Uniqueness for an elliptic-parabolic problem with Neumann boundary condition

We consider the problem $$ b\left( u \right) - \Delta u + div F\left( u \right) = f$$ in a smooth bounded domain $$ \Omega \subset {\mathbb{R}^N}$$ , as well as the corresponding evolution equation $$ b{\left( u \right)_t} - \Delta u + div F\left( u \right) = f$$ , $$ b\left( {u\left( {0,.} \right)} \right) = {b^0}.$$ . For the stationary equation we show existence results, then we adapt the techniques of doubling of variables to the case of the homogeneous Neumann boundary conditions and obtain the appropriate L 1-contraction principle and uniqueness. Subsequently, we are able to apply the nonlinear semigroup theory and prove the L 1 -contraction principle for the associated evolution equation.