To the uniqueness problem for nonlinear parabolic equations. (Q1433924)

The authors prove a priori estimates, existence and uniqueness of weak solutions to initial boundary value problems of the form \[ \begin{cases} {\partial\over\partial t}\sigma(u)- \sum^n_{i=1}{\partial\over\partial x_i} \bigl(\rho(u)b_i(t,x,{\partial u\over\partial x})\bigr)+ a(t,x,u,{\partial u\over\partial x})= 0,\quad &(t,x)\in Q,\\ u(t,x)= f(t,x),\quad &(t,x)\in(0,T)\times \partial\Omega,\\ u(0,x)= g(x),\quad & x\in\Omega,\end{cases} \] where \(\rho(u)= {d\sigma(u)\over du}\), \(\Omega\) is a bounded open set in \(\mathbb{R}^d\) and \(Q= (0,T)\times\Omega\). The authors' nonstandard assumption is that \(\log\rho(u)\) is concave.