Self-similarity and the singular Cauchy problem for the heat equation with cubic absorption (Q5938870)

This note is devoted to the construction of self-similar solutions of the 1D heat equation with absorption NEWLINE\[NEWLINEu_t- u_{xx}+ u^3= 0,\quad x\in\mathbb{R},\quad t>0,NEWLINE\]NEWLINE and study its limiting behaviour as \(t\to 0+\). The author proves that, for any \(c\in \mathbb{R}\), there exists a solution of the singular Cauchy problem for the semilinear heat equation NEWLINE\[NEWLINEu_t- u_{xx}+ u^3= 0,\qquad u(x,0)= \text{p.v. }{c\over x},NEWLINE\]NEWLINE where the initial value being given in the sense of distributions.