Continuum Mech. Thermodyn. (2019) 31:331–340 https://doi.org/10.1007/s00161-018-0731-x ORIGINAL ARTICLE Marin Marin · Andreas Öchsner · Vicentiu Radulescu A polynomial way to control the decay of solutions for dipolar bodies Received: 27 September 2018 / Accepted: 30 October 2018 / Published online: 14 November 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract In our paper, we consider a combination of two sub-cylinders coupled by an interface in a semi- infinite cylinder. Both sub-cylinders are made of dipolar elastic materials. For one of the two sub-cylinders, we will consider the elastostatic problem, and for the other the elastodynamic problem. Thus, the spatial behaviors of the sub-cylinders are of different kind and the question arises whether the evolution of this combination can be controlled. By using a polynomial way, we prove that the decay of solutions for the two problems can be controlled. Keywords Dipolar bodies · Elastostatics · Elastodynamics · Spatial estimates · Upper bound · Polynomial decay 1 Introduction In the last decades, many studies have been published in which various spatial estimates on decay or growth of solutions were made. But these estimates refer to the solutions of some elliptical, parabolic or hyperbolic equations. However, some specific situations have forced the combination of different materials and the cor- responding mathematical models must be based on combinations of different types of equations. Fortunately, research has proven that some combinations of two types of equations do not create any difficulty. Thus, in the case of a mixture of a parabolic equation and an elliptic equations, it was found that the spatial decay of solutions can be described by means of estimations specific to elliptical equations, see [1, 2]. Analogously, in the case of the combination of hyperbolic equations with parabolic equations, it has been shown that the behavior of solutions is very similar to the behavior of solutions of parabolic equations. But, until now, there are not well-clarified the questions regarding the behavior of solutions when combining some elliptic equations with some hyperbolic equations [3, 4]. In the case of an equation with delay, a spatial decay can be found in the Communicated by Andreas Öchsner. M. Marin (B ) Department of Mathematics and Computer Science, Transilvania University of Brasov, 500093 Brasov, Romania E-mail: m.marin@unitbv.ro A. Öchsner Faculty of Mechanical Engineering, Esslingen University of Applied Sciences, 73728 Esslingen, Germany V. Radulescu Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland V. Radulescu Department of Mathematics, University of Craiova, 200585 Craiova, Romania