Regularity for solutions of parabolic equations with anisotropic growth

We prove the integrability for solutions of the nonlinear parabolic problem with anisotropic growth$\{\begin{array}{cc}{u}_t-\sum _{i=1}^n{D}_i\left(a_i\right(x,t,Du\left)\right)=f(x,t)& \text{in}{Q}_T=\Omega \times (0,T),\\ u(x,0)=0& \text{in}\Omega ,\\ u(x,t)=0& \text{on}\Gamma =\partial \Omega \times (0,T)\end{array}$ for the model case of $a_i(x,t,\xi )\approx |{\xi }_i{|}^{p_i-2}{\xi }_i,p_i>1$ for $i=1,\cdots ,n$.