In this paper, we consider solutions to the following chemotaxis system with general sensitivity $$\left\{\begin{array}{c}\tau u_t=\Delta u-\nabla ·\left(u\nabla \chi \left(v\right)\right)\text{in}\Omega \times (0,\infty ),\hfill \\ \eta v_t=\Delta v-v+u\text{in}\Omega \times (0,\infty ),\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }=\frac{\partial u}{\partial \nu }=0\text{on}\partial \Omega \times (0,\infty ).}\hfill \end{array}\right.$$ Here, $\tau$ and $\eta$ are positive constants, $\chi$ is a smooth function on $(0,\infty )$ satisfying ${\chi }^{\text{'}}(·)>0$ and $\Omega$ is a bounded domain of ${𝐑}^n$ ($n\ge 2$). It is well known that the chemotaxis system with direct sensitivity ($\chi \left(v\right)={\chi }_{0}v$, ${\chi }_0>0$) has blowup solutions in the case where $n\ge 2$. On the other hand, in the case where $\chi \left(v\right)={\chi }_{0}logv$ with $0<{\chi }_0\ll 1$, any solution to the system exists globally in time and is bounded. We present a sufficient condition for the boundedness of...

In this paper, we propose a computational model to investigate the coupling between cell’s adhesions and actin fibres and how this coupling affects cell shape and stability. To accomplish that, we take into account the successive stages of adhesion maturation from adhesion precursors to focal complexes and ultimately to focal adhesions, as well as the actin fibres evolution from growing filaments, to bundles and finally contractile stress fibres.We use substrates with discrete patterns of adhesive...

This review is dedicated to recent results on the 2d parabolic-elliptic Patlak-Keller-Segel model, and on its variant in higher dimensions where the diffusion is of critical porous medium type. Both of these models have a critical mass $M_c$ such that the solutions exist globally in time if the mass is less than $M_c$ and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out. A number of open questions are also...

We develop the qualitative theory of the solutions of the McKendrick partial differential equation of population dynamics. We calculate explicitly the weak solutions of the McKendrick equation and of the Lotka renewal integral equation with time and age dependent birth rate. Mortality modulus is considered age dependent. We show the existence of demography cycles. For a population with only one reproductive age class, independently of the stability of the weak solutions and after a transient time,...

We consider optimal control problems for the bidomain equations of cardiac electrophysiology together with two-variable ionic models, e.g. the Rogers–McCulloch model. After ensuring the existence of global minimizers, we provide a rigorous proof for the system of first-order necessary optimality conditions. The proof is based on a stability estimate for the primal equations and an existence theorem for weak solutions of the adjoint system.

We present a model for describing the spread of an infectious disease with public screening measures to control the spread. We want to address the problem of determining an optimal screening strategy for a disease characterized by appreciable duration of the infectiveness period and by variability of the transmission risk. The specific disease we have in mind is the HIV infection. However the model will apply to a disease for which class-age structure...

A mathematical model for a problem of blood perfusion in a living tissue through a system of parallel capillaries is studied. Oxygen is assumed to be transported in two forms: freely diffusing and bounded (to erytrocytes in blood, to myoglobin in tissue). Existence of a weak solution is proved and a homogensation procedure is carried out in the case of randomly distribuited capillaries.