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

Abstract

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