Real analytic solutions of parabolic equations with time-measurable coefficients

Abstract:

We use Bernstein’s technique to show that for any fixed $t$, strong solutions $u(t,x)$ of the uniformly parabolic equation $Lu:=a^{ij}(t)u_{x_{i}x_{j}}-u_{t}=0$ in $Q$ are real analytic in $Q(t)=\{x:(t,x)\in Q\}$. Here, $Q\subset \mathbb {R}^{d+1}$ is a bounded domain and the coefficients $a^{ij}(t)$ are measurable. We also use Bernstein’s technique to obtain interior estimates for pure second derivatives of solutions of the fully nonlinear, uniformly parabolic, concave equation $F(D^{2}u,t)-u_{t}=0$ in $Q$, where $F$ is measurable in $t$.