Spatial decay of static problems as evolutive second-order equations

Abstract

The aim of this work is to show how we can transform a couple of problems involving equations that describe static processes into dynamic problems with second derivatives with respect to the evolution variable. To demonstrate this effect, we will need to work with the double Laplacian operator. Specifically, we will apply these arguments to the study of a static heat equation of the second gradient type and to the study of thin thermoelastic plates. In both cases, we will demonstrate that we obtain a dissipative analytic semigroup and, consequently, the exponential decay of the solutions. It is worth noting that, through this approximation, we can work with solutions having weaker regularity than the classical solutions, which are the usual type of solutions considered when studying the spatial decay of solutions. Therefore, our approximation also proposes an innovation from this point of view.