Local boundedness for solutions to parabolic double phase problems

Abstract

In this paper, we investigate the local boundedness of weak solutions to parabolic double phase equations of type $$u_t-\mathrm{div}(|Du{|}^{p-2}Du+a(x,t)|Du{|}^{q-2}Du)=0\text{in }{\Omega }_T:=\Omega \times (0,T),$$ where $0\le a(\cdot )\in L^{\infty }({\Omega }_T)$ and $\frac{2n}{n+2}<p<q<p\left(\frac{n+2}{n}\right)$, so that both the degenerate range $p\ge 2$ and the singular range $p<2$ are covered. To this end, we derive the Caccioppoli inequality and recall a parabolic embedding theorem, which are then utilized in an iteration method.