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

TL;DR

This paper proves higher gradient integrability for solutions to degenerate parabolic double phase equations under specific gap conditions, extending the understanding of regularity in such problems.

Contribution

It establishes new gradient higher integrability results for degenerate parabolic double phase equations under optimal gap conditions, including for solutions with certain continuity properties.

Findings

Higher integrability holds under the gap condition q ≤ p + α for bounded solutions.

For solutions in C(0,T;L^s(Ω)), higher integrability is valid if q ≤ p + (sα)/(n+s).

Results interpolate between known gap bounds in the parabolic double phase setting.

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.