Local boundedness for weak solutions to strongly degenerate orthotropic parabolic equations

We prove the local boundedness of local weak solutions to the parabolic equation \[ ∂ₜu\,=\,∑ᵢ₌₁ⁿ∂_{xᵢ}[( u_{xᵢ}-δᵢ)₊ᵖ⁻¹{u_{xᵢ}}{ u_{xᵢ}}]\,\,\,\,\,\,\,\,\,\,in\,\,\,ΩT=Ω×(0,T]\,, \] where $Ω$ is a bounded domain in Rⁿ with n≥2, p≥2, δ₁,…,δₙ are non-negative numbers and (\,·\,)₊ 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.