Solution of a nonlinear heat equation with arbitrarily given blow‐up points

We consider the equation\documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c} {(1)} \hfill & {} \hfill & {\left\{ {\begin{array}{*{20}c} {u(0,x)} \hfill & { = \varphi (x),} \hfill & {} \hfill \\ {u_t } \hfill & { = u_{xx} + \left| u \right|^{p - 1} u} \hfill & {{\rm on}\ [0,T) \times I,} \hfill \\ u \hfill & { = 0} \hfill & {{\rm on}\ [0,T) \times \partial I,} \hfill \\ \end{array}} \right.} \hfill \\ \end{array} $$\end{document} where I ⊂ ℝ u is scalar‐valued and p > 1. It has been proven that if u(t) blows up at time T , the blow‐up points are finite in number and located in I °. Our aim is to prove that this result is optimal. That is, for any given points x 1 ,…, x k in I ° there is a solution u such that its blow‐up points are exactly x 1 ,…, x k .