Equations with Gradient Terms

Part of the Birkhäuser Advanced Texts / Basler Lehrbücher book series (BAT)


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).


Maximum Principle Global Solution Model Problem Global Existence Comparison Principle 
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© Birkhäuser Verlag AG 2007

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