Optimal Schauder estimates for parabolic problems with data measurable with respect to time (Q2706202)

This paper is concerned with estimates for solutions of the Cauchy problem for the second-order linear parabolic differential equation NEWLINE\[NEWLINE -u_t +a^{ij}D_{ij}u + b^iD_iu + cu = f NEWLINE\]NEWLINE under weak hypotheses on the coefficients \(a^{ij}\), \(b^i\), and \(c\) as well as on the inhomogeneous term \(f\). The main way in which the usual assumptions are weakened is that these functions are only assumed to be measurable with respect to time but they are Hölder continuous with respect to space. In addition, the function \(f\) is allowed to grow at a certain rate (either like \(|x|^{2m}\) for some nonnegative integer \(m\) or like \(\exp(K|x|^2)\) for some positive constant \(K\); in addition some related growth is placed on the Hölder quotients of \(f\)). When the coefficients are also Hölder with respect to time, a corresponding theory was developed in [\textit{C. P. Mawata}, Differ. Integral Equ. 2, No. 3, 251-274 (1989; Zbl 0742.35030)]. The main part of the effort is to prove suitable a priori estimates when the coefficients depend only on time and \(f=0\). These estimates are proved by using a representation of Green's function in this case and applying a suitable interpolation lemma. The estimates for general equations follow by a simple perturbation argument.