Pointwise gradient estimates of solutions to one-dimensional nonlinear parabolic equations (Q1880303)

A general method to obtain pointwise gradient estimates for solutions to (possibly) degenerate parabolic equations has been introduced by Ph. Bénilan in 1981, and applications of this method are presented and reviewed here in a one-dimensional setting. More precisely, inequalities of the form \(\sigma(u,u_x)\leq k(t,x,u)\) are obtained for the solutions \(u\) to the general parabolic equation \(u_t=(\sigma(u,u_x) )_x\), \(t\geq 0\), \(x\in{\mathbb R}\), provided that \(k\) satisfies a suitable differential inequality. More explicit estimates are given for the convection-diffusion equation \(u_t=\varphi(u)_{xx}+\psi(u)_x\), and their optimality is discussed too. The case of homogeneous Dirichlet boundary conditions is also considered. Applications to smoothing effects and to the evolution of free boundaries are also given.