Self-similar blow-up solutions of the KPZ equation (Q274814)

Consider the partial differential equation NEWLINE\[NEWLINE{\partial u\over\partial t}= {\partial^2 u\over\partial x^2}+\Biggl|{\partial u\over\partial x}\Biggr|^q\quad\text{for }(x,t)\in \mathbb{R}\times (0,T).\tag{\(*\)}NEWLINE\]NEWLINE The existence of self-similar blow-up solutions of \((*)\) with \(q>2\) having the form NEWLINE\[NEWLINEu(x,t)= (T- t)^\alpha f(\xi)NEWLINE\]NEWLINE with \(\xi=|x|(T-t)^\beta\), \(0<t<T\), leads to the initial value problem NEWLINE\[NEWLINE\begin{gathered} f''+|f'|^q- {1\over 2}\xi f'+\alpha f=0,\\ f(0)=-f_0<0,\;f'(0)= 0,\end{gathered}\tag{\(**\)}NEWLINE\]NEWLINE where \(\alpha= {q-2\over 2(q-1)}\), \(\beta=-{1\over 2}\).NEWLINENEWLINE The author proves that in case \(q>2\) a solution of \((**)\) satisfies NEWLINE\[NEWLINE\lim_{\xi\to\infty} {f(\xi)\over\xi^{q/(q-1)}}= \Biggl[{1\over q-1} \Biggl({q-1\over q}\Biggr)^q\Biggr]^{{1\over q-1}}.NEWLINE\]