Backward motion and waiting time phenomena for degenerate parabolic equations with nonlinear gradient absorption (Q644980)

Some properties of the time evolution of the support of nonnegative and compactly supported solutions to the Cauchy problem \(\partial_t u - \Delta_p u + |\nabla u|^q = 0\) in \((0,\infty)\times\mathbb{R}^N\) with initial condition \(u_0\) are studied according to the values of \(p>2\), \(q>0\), and the local behaviour of \(u_0\) at a point at the edge of its support. More precisely, if \(y\) belongs to the boundary of the support of \(u_0\) and \(u_0\) is appropriately flat in a neighbourhood of \(y\), then there is \(T_*\in (0,\infty]\) such that \(u(t,y)=0\) for \(t\in [0,T_*]\) (waiting time phenomenon). In addition, if \(q\in (0,p-1)\), conditions are given that guarantee that \(T_*=\infty\) (infinite waiting time). Backward motion of the support is also shown to occur when \(q\in (0,1)\), that is, the support of \(u(t)\) shrinks in the neighbourhood of a point of its boundary where \(u_0\) is sufficiently flat.