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The goal of the article is to improve constants in the infimum convolution inequalities (IC for short) which were introduced by R. Lata{\l}a and J.O. Wojtaszczyk. We show that the exponential distribution satisfies IC with constant $2$ but not with constant $1$, which implies that linear functions are not extremal in Maurey's property $(\tau)$. Using transport of measure we use this result to better constants in the IC inequalities for product symmetric log-concave measures as well as in the Talagrand's two level concentration inequality for the exponential distribution.

We analyse a stationary problem for the two dimensional model of morphogen transport introduced by Hufnagel et al. The model consists of one linear elliptic PDE posed on $(-1,1)\times(0,h)$ which is coupled via a nonlinear boundary condition with a nonlinear elliptic PDE posed on $(-1,1)\times\{0\}$. The main result is that the system has a unique steady state for all ranges of parameters present in the system. Moreover we consider the problem of the dimension reduction. After introducing an appropriate scaling in the model we prove that, as $h\to0$, the stationary solution converges to the unique steady state of the one dimensional simplification of the model which was analysed in the first part of the paper. The main difficulty in obtaining appropriate estimates stems from the presence of a measure source term in the boundary condition.

A model of morphogen transport consisting of two evolutionary PDEs of reaction-diffusion type and three ODEs posed on a rectangular domain is analysed. We prove that the problem is globally well-posed and that the corresponding solutions converge, as the width of the rectangle tends to zero, to the unique solution of the one dimensional system which was analyzed in the first paper of the series. Main difficulties in the analysis stem from the presence of a singular source term - a Dirac Delta combined with no smoothing effect in the ODE part of the system.

We analyze a one dimensional version of a model of morphogen transport, a biological process governing cell differentiation. The model was proposed by Hufnagel et al. to describe the forming of morphogen gradient in the wing imaginal disc of the fruit fly. In mathematical terms the model is a system of reaction-diffusion equations which consists of two parabolic PDE's and three ODE's. The source of ligands is modelled by a Dirac Delta. Using semigroup approach and $L_1$ techniques we prove that the system is well-posed and possesses a unique steady state. All results are proved without imposing any artificial restrictions on the range of parameters.

Morphogen transport is a biological process, occurring in the tissue of living organisms, which is a determining step in cell differentiation. We present rigorous analysis of a simple model of this process, which is a system coupling parabolic PDE with ODE. We prove existence and uniqueness of solutions for both stationary and evolution problems. Moreover we show that the solution converges exponentially to the equilibrium in $C^1\times C^0$ topology. We prove all results for arbitrary dimension of the domain. Our results improve significantly previously known results for the same model in the case of one dimensional domain.