing and bounded. We prove existence of suitably defined nonnegative Radon measure-valued solutions. The solution class is natural since smooth initial data may generate solutions which become measure-valued after finite time. (C) 2017 Elsevier Ltd. All rights reserved
A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic Fisher KPP equation
No full textWe consider a mathematical model describing population dynamics of normal and abnormal cell densities with contact inhibition of cell growth from a theoretical point of view. In the first part of this paper, we discuss the global existence of a solution satisfying the segregation property in one space dimension for general initial data. Here, the term segregation property means that the different types of cells keep spatially segregated when the initial densities are segregated. The second part is devoted to singular limit problems for solutions of the PDE system and traveling wave solutions, respectively. Actually, the contact inhibition model considered in this paper possesses quite similar properties to those of the Fisher-KPP equation. In particular, the limit problems reveal a relation between the contact inhibition model and the Fisher-KPP equation
Standing and travelling waves in a parabolic-hyperbolic system
No full textWe consider a nonlinear system of partial differential equations which describes the dynamics of two types of cell densities with contact inhibition. After a change of variables the system turns out to be parabolic-hyperbolic and admits travelling wave solutions which solve a 3D dynamical system. Compared to the scalar Fisher-KPP equation, the structure of the travelling wave solutions is surprisingly rich and to unravel part of it is the aim of the present paper. In particular, we consider a parameter regime where the minimal wave velocity of the travelling wave solutions is negative. We show that there exists a branch of travelling wave solutions for any nonnegative wave velocity, which is not connected to the travelling wave solution with minimal wave velocity. The travelling wave solutions with nonnegative wave velocity are strictly positive, while the solution with minimal one is segregated in the sense that the product uv vanishes
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