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In this paper, we prove the existence of the initial-nonlinear nonlocal solutions for a parabolic system in a weighted Sobolev space. The methods applied are the Faedo–Galerkin approximation and the general theory of weak compactness in appropriate weighted Sobolev spaces together with using the Poincaré-type operator for dealing nonlinear nonlocal conditions. Furthermore, the boundedness and positivity of solutions depending on the boundedness and positivity of given data are also discussed by using suitable test functions.

The initial-nonlinear nonlocal solutions for a parabolic system in a weighted Sobolev space