Existence and uniqueness of solutions of nonlinear infinite systems of parabolic differential-functional equations (Q2759180)

The author considers an infinite system of weakly coupled nonlinear differential-functional equations of the form \({\mathcal F}^i [z^i](t,x)= f^i(t,x,z)\), \(i\in S\), where NEWLINE\[NEWLINE{\mathcal F}^i= \partial/\partial t-{\mathcal A}^i, \quad {\mathcal A}^i= \sum_{j,k=1}^m a_{jk}^i(t,x) \partial^2/\partial x_j\partial x_k,NEWLINE\]NEWLINE \(x= (x_1,\dots, x_m)\), \((t,x)\in (0,T]\times G\equiv D\), \(T< \infty\), \(G\subset \mathbb{R}^m\), \(G\) is an open bounded domain with boundary \(\partial G\in C^{2+\alpha}\cap C^{2-0}\) \((0< \alpha\leq 1)\), \(S\) is a set of indices and \(z\) stands for the mapping \(z:S\times \overline{D}\to\mathbb{R}\), that is, \((i,t,x)\in\overline D\mapsto z^i(t,x)\in \mathbb{R}\), \(z^i\) are the unknown functions. The Fourier first initial-boundary value problem is considered for the above stated infinite system of weakly coupled nonlinear differential-functional equations of parabolic type. The right-hand sides of this system are functionals of unknown functions. Existence and uniqueness of the solution are proved by the Banach fixed point theorem.