  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 $q <
p\frac{N+1}{N-1}$.
