Sharp pointwise estimates for solutions of weakly coupled second order parabolic system in a layer

Abstract

We deal with $m$-component vector-valued solutions to the Cauchy problem for linear both homogeneous and nonhomogeneous weakly coupled second order parabolic system in the layer ${\mathbb{R}}_T^{n+1}={\mathbb{R}}^n\times (0,T)$. We assume that coefficients of the system are real and depending only on $t$, $n\ge 1$ and $T<\infty$. The homogeneous system is considered with initial data in $[L^p({\mathbb{R}}^n){]}^m$, $1\le p\le \infty$. For the nonhomogeneous system we suppose that the initial function is equal to zero and the right-hand side belongs to $[L^p({\mathbb{R}}_T^{n+1}){]}^m\cap [C^{\alpha }\left(\overline{{\mathbb{R}}_T^{n+1}}\right){]}^m$, $\alpha \in (0,1)$. Explicit formulas for the sharp coefficients in pointwise estimates for solutions of these problems and their directional derivative are obtained.

18 pages. arXiv admin note: text overlap with arXiv:1909.01873