Singular solutions for a convection diffusion equation with absorption (Q1191849)

(From author's summary.) The author proves the existence of a very singular solution of the Cauchy problem \[ u_ t=\Delta u+a\cdot \nabla u^ q-u^ p, \qquad u(x,0) \quad\text{if}\quad x\neq 0 \quad \text{(a constant)}, \] which is more singular at \((0,0)\) than the fundamental solution of the heat equation if \(1<p<(N+2)/N\) and \(1\leq q\leq(p+1)/2\). He also proves the nonexistence of singular solutions if \(p\geq(N+2)/N\) and \(1\leq q\leq(p+1)/2\).