Operator equations in orderd sets and discontinuous implicit parabolic equations (Q5925828)

The authors study the implicit parabolic initial-boundary value problem NEWLINE\[NEWLINEF(x,t,u,{\mathcal L}u)= 0\quad\text{in }Q,\quad u= 0\quad\text{on }\Gamma,\quad u(x,0)= \psi(x)\quad\text{in }\Omega\times \{0\},\tag{\(*\)}NEWLINE\]NEWLINE where \(\Omega\subset \mathbb{R}^N\) is a bounded domain with Lipschitz boundary \(\partial\Omega\), \(Q:= (0,T)\times \Omega\), and \(\Gamma:= (0,T)\times \partial\Omega\). The operator \({\mathcal L}\) is assumed to be a semilinear parabolic differential operator in the form NEWLINE\[NEWLINE{\mathcal L}u(x, t):= {\partial u(x,t)\over\partial t}+ Au(x, t)+ g(x, t,u(x,t)),NEWLINE\]NEWLINE where \(A\) is a second-order strongly elliptic differential operator given by NEWLINE\[NEWLINEAu(x, t)= -\sum^N_{i,j=1} {\partial\over\partial x_i} \Biggl(a_{ij}(x, t){\partial u\over\partial x_j}\Biggr),NEWLINE\]NEWLINE with coefficients \(a_{ij}\in L^\infty(Q)\) satisfying for all \(\xi= (\xi_1,\dots, \xi_N)\in \mathbb{R}^N\) NEWLINE\[NEWLINE\sum^N_{i,j=1} a_{ij}(x, t)\xi_i\xi_j\geq \mu|\xi|^2,\text{ for a.e. }(x,t)\in Q\text{ with some constant }\mu> 0.NEWLINE\]NEWLINE The main goal of the present paper is to prove the existence of extremal solutions of the discontinuous implicit IBVP \((*)\) by deriving first an abstract existence result for an operator equation in the form \(Lu= Nu\), where \(L\) and \(N\) are operators from a partially ordered set to an ordered metric space.