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 $$ u_t-\operatorname{div} \left(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du\right)=0 $$ in $\Omega_T := \Omega\times (0,T)$, where $a(\cdot)\in C^{\alpha,\frac{\alpha}{2}}(\Omega_T)$. For bounded solutions, we prove that the result holds under the gap condition $$ q \leq p + \alpha. $$ Moreover, for solutions with $$ u\in C(0,T;L^s(\Omega)), \quad s \geq 2, $$ we obtain higher integrability under the gap condition $$ q \leq p + \frac{s\alpha}{n+s}. $$ These results provide an interpolation between the gap bounds in the parabolic double phase setting.