The Cauchy problem for $u_ t=\Delta u^ m$ when $0<m<1$

Abstract:

This paper deals with the Cauchy problem for the nonlinear diffusion equation $\partial u/\partial t - \Delta (u|u{|^{m - 1}}) = 0$ on $(0,\infty ) \times {{\mathbf {R}}^N},u(0, \cdot ) = {u_0}$ when $0 < m < 1$ (fast diffusion case). We prove that there exists a global time solution for any locally integrable function ${u_0}$: hence, no growth condition at infinity for ${u_0}$ is required. Moreover the solution is shown to be unique in that class. Behavior at infinity of the solution and $L_{\operatorname {loc} }^\infty$-regularizing effects are also examined when $m \in (\max \{ (N - 2)/N,0\} ,1)$.