In this paper we examine self-similar solutions to the system $u_{it}-d_i\Delta u_i={\prod }_{k=1}^m{u}_k^{p_k^i}$, i = 1,…,m, $x\in {ℝ}^N$, t > 0, $u_i(0,x)=u_{0i}\left(x\right)$, i = 1,…,m, $x\in {ℝ}^N$, where m > 1 and $p_k^i>0$, to describe asymptotics near the blow up point.

We consider a system which describes the scaling limit of several chemotaxis systems. We focus on self-similarity, and review some recent results on forward and backward self-similar solutions to the system.

We study systems of reaction-diffusion equations with discontinuous spatially distributed hysteresis on the right-hand side. The input of the hysteresis is given by a vector-valued function of space and time. Such systems describe hysteretic interaction of non-diffusive (bacteria, cells, etc.) and diffusive (nutrient, proteins, etc.) substances leading to formation of spatial patterns. We provide sufficient conditions under which the problem is well posed in spite of the assumed discontinuity of...