This paper deals with a nonlinear parabolic problem of the following form \[ \frac{\partial u}{\partial t}- \sum_{|\alpha |= 2m} a_{\alpha}(x,u,\dots, D^{\beta}u) D^{\alpha } u= f (x, u,\dots, D^{\beta}u), \quad |\beta |\leq 2m-1, \tag{1} \] where \( \alpha =(\alpha_{1},\dots, \alpha_{n}) \) is a multi-index and \( D^{\alpha } = \partial^{|\alpha |} / \partial x_{1}^{\alpha_{1}} \dots \partial x_{n}^{\alpha_{n} }\); \(x \in {\mathbb{R}}^{n}. \) The authors investigate a problem of existence of global solutions to the linear system associated with problem (1). It allows them to prove an existence of \(r\)-parametric invariant manifold for problem (1). Than the authors investigate smooth properties of invariant manifolds and prove the reduction principle for system (1). The authors prove also that the local smooth center manifold of system (1) can be represented by asymptotic series.