Local boundedness for weak solutions to strongly degenerate orthotropic parabolic equations

We prove the local boundedness of local weak solutions to the parabolic equation $${\mathrm{\partial }}_{t}u=\sum _{i=1}^n{\mathrm{\partial }}_{x_i}[(\mathrm{\mid }{u}_{x_i}\mathrm{\mid }-{\delta }_i{)}_{+}^{p-1}\frac{u_{x_i}}{\mathrm{\mid }{u}_{x_i}\mathrm{\mid }}]\mathrm{i}\mathrm{n}{\mathrm{\Omega }}_T=\mathrm{\Omega }\times (0,T],$$ where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^n$ with $n\ge 2$, $p\ge 2$, ${\delta }_1,\dots ,{\delta }_n$ are non-negative numbers and ${(\cdot )}_{+}$ 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.