Asymptotical self-similarity in nonlinear parabolic systems (Q814415)

The author deals with the structure of the blow-up for \[ \begin{gathered} u_{jt}- d_j\Delta u_j= \prod^m_{k=1} u^{p^j_k}_k,\quad j=1,\dots, m,\quad x\in\mathbb{R}^N,\quad t> 0,\\ u_j(0,x)= u_{0,j}(x),\quad j= 1,\dots, m,\quad x\in\mathbb{R}^N,\quad m> 1, \quad p^j_k\geq 0,\end{gathered}\tag{1} \] with nonnegative, continuous, bounded initial data and positive \(d_j\). The author shows that, blowing up solutions of (1) behave like self-similar solutions to this system near the blow-up point. The deep analysis in the case \(m=2\) specifies the obtained result.