Nonlinear nonautonomous math models for drug concentration in the body

Abstract

A nonautonomous $p(x)$-Laplacian evolution equation is proposed as a model for administration of a medicament. The equation is of the form $$\dfrac{\partial
u}{\partial t}(t)-D\textrm{div}\left(|\nabla
u(t)|^{p(x)-2}\nabla
u(t)\right)+k(t)|u(t)|^{p(x)-2}u(t)=f(u(t))$$
on a bounded smooth domain $\Omega$ in $\mathbb{R}^3$, with a homogeneous Neumann boundary condition, where the exponent $p(\cdot)\in C(\bar{\Omega},\mathbb{R}^+)$ satisfies $p^-$ $:=$ $\min p(x)$ $\geq$ $2$. By considering large diffusion and a uniform and linear spreading of the drug through the human body, a simplification can be done in order to obtain the following handy ordinary differential equation
$$\frac{du}{dt}(t) + k(t)u = f(u).$$