Stability of solutions to neutral type parabolic systems with time delay (Q674699)

The paper deals with the neutral-type parabolic system with time delay \[ {\partial\over\partial t} (Q(x,t)- PQ(x,t-\tau))=D\Delta Q(x,t)+ AQ(x,t)+ BQ(x,t-\sigma),\quad (x,t)\in\Omega\times\mathbb{R}_+,\tag{1} \] subject to the boundary condition \[ {\partial Q(x,t)\over\partial n}=0,\quad (x,t)\in\partial\Omega\times[-\tau^*,+\infty),\quad\tau^*=\max\{\tau,\sigma\},\tag{2} \] where \(Q(x,t)\in\mathbb{R}^n\), \(A\), \(B\) are \(n\times n\) constant matrices, and \(P\), \(D\) are \(n\times n\) constant diagonal matrices; \(\tau\), \(\sigma>0\), \(\Omega\subset\mathbb{R}^m\) is a bounded domain. The authors construct several auxiliary functionals and using \(L_p\)-estimates they prove stability of solutions of (1), (2).