Sharp pointwise estimates for the gradients of solutions to linear parabolic second-order equation in the layer

We deal with solutions of the Cauchy problem to linear both homogeneous and nonhomogeneous parabolic second-order equations with real constant coefficients in the layer ℝn+1 T=ℝn×(0,T), where n≥1 and T<∞. The homogeneous equation is considered with initial data in Lp(ℝn),1 ≥ p ≥ ∞. For the nonhomogeneous equation we suppose that initial function is equal to zero and the function in the right-hand side belongs to (Formula presented.), p>n + 2 and α ∈ (0,1). Explicit formulas for the sharp coefficients in pointwise estimates for the length of the gradient to solutions to these problems are obtained. © 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group.