We study the existence of sign-changing solutions to the nonlinear heat equation $partial _t u = Delta u + |u|^alpha u$ on ${mathbb R}^N $, $Nge 3$, with $frac {2} {N-2} < alpha <alpha _0$, where $alpha _0=frac {4} {N-4+2sqrt{ N-1 } }in (frac {2} {N-2}, frac {4} {N-2})$, which are singular at $x=0$ on an interval of time. In particular, for certain $mu >0$ that can be arbitrarily large, we prove that for any $u_0 in mathrm{L} ^infty _{mathrm{loc}} ({mathbb R}^N setminus { 0 }) $ which is bounded at infinity and equals $mu |x|^{- frac {2} {alpha }}$ in a neighborhood of $0$, there exists a local (in time) solution $u$ of the nonlinear heat equation with initial value $u_0$, which is sign-changing, bounded at infinity and has the singularity $beta |x|^{- frac {2} {alpha }}$ at the origin in the sense that for $t>0$, $ |x|^{frac {2} {alpha }} u(t,x) to beta $ as $ |x| to 0$, where $beta = frac {2} {alpha } ( N -2 - frac {2} {alpha } ) $. These solutions in general are neither stationary nor self-similar.
Download