We prove the local boundedness of local weak solutions to the parabolic equation \( \partial _{t}u\,=\,\sum _{i=1}^{n}\partial _{x_{i}}\left[ (\vert u_{x_{i}}\vert -\delta _{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert u_{x_{i}}\vert }\right] \,\,\,\,\,\,\,\,\,\,\textrm{in}\,\,\,\Omega _{T}=\Omega \times (0,T]\,, \) where \(\Omega \) is a bounded domain in \({\mathbb {R}}^{n}\) with \(n\ge 2\) , \(p\ge 2\) , \(\delta _{1},\ldots ,\delta _{n}\) are non-negative numbers and \(\left( \,\cdot \,\right) _{+}\) denotes the positive part. The main novelty here is that the above equation combines an orthotropic structure with a strongly degenerate behavior. The core result of this paper thus extends a classical boundedness theorem, originally proved for the parabolic p-Laplacian, to a widely degenerate anisotropic setting. As a byproduct, we also obtain the local boundedness of local weak solutions to the isotropic counterpart of the above equation.
