Decay of semilinear damped wave equations:cases without geometric control condition

We consider the semilinear damped wave equation $\partial_{tt}^2 u(x,t)+\gamma(x)\partial_t u(x,t)=\Delta u(x,t)-\alpha u(x,t)-f(x,u(x,t))$. In this article, we obtain the first results concerning the stabilization of this semilinear equation in cases where $\gamma$ does not satisfy the geometric control condition. When some of the geodesic rays are trapped, the stabilization of the linear semigroup is semi-uniform in the sense that $\|e^{At}A^{-1}\|\leq h(t)$ for some function $h$ with $h(t)\rightarrow 0$ when $t\rightarrow +\infty$. We provide general tools to deal with the semilinear stabilization problem in the case where $h(t)$ has a sufficiently fast decay

Asymptotic profiles for a travelling front solution of a biological equation

We are interested in the existence of depolarization waves in the human brain. These waves propagate in the grey matter and are absorbed in the white matter. We consider a two-dimensional model u_t=\Delta u + f(u) \1_{|y|\leq R} - \alpha u \1_{|y|>R}, with $f$ a bistable nonlinearity taking effect only on the domain \Rm\times [-R,R], which represents the grey matter layer. We study the existence, the stability and the energy of non-trivial asymptotic profiles of possible travelling fronts. For this purpose, we present dynamical systems technics and graphic criteria based on Sturm-Liouville theory and apply them to the above equation. This yields three different behaviours of the solution $u$ after stimulation, depending of the thickness $R$ of the grey matter. This may partly explain the difficulties to observe depolarization waves in the human brain and the failure of several therapeutic trials

The determinants of urban public transport: an international comparison and econometric analysis

The analysis presented in this paper is based on the database created by the UITP (International Association of Public Transport), "The Millennium Cities Database", which covers the public transport systems in 100 of the world's cities. It contains data on demography, urban structure, transport networks, daily mobility, environmental impacts, etc. Our analysis demonstrates the contrasts between European and American travel practices. It explores possible links between public transport market share and geographical and economic conditions on the one hand and the characteristics and performances of public transport systems on the other. Our research has generated an explanatory econometric model for public transport market share. To end with, a consideration of the levers that can be used to influence the public transport system leads into a discussion about the future of cities with ''European urban mobility'' and the danger of a slide towards ''American urban mobility'' taking place.Transport systems ; Urban mobility ; Transport policy ; Public transport

Stabilization for the semilinear wave equation with geometric control condition

In this article, we prove the exponential stabilization of the semilinear wave equation with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. The main novelty compared to previous results, is the proof of a unique continuation result in large time for some undamped equation. The idea is to use an asymptotic smoothing effect proved by Hale and Raugel in the context of dynamical systems. Then, once the analyticity in time is proved, we apply a unique continuation result with partial analyticity due to Robbiano, Zuily, Tataru and H\"ormander. Some other consequences are also given for the controllability and the existence of a compact attractor

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

International audienceWe consider the damped wave equation αu tt +u t = u xx −V ′ (u) on the whole real line, where V is a bistable potential. This equation has travelling front solutions of the form u(x, t) = h(x − st) which describe a moving interface between two different steady states of the system, one of which being the global minimum of V. We show that, if the initial data are sufficiently close to the profile of a front for large |x|, the solution of the damped wave equation converges uniformly on R to a travelling front as t → +∞. The proof of this global stability result is inspired by a recent work of E. Risler [38] and relies on the fact that our system has a Lyapunov function in any Galilean frame