Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements

Abstract

This paper is devoted to the study of long-time behavior of the solutions to a one-dimensional full model for the first order phase transitions. Our system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature $\theta$, which is coupled with an evolution equation for the phase change parameter $f$ with a third-order nonlinearity $G_2'(f)$ in place of the customarily constant latent heat. The main novelty of this paper is that we perform an argument to establish Lemma 3.1 which enables us to obtain uniform estimates of the global solutions with respect to time. Asymptotic behavior of the solutions as time goes to infinity and the compactness of the orbit are obtained. Furthermore, we investigate the dynamics of the system and prove the existence of global attractors.