Analysis of fundamental properties of elastic equations

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In this research project we aimed initially to know properties of solutions of the elastic equations and to clarify roots of those properties. And, using the clarification, we intended to study individual topics concerned with elastic waves.We have seen that the elastic wave equations are near the scalar-valued wave equation although they are one of kinds of hyperbolic systems, and that this is because of positivity and symmetry of the elastic operators. As one of main results concerning this, we have proved that the elastic operators can be expressed of product form of first order operators in the same way as the scalar-valued elliptic operators. This expression cannot be expected for systems of the general form.It is known that there exists the Rayleigh wave in the elastic equations, which does not occur in the scalar-valued equations. We have shown synthetically how this existence affects scattering theories and have constructed a general scattering theory of the Lax-Phillips type accounting the Rayleigh wave.Furthermore, we have studied individual topics in this theory, and have obtained a representation of the scattering kernel. This representation is a fundamental and useful formula to solve inverse scattering problems.A special result on the energy decay has been got also for the scalar-valued equation. Namely it is proved that the total energy does not decay necessarily if the dissipative term is added of non isotropy. This gives an interesting suggestion on behavior of the elastic waves.