Bounded solutions and interpolative gap bounds for degenerate parabolic double phase problems

We establish gradient higher integrability results for weak solutions to degenerate parabolic equations of double phase type $$u_t-\mathrm{div}(\mathrm{\mid }Du{\mathrm{\mid }}^{p-2}Du+a(x,t)\mathrm{\mid }Du{\mathrm{\mid }}^{q-2}Du)=0$$ in ${\mathrm{\Omega }}_T:=\mathrm{\Omega }\times (0,T)$, where $a(\cdot )\in C^{\alpha ,\frac{\alpha }{2}}({\mathrm{\Omega }}_T)$. For bounded solutions, we prove that the result holds under the gap condition $$q\le p+\alpha \mathrm{.}$$ Moreover, for solutions with $$u\in C(0,T;L^s(\mathrm{\Omega })),s\ge 2,$$ we obtain higher integrability under the gap condition $$q\le p+\frac{s\alpha }{n+s}\mathrm{.}$$ These results provide an interpolation between the gap bounds in the parabolic double phase setting.