The author obtains an estimate for the spatial gradient of solutions of the heat equation, subject to a homogeneous Neumann boundary condition, in terms of the gradient of the initial data. The proof is accomplished via the maximum principle; the main assumption is that the sufficiently smooth boundary be convex.
A condition for solvability of an integral equation which is connected with the first boundary value problem for the heat equation is investigated. It is shown that if this condition is fulfilled then the boundary considered is 1⁄2-Hölder. Further, some simple concrete examples are examined.
Steady-state system of equations for incompressible, possibly non-Newtonean of the p-power type, viscous flow coupled with the heat equation is considered in a smooth bounded domain Ω ⊂ ℝn, n = 2 or 3, with heat sources allowed to have a natural L1- structure and even to be measures. The existence of a distributional solution is shown by a fixed-point technique for sufficiently small data if p > 3/2 (for n = 2) or if p > 9/5 (for n = 3).