Very singular solutions to a nonlinear parabolic equation with absorption. I: Existence (Q2710418)

The authors deal with the existence of very singular solutions for a parabolic equation with absorption when the absorption term is a nonnegative function of \(\nabla u\). The main goal of the authors is to prove the existence of a very singular solution at the origin to the following viscous Hamilton-Jacobi equation NEWLINE\[NEWLINEu_t-\Delta u+|\nabla u|^p= 0\quad\text{in }(0,\infty)\times \mathbb{R}^N.NEWLINE\]NEWLINE They present existence of such solutions when \(1< p<{N+2\over N+1}\). The main tool of the proof is to obtain an \(L^\infty\)-estimate for the fundamental solutions which does not depend on the initial mass.