On a nonlinear renewal equation with diffusion

Abstract

In this paper, we consider a nonlinear age structured McKendrick-von Foerster population model with diffusion term. Here we prove existence and uniqueness of the solution of the equation. We consider a particular type of nonlinearity in the renewal term and prove Generalized Relative Entropy type inequality. Longtime behavior of the solution has been addressed for both linear and nonlinear versions of the equation. In linear case, we prove that the solution converges to the first eigenfunction with an exponential rate. In nonlinear case, we have considered a particular type of nonlinearity that is present in the mortality term in which we can predict the longtime behavior.

A note on a neuron network model with diffusion

“…and finally by the continuity of p we obtain thatp is a solution to (12). Using the standard arguments it is straightforward to show the existence and uniqueness of a solution n ∈ X , to (11) (see [11], for instance). Hence the map T is well defined.…”

“…The key ingredient in the proof of uniqueness result is the L 1 -stability estimate that we prove here. Let (m,q) and (n,p) be solutions to (11) with m(0, s; σ) = m 0 (s; σ), n(0, s; σ) = n 0 (s; σ), By the mean value theorem we obtain…”

“…In fact, we make some technical assumptions on p to establish the existence of solution to this problem. In fact, the role that X plays in the model is similar to that of the weighted population (see [11], [28]). Now, on integrating (3) with respect to s, we get…”

A numerical scheme to the McKendrick–von Foerster equation with diffusion in age

“…Similarly, denote the first, second and third terms on the right-hand side by J 1 , J 2 , and J 3 , respectively. With this notation Equation 13 becomes…”

“…Moreover, the long‐time behavior of the solution to (1) was also studied in detail using the Generalized Relative Entropy inequality. Existence, uniqueness, and asymptotic behavior of a solution to a nonlinear version of (1) were studied in . In , Iannelli et al considered the McKendrick–von Foerster equation without the diffusion term.…”