A very singular solution of a quasilinear degenerate diffusion equation with absorption

Abstract

The object of this paper is to study the existence of a nonnegative solution of the Cauchy problem \[ u t = div ⁡ ( | ∇ u | p − 2 ∇ u ) − u q , u ( x , 0 ) = 0 if x ≠ 0 , {u_t} = \operatorname {div} (|\nabla u{|^{p - 2}}\nabla u) - {u^q},\qquad u(x,\,0) = 0\quad {\text {if}}\;x \ne 0, \] which is more singular at ( 0 , 0 ) (0,\,0) than the fundamental solution of the corresponding equation without the absorption term.