Generalized principle of maximum for semilinear parabolic systems (Q1174047)

It is known that for the semilinear parabolic system: \[ u_ t^ \ell- \sum^ m_{i,j=1}A^ \ell_{ij}(t,x)u^ \ell_{x_ ix_ j}=f^ \ell(t,x,u),\quad \ell=1,\ldots,n,\leqno (1) \] the maximum principle is valid, if \(f^ \ell(t,x,u)\) are monotone non-decreasing functions with respect to the variables \(u^ i\), \(i=1,\ldots,n\); \(i\neq\ell\). In the general case the maximum principle does not exist for the system (1). In this paper a generalized maximum principle for the system (1) is investigated. An auxiliary semilinear parabolic system with monotone right-hand side is constructed such that all solutions of the system (1) are solutions of this auxiliary system as well. By this process, the sufficient conditions for the existence of the solution of the initial value problem for every \(t>0\), are obtained.