Existence and regularity theory and qualitative analysis

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The adequate modeling of dynamical behavior in piezoceramic materials of the considered type requires detailed knowledge about certain classes of partial differential equations which appropriately account for both the dispersive and the dissipative among the mechanisms influencing the respective real system.The project part ANA firstly intends to develop fundamental solution theories for various models relevant for the work in the project part MESS; here, a focus is to be set not only on basic questions related to solvability and well-posedness, but beyond this also on aspects of immediate relevance for the investigations in the project parts OPT and SIM, inter alia in the course of a suitably far-reaching regularity analysis. The classes of evolution equations under consideration are to be selected in such a way that within a hierarchy of models with increasing level of complexity, both simplifying systems concentrating on particular mechanisms and comprehensive models including all essential components will be studied.Apart from that, a second central part of this project aims at identifying key characteristics of the considered models by an analysis of qualitative solution behavior. Here in line with the application contexts considered in the project part MESS, this especially includes a qualitative analysis of nonlinear and particularly thermal effects arising in the examined framework of interaction between mechanical and thermal fields. A predominant focus in this direction is to be laid on analytical descriptions of either thespontaneous occurrence or a possible exclusion of thermal "hot spot" phenomena in the shape of locally concentrated temperature distributions.