Asymptotic profiles of solutions for the isothermal Falk-Konopka system of shape memory alloys with weak damping

摘要

The aim of this paper is to investigate asymptotic behavior of solutions to the Cauchy problem of certain fourth-order semilinear elastic system with weak damping in three space dimension. The system corresponds to the isothermal case of the thermoelastic system representing the martensitic phase transitions on shape memory alloys. We give several asymptotic profiles of solutions. By considering the second-order expansion of solutions, we obtain more precise information about asymptotic behavior of solutions as t →∞, and observe the contribution of the nonlinear term to solutions. Moreover, we give the third-order expansion formula which enable us to clarify the relation between the fourth- and the second-order derivative term.