Real analytic solutions of parabolic equations with time-measurable coefficients (Q2781252)

Using the Bernstein technique the author shows that for any fixed \(t,\) the strong solutions of the linear uniformly parabolic equation \(Lu:= a^{ij}(t)u_{x_ix_j}-u_t=0\) are real analytic in \(Q(t)=\{x: (t,x)\in Q\}\) where \(Q\subset {\mathbb R}^{d+1}\) is a bounded domain and the coefficients \(a^{ij}(t)\) are measurable. It is also considered the fully nonlinear uniformly parabolic equation \(F(D^2u,t)-u_t=0\) in \(Q\) where the operator \(F(M,t)\) is smooth and concave in \(M\in {\mathbb R}^{d^2}\) and measurable in \(t.\) Using the Bernstein technique there are obtained interior estimates for pure second derivatives of the solutions of the nonlinear equation.