Continuity results for degenerate diffusion equations with $L^{p}_t L^{q}_{x}$ drifts

Abstract

In this paper, we study local uniform continuity of nonnegative weak solutions to degenerate diffusion-drift equations in the form \[ u_{t} = \Delta u^{m} + \nabla\cdot \left( B (x,t) \, u\right), \quad \text{for } m \geq 1 \] assuming a vector field $B \in L^{p}_t L^{q}_{x}$. Regarding local H\"{o}lder continuity, we provide a sharp condition on $p$ and $q$, which is referred to as the subcritical region. In the critical region, the divergence-free condition is essential to providing uniform continuity which depends on the modulus continuity of $B$.