Steel corrosion plays a central role in different technological fields. In this article, we consider a simple case of a corrosion phenomenon which describes a pure iron dissolution in sodium chloride. This article is devoted to prove rigorously that under rather general hypotheses on the initial data, the solution of this iron dissolution model converges to a self-similar profile as $t\rightarrow +\infty$. We will do so for an equivalent formulation as presented in the book of Avner Friedman about parabolic equations (Friedman (1964) Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ.). In order to prove the convergence result, we apply a comparison principle together with suitable upper and lower solutions.

The study of radially symmetric motion is important for the theory of explosion waves. We construct rigorously self-similar entropy solutions to Riemann initial-boundary value problems for the radially symmetric relativistic Euler equations. We use the assumption of self-similarity to reduce the relativistic Euler equations to a system of nonlinear ordinary differential equations, from which we obtain detailed structures of solutions besides their existence. For the ultra-relativistic Euler equations, we also obtain the uniqueness of the self-similar entropy solution to the Riemann initial-boundary value problems.

We consider the Cauchy problem

$$\left\{ {\matrix{ {u_t = \Delta u + u^p,\quad } \hfill & {x\in {\bf R}^N,\;t \leq 0,} \hfill \cr {u(x,0) = u_0(x),\quad } \hfill & {x\in {\bf R}^N,} \hfill \cr } } \right.$$

Deformable cell model is developed to study pattern formation and to simulate planttissue growth. Each cell represents a polygon with a number of vertices connected bysprings. Some cells in the tissue can grow and divide, other cells are differentiated anddo not grow or divide but remain deformable. The model is used to investigate formation ofself-similar structures which reproduce the same cell organization during their growth. Innumerical experiments we observed that self-similar solutions can exist for a ratherprecise choice of plant structure and mechanical properties of cell walls. We test themodel for simulation of apical meristems functioning which represent self-similar cellstructures in plants. At the next stage of modelling, auxin distribution is introduced bymeans of diffusion and polar transport mechanisms. The existence of steady auxindistribution in a growing root is investigated. Single as well as multiple auxin maximahave been observed in model solutions.