Bounded solutions and interpolative gap bounds for degenerate parabolic double phase problems
Bogi Kim, Jehan Oh
TL;DR
This work addresses gradient higher integrability for weak solutions of the parabolic double-phase equation $u_t - \operatorname{div} \left(|Du|^{p-2}Du + a(z)|Du|^{q-2}Du\right)=0$ in $\Omega_T$, with $a(\cdot)\in C^{\alpha,\frac{\alpha}{2}}$. The authors develop an intrinsic-scaling framework using stopping-time arguments, parabolic Sobolev-Poincaré inequalities, and reverse Hölder estimates, then employ a Vitali-type covering to glue local estimates into a global result. They prove gradient higher integrability under two interpolation gap bounds: (i) bounded solutions satisfy $q \le p+\alpha$, and (ii) solutions with $u\in C(0,T;L^s(\Omega))$ satisfy $q \le p+\frac{s\alpha}{n+s}$, thereby unifying and extending prior elliptic and parabolic double-phase regularity results. The results yield quantitative higher integrability of the gradient, enabling an interpolation between gap bounds and providing insight into regularity thresholds 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 $Ω_T := Ω\times (0,T)$, where $a(\cdot)\in C^{α,\fracα{2}}(Ω_T)$. For bounded solutions, we prove that the result holds under the gap condition $$ q \leq p + α. $$ Moreover, for solutions with $$ u\in C(0,T;L^s(Ω)), \quad s \geq 2, $$ we obtain higher integrability under the gap condition $$ q \leq p + \frac{sα}{n+s}. $$ These results provide an interpolation between the gap bounds in the parabolic double phase setting.