A priori estimates for almost similar similar pde's

Question

Let $\Omega\subset\mathbb{R}^2$ a sufficiently smooth domain. Let the semi-linear pde problem \begin{align} u_t-\Delta u &= f(u),\;\;\;on\;\;\;\Omega\subset\mathbb{R}^2\\ \nabla u\cdot n & = 0\;\;\;on\;\;\; \partial\Omega. \end{align} Let that we have some a priori estimates for the solution of a pde problem, such that \begin{equation} \|u(t)\|_{H^2(\Omega)} \leq C, \end{equation} where the constant $C$ depends only on problem data. Let now, the following semi-linear pde problem \begin{align} v_t-\Delta v + v &= f(v),\;\;\;on\;\;\;\Omega\subset\mathbb{R}^2\\ \nabla v\cdot n & = 0\;\;\;on\;\;\; \partial\Omega. \end{align} Is there any connection between the apriori estimates for the above semi-linear pde problems? In fact, can I immediately conclude to \begin{equation} \|v(t)\|_{H^2(\Omega)} \leq C, \end{equation} where the constant $C$ depends only on problem data?