The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model

We study a generalization of the fully overdamped Frenkel-Kontorova model in dimension $n\geq 1.$ This model describes the evolution of the position of each atom in a crystal, and is mathematically given by an infinite system of coupled first order ODEs. We prove that for a suitable rescaling of this model, the solution converges to the solution of a Peierls-Nabarro model, which is a coupled system of two PDEs (typically an elliptic PDE in a domain with an evolution PDE on the boundary of the domain). This passage from the discrete model to a continuous model is done in the framework of viscosity solutions

A competition on blow-up of solutions to semilinear wave equations with scale-invariant damping and nonlinear memory term

In this paper, we investigate blow-up of solutions to the Cauchy problem for semilinear wave equations with scale-invariant damping and nonlinear memory term, which can be represented by the Riemann-Liouville fractional integral of order $1-\gamma$ with $\gamma\in(0,1)$. Our main interest is to study mixed influence of various kinds from damping term and the nonlinear memory kernel on the blow-up condition for the power of nonlinearity by using test function method or generalized Kato's type lemma. We find a new competition, particularly for the small value of $\gamma$, on the blow-up between the effective case and the non-effective case