Spatial regularity for several nonstandard dynamical problems

The aim of this article is to provide a functional structure for certain nonstandard problems in semi-infinite cylinders with time dependence. These issues are related to Saint-Venant's principle and determine problems that are ill-posed in the Hadamard sense, since, in general, there will not be unique solutions. We will restrict our attention to bounded solutions and will be able to formulate our problem as a “well-posed” problem. This task has been carried out in a couple of static situations, but we will do it here in dynamic cases, provided that certain conditions regarding the formulation of the problem are satisfied. Specifically, we will perform this task for the parabolic heat equation (Fourier) and also for the so-called two-temperature theory. Since both of these determine first-order problems in the evolutionary variable, we will also consider the second-gradient type of heat equation, since this determines a second-order equation in the evolutionary variable.