We prove a global weighted \(L^{p(\cdot )}\)-regularity for the Hessian of strong solution to the Cauchy–Dirichlet problem for fully nonlinear parabolic equations in a bounded \(C^{1,1}\)-domain, where the associated nonlinearity is \((\delta ,R)\)-vanishing in independent variables, the variable exponent \(p(\cdot )\) is \(\log \)-Hölder continuous, and the weight \(\omega \) is of the \(A_{p(\cdot )/(n+1)}\) class. As a consequence, we also derive Morrey’s regularity for the Hessian of strong solution to this problem under consideration, which implies a global Hölder continuity of the spatial gradient under the assumption of higher regular datum.

我们证明了有界\（C ^ {1,1} \的完全非线性抛物方程组Cauchy-Dirichlet问题的强解的Hessian的全局加权\（L ^ {p（\ cdot）} \） -正则性）域，其中相关的非线性为\（（\ delta，R）\） -在自变量中消失，变量指数\（p（\ cdot）\）为\（\ log \）- Hölder连续，并且权重\（\ omega \）属于\（A_ {p（\ cdot）/（n + 1）} \）类。因此，我们还针对正在考虑的问题的强解的Hessian导出了Morrey的正则性，这意味着在假定较高正则基准的情况下，空间梯度的全局Hölder连续性。