The present study investigates a class of variational inequality problems under the framework of the parabolic Kirchhoff operator from the financial contract problem. This particular issue stems from the financial contract problem. By utilizing the energy inequality of the obtained solutions, the energy inequality of the solution gradients, and the Caffarelli–Kohn–Nirenberge inequality, an estimation of the infinite norm of the solution gradients is obtained.

This paper analyzes a class of variational inequalities involving a double-phase degenerate parabolic operator with variable exponents, which arises in the valuation of American put options. By establishing energy estimates for the associated penalty problem and applying a limiting argument, the existence of a weak solution is obtained. The uniqueness of the weak solution is also discussed.

This paper investigates local estimates for the spatial gradient of solutions to variational inequalities within the framework of a parabolic Kirchhoff operator, which arises from mortgage problems. By utilizing the integral inequality for the gradient of the solutions derived in this study, together with the Caffarelli–Kohn–Nirenberg inequality, we establish an $L^{\mathrm{\infty }}$ norm estimate for the gradient of the solution in a local cylindrical region. This $L^{\mathrm{\infty }}$ estimate is formulated in terms of the $L^p$ norm of the solution.