Parabolic Boundary Harnack Inequalities with Right-Hand Side

We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing, for the first time, a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side \(f \in L^q\) for \(q > n+2\). In the case of the heat equation, we also show the optimal \(C^{1-\varepsilon }\) regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are \(C^{1,\alpha }\) in the parabolic obstacle problem and in the parabolic Signorini problem.