Equations with Gradient Terms

Abstract

In Chapter IV, we consider problems with nonlinearities depending on u and its space derivatives:

$$ \left. {\begin{array}{*{20}c} {u_t - \Delta u = F\left( {u,\nabla u} \right),{\mathbf{ }}x \in \Omega ,t > 0,} \\ {u = 0,{\mathbf{ }}x \in \partial \Omega ,t > 0,} \\ {u\left( {x,0} \right) = u_0 \left( x \right),{\mathbf{ }}x \in \Omega .} \\ \end{array} {\mathbf{ }}} \right\} $$

Here F = F(u, ξ):ℝ×ℝⁿ → lo is a C¹-function (except for problem (34.4) with 1 < q > 2, see below).