In this paper, we study variation-inequality initial-boundary value problems with fouth order $p(x)$-Kirchhoff operators. First, an operator is constructed based on the Leray Schauder principle, and the existence of solutions is obtained. Secondly, the stability and uniqueness of the solution are analyzed after the conditions are appropriately relaxed on the Kirchhoff operators.

This article studies a class of variational inequality problems composed of non-divergence type parabolic operators. In comparison with traditional differential equations, this study focuses on overcoming inequality constraints to obtain Hölder and Schauder estimates for weak solutions. The results indicate that the weak solution of the variational inequality possesses the $C^{\alpha }$ continuity and the Schauder estimate on the $W^{1,p}$ space, where $\alpha \in (0,1)$ and $p\ge 2$.