Journal d'Analyse Mathematique

We study the implications of the conformal conservation law for the time decay of solutions of nonlinear wave equations and present some improvements over previous work of Ginibre and Velo. We also consider the theory of nonlinear scattering and prove asymptotic completeness in a weighted Sobolev space.

In this article, we investigate the parabolic logistic equation with blow-up initial and boundary values $${u_t} - \Delta u = a(x,t)u - b(x,t){u^p}in\Omega \times (0,T),$$ $$u = \infty on\partial \Omega \times (0,T) \cup \overline \Omega \times \{ 0\} ,$$ where Ω is a smooth bounded domain, T > 0 and p > 1 are constants, and a and b are continuous functions, b > 0 in Ω × [0, T) and b(x, T) ≡ 0. We study the existence and uniqueness of positive solutions and their asymptotic behavior near the parabolic boundary. We show that under the extra condition that $$b(x,t) \ge c{(T - t)^\theta }d{(x,\partial \Omega )^\beta } on \Omega \times \left[ {0,T} \right)$$ for some constants c > 0, θ > 0, and β > −2, such a solution stays bounded in any compact subset of Ω as t increases to T, and hence solves the equation up to t = T.

We prove optimal local large deviations for the periodic infinite horizon Lorentz gas viewed as a ℤd-cover (d = 1,2) of a dispersing billiard. In addition to this specific example, we prove a general result for a class of nonuni-formly hyperbolic dynamical systems and observables associated with central limit theorems with nonstandard normalisation.