Second-order time degenerate parabolic equations

Abstract:

We study the degenerate parabolic operator $Lu = \sum \nolimits _{i,j = 1}^n {{a^{ij}}{u_{{x_j}{x_j}}}} + \sum \nolimits _{i = 1}^n {{b^i}{u_{{x_i}}}} - c{u_t} + du$ where the coefficients of $L$ are bounded, real-valued functions defined on a domain $D = \Omega \times (0,T] \subset {R^{n + 1}}$. Classically, $c(x,t) \equiv 1$ or, equivalently, $c(x,t) \geq \eta > 0$ for all $(x,t) \in \bar D$. We assume only that $c$ is non-negative. We prove weak maximum principles and Harnack inequalities. Assume that ${a^{ij}}$ is constant, the coefficients of $L$ and $f$ and their derivatives with respect to time are uniformly Hölder continuous (exponent $\alpha$) in $\bar D,\bar D$ has sufficiently nice boundary, $c > 0$ on the normal boundary of $D$, $\psi \in {\bar C_{z + \alpha }}$, and $L\psi = f$ on $\partial B = \partial (\bar D \cap \{ t = 0\} )$. Then there exists a unique solution $u$ of the first initial-boundary value problem $Lu = f,u = \psi$ on $\bar B + (\partial B \times [0,T])$; and, furthermore, $u \in {\bar C_{2 + \alpha }}$. All results require proofs that differ substantially from the classical ones.