Initial trace of positive solutions of a class of degenerate heat equation with absorption

We study the initial value problem with unbounded nonnegative functions or measures for the equation \prt_tu-\Gd_p u+f(u)=0 in \BBR^N\ti(0,\infty) where $p>1$, \Gd_p u = \text{div}(\abs {\nabla u}^{p-2} \nabla u) and $f$ is a continuous, nondecreasing nonnegative function such that $f(0)=0$. In the case $p>\frac{2N}{N+1}$, we provide a sufficient condition on $f$ for existence and uniqueness of the solutions satisfying the initial data k\gd_0 and we study their limit when $k\to\infty$ according $f^{-1}$ and $F^{-1/p}$ are integrable or not at infinity, where F(s)=\int_0^s f(\gs)d\gs. We also give new results dealing with non uniqueness for the initial value problem with unbounded initial data. If $p>2$, we prove that, for a large class of nonlinearities $f$, any positive solution admits an initial trace in the class of positive Borel measures. As a model case we consider the case f(u)=u^\ga \ln^\gb(u+1), where \ga>0 and \gb\geq 0

Wiener criteria for existence of large solutions of quasilinear elliptic equations with absorption

We obtain sufficient conditions expressed in terms of Wiener type tests involving Hausdorff or Bessel capacities for the existence of large solutions to equations (1) -\Gd_pu+e^{\lambda u}+\beta=0 or (2) -\Gd_pu+\lambda |u|^{q-1}u+\beta=0 in a bounded domain \Gw when $q>p-1>0, \lambda>0$ and $\beta\in\mathbb{R}$. We apply our results to equations (3) -\Gd_pu+a\abs{\nabla u}^{q}+bu^{s}=0, (4) \Gd_p u+u^{-\gamma}=0 with $10, b\geq 0$ and $(q-p+1)+b(s-p+1)>0$, $\gamma>0$.Comment: To appear in Potential Analysi

Local and global properties of solutions of heat equation with superlinear absorption

We study the limit, when $k\to\infty$ of solutions of $u_t-\Delta u+f(u)=0$ in $R^N\times(0,\infty)$ with initial data k\gd, when $f$ is a positive increasing function. We prove that there exist essentially three types of possible behaviour according $f^{-1}$ and $F^{-1/2}$ belong or not to $L^1(1,\infty)$, where $F(t)=\int_0^t f(s)ds$. We emphasize the case where f(u)=u((\ln u+1))^{\alpha}. We use these results for giving a general result on the existence of the initial trace and some non-uniqueness results for regular solutions with unbounded initial data

Reconstruction and stability in acousto-optic imaging for absorption maps with bounded variation

The aim of this paper is to propose for the first time a reconstruction scheme and a stability result for recovering from acoustic-optic data absorption distributions with bounded variation. The paper extends earlier results on smooth absorption distributions. It opens a door for a mathematical and numerical framework for imaging, from internal data, parameter distributions with high contrast in biological tissues

Proofs of Control of a Quadrotor and a Ground Vehicle Manipulating an Object

This paper focuses on the control of a cooperative system composed of an Unmanned Aerial Vehicle (UAV) and an Unmanned Ground Vehicle (UGV) manipulating an object. The two units are subject to input saturations and collaborate to move the object to a desired pose characterized by its position and inclination. The dynamics are derived using Euler-Lagrange method. A pre-stabilizing control law is proposed where the UGV is tasked to deploy the object to a certain position whereas the UAV adjusts its inclination. In particular, a proportional-derivative control law is proposed for the UGV, and a cascade control approach is used for the UAV, where the inner loop controls the attitude of the UAV and the outer loop stabilizes the inclination of the object. Then, we prove the stability of the points of equilibrium using small gain arguments. To ensure constraints satisfaction at all times, a reference governor unit is added to the pre-stabilizing control scheme. Finally, numerical results combined with experimental results are provided to validate the effectiveness of the proposed control scheme in practice.Comment: 16 pages, 7 figure

XQ2P: Efficient XQuery P2P Time Series Processing

In this demonstration, we propose a model for the management of XML time series (TS), using the new XQuery 1.1 window operator. We argue that centralized computation is slow, and demonstrate XQ2P, our prototype of efficient XQuery P2P TS computation in the context of financial analysis of large data sets (>1M values)