POINTWISE BOUNDS AND SPATIAL DECAY-ESTIMATES IN HEAT-CONDUCTION PROBLEMS

In this paper we derive a new maximum principle for the absolute value of the gradient of a solution to the heat equation. We then apply this principle to obtain explicit bounds in the associated Dirichlet problem. Finally we derive explicit pointwise St-Venant type spatial decay estimates for solutions of certain initial-boundary value problems and their gradients in the case of unbounded domains.