37 research outputs found
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 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 with . 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 , on the blow-up between the
effective case and the non-effective case