Model Parabolic Problems

Abstract

In Chapter II, we mainly consider semilinear parabolic problems of the form

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

where f is a C¹-function with a superlinear growth. For simplicity, we formulate most of our assertions for the model case f(u) = |u|^p-1u with p 1, but the methods of our proofs can be applied to more general parabolic problems (not necessarily of the form (14.1)). Some of possible generalizations and modifications will be mentioned as remarks, other can be found in the subsequent chapters.