RESUMEN

In [H. A.\ Levine and J. Renc{\l}awowicz, {\em SIAM J.\ Appl.\ Math.}, 65 (2004), pp. 336--360] we considered the problem $u_t=u_{xx}-(uv_x)_x, v_t=u-av$ on the interval $I=[0,1]$, where $u_x, v_x =0$ at the end points, $u(x,0), v(x,0)$ are prescribed, and $a>0$. (It was claimed in that article that there were solutions that blow up in finite time in every neighborhood of the spatially homogeneous steady state $(u,v)=(\mu, \mu/a)$ if $\mu>a$.) Here we correct an estimate and reduce Nagai's conjecture to the following statement. Let $\sigma=a/(\mu-a), \rho_1=1.$ If $\lim_{n\to +\infty}\rho_n$ exists, where for $n\ge 2$, $\rho_n^n\equiv 1/(n-1)\sum_{j=1}^{n-1}(1+\sigma/j)\rho_j^j\rho_{n-j}^{n-j},$ then the blow up assertion holds.