An optimal boundedness result for weak solutions of double phase quasilinear parabolic equations

We obtain local boundedness of weak solutions of double phase quasilinear parabolic equations of the form $$u_t-\text{div} \left(|\nabla u|^{p-2}\nabla u+a(x,t)|\nabla u|^{q-2}\nabla u\right)=0,$$ where, we have imposed the restrictions $\frac{2N}{N+2}<p<\infty$, $0\leq a(x,t)\leq M$ is measurable and $q < p\frac{N+1}{N-1}$.