Pointwise bounds for linear reaction-diffusion systems and an extension to nonlinear problems (Q1118754)

The author considers systems of the form \[ \partial u^ i/\partial t- div(d_ iA(\nabla u^ i))+C^ i_ ju^ j=r^ i\quad in\quad \Omega \times (0,\infty),\quad i=1,...,n, \] where \(d_ i>0\) are diffusion coefficients, A is a uniformly positive definite symmetric matrix, \(C^ i_ j(x,t)\) and \(r^ i(x,t)\) are continuous functions. \(\Omega\) is a bounded domain in \({\mathbb{R}}^ m\), \(u^ i(x,0)\) is given and the boundary condition is \(Bu^ i=0\) on \(\partial \Omega \times (0,\infty)\) where B is the identity or \(Bu^ i=\nabla u^ iA\nu +bu^ i,\) \(\nu =unit\) outer normal and b a nonnegative scalar function. The main result of the paper is a bound for \(\max | u^ i(x,t)|\) (x\(\in \Omega\), \(i=1,...,n\), \(t\geq 0\}\) in terms of the data.