Related papers: Quantitative unique continuation and observability…

We propose a generalized diffusion equation for a flat Euclidean space subjected to a continuous infinitesimal scale transform. For the special cases of an algebraic or exponential expansion/contraction, governed by time-dependent scale…

This paper is concerned with the spatio-temporal dynamics of nonnegative bounded entire solutions of some reaction-diffusion equations in R N in any space dimension N. The solutions are assumed to be localized in the past. Under certain…

In this article, we develop a new method to prove both global propagation of analyticity and unique continuation in finite time for solutions of semilinear wave-type equations with analytic nonlinearity. It combines control theory…

Let T be an exterior modular correspondence on an irreducible locally symmetric space X. In this note, we show that the isolated fixed points of the power T^n are equidistributed with respect to the invariant measure on X as n tends to…

Existence and uniqueness of radially symmetric self-similar very singular solutions are proved for the singular diffusion equation with gradient absorption {equation*} \partial_t u -\Delta_{p}u+|\nabla u|^q=0, \ \hbox{in} \…

Solutions of a system of wave equations are constructed for both homogeneous and inhomogeneous Dirichlet boundary conditions at every regularity level. We prove that boundary observability, and thus boundary exact controllability, at some…

This paper is concerned with the uniqueness, existence, comparison principle and long-time behavior of solutions to the initial-boundary value problem for a unidirectional diffusion equation. The unidirectional evolution often appears in…

This paper studies the large time behavior of aggregation-diffusion equations. For one spatial dimension with certain assumptions on the interaction potential, the diffusion index $m$, and the initial data, we prove the convergence to the…

This work is devoted to the analysis of the quantum drift-diffusion model derived by Degond et al. The model is obtained as the diffusive limit of the quantum Liouville-BGK equation, where the collision term is defined after a local quantum…

Numerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension. The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose…

We derive a fundamental solution $\mathscr{E}$ to a space-fractional diffusion problem on the half-line. The equation involves the Caputo derivative. We establish properties of $\mathscr{E}$ as well as formulas for solutions to the…

This paper revisits the problems of estimating the position of an object moving in $n$ ($\geq 2$)-dimensional Euclidean space using velocity measurements and either direction or range measurements of one or multiple source points. The…

Hydrodynamic projections, the projection onto conserved charges representing ballistic propagation of fluid waves, give exact transport results in many-body systems, such as the exact Drude weights. Focussing one one-dimensional systems, I…

This work contributes to an understanding of the domain size's effect on the existence and uniqueness of the linear convection--diffusion equation with integral-type boundary conditions, where boundary conditions depend non-locally on…

We investigate the quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients. Quantitative unique continuation described by the vanishing order is a quantitative form of strong unique…

Let $\Omega\subset \mathbb{R}^N$, $N=1,2,3$, be an open bounded and connected set with continuous piecewise $\mathrm{C}^{\infty}$ boundary. Here we deal with almost periodic distributions of the form $u(t,x)=\sum_{n=0}^{+\infty} c_n S_n(x)…

We quantify the uniqueness of continuation from Cauchy or interior data. Our approach consists in extending the existing results in the linear case. As by product we obtain a new stability estimate in the linear case. We also show the…

The $\nu$-metric used in robust control is computed for control systems with parametric uncertainty, governed by a diffusion equation in a bounded one-dimensional spatial region with boundary control and point observation.

We scrutinize the diagrammatic perturbation theory of noninteracting electrons in a random potential with the aim to accomplish a consistent comprehensive theory of quantum diffusion. Ward identity between the one-electron self-energy and…

We consider a class of multidimensional conservation laws with vanishing nonlinear diffusion and dispersion terms. Under a condition on the relative size of the diffusion and dispersion coefficients, we establish that the…