Atherosclerosis is a chronic inflammatory disease that poses a serious threat to human health. It starts with the buildup of plaque in the artery wall, which results from the accumulation of pro-inflammatory factors and other substances. In this paper, we propose a mathematical model of early atherosclerosis with a free boundary and time delay. The time delay represents the transformation of macrophages into foam cells. We obtain an explicit solution and analyze the stability of the model and the effect of the time delay on plaque size. We show that for non-radial symmetric perturbations, when $n = 0$ or $1$, the steady-state solution $(M_*,p_*,r_*)$ is linearly stable; when $n \ge 2$, there exists a critical parameter $L_*$ such that the steady-state solution is linearly stable for $L < L_*$ and unstable for $L > L_*$. Moreover, we find that smaller plaque are associated with the presence of time delay.

In this paper, we study the stabilization of a coupled wave system formed by one localized non-regular fractional viscoelastic damping of Kelvin-Voigt type and localized non-smooth coefficients. Our main aim is to prove that the C0-semigroup associated with this model is strong stability and decays polynomially at a rate of t−1. By introducing a new system to deal with fractional Kelvin-Voigt damping, we obtain a new equivalent augmented system, so as to show the well-posedness of the system based on Lumer-Phillips theorem. We achieve the strong stability for the C0-semigroup associated with this new model by using a general criteria of Arendt-Batty, and then turn out a polynomial energy decay rate of order t−1 with the help of a frequency domain approach.

In this paper, we consider the estimates of Dirichlet eigenvalues for Grushin type degenerate elliptic operator in weighted divergence form with a potential $-{\rm div}_{G}(A\nabla_{G})+\langle A\nabla_{G}\phi,\nabla_{G}\rangle-V$. Using the method of Fourier transformation, we get precise lower bound estimates for the eigenvalues. Then, through the way of trail function, we obtain Yang-type inequalities which give upper bounds of eigenvalues.