Existence of a solution of a Fourier nonlocal quasilinear parabolic problem (Q1186524)

Summary: The aim of this paper is to give a theorem about the existence of a classical solution of a Fourier third nonlocal quasilinear parabolic problem. To prove this theorem, Schauder's theorem is used. The paper is a continuation of earlier papers [e.g.: the author and \textit{V. Lakshmikantham}, Appl. Anal. 40, No. 1, 11-19 (1991; Zbl 0694.34001); the author, Uniqueness of solutions of parabolic semilinear nonlocal-boundary value problems (to appear)] and the generalizations of some results e.g. from \textit{J. Chabrowski} [Funkc. Ekvacioj, Ser. Int. 27, 101-123 (1984; Zbl 0568.35046)]. The theorem established in this paper can be applied to describe some phenomena in the theories of diffusion and heat conduction with better effects than the analogous classical theorem about the existence of a solution of the Fourier third quasilinear parabolic problem.