Related papers: Quantitative unique continuation and observability…
Given an equidistributed set in the whole Euclidean space, we have established in [1] that there exists a constant positive $C$ such that the observability inequality of diffusion equations holds for all $T\in]0,1[$, with an observability…
We study the propagation properties of the solutions of the finite-difference space semi-discrete wave equation on an uniform grid of the whole Euclidean space. We provide a construction of high frequency wave packets that propagate along…
In this paper, a quantitative estimate of unique continuation for the stochastic heat equation with bounded potentials on the whole Euclidean space is established. This paper generalizes the earlier results in [29] and [17] from a bounded…
In this paper we establish an observability inequality for the heat equation with bounded potentials on the whole space. Roughly speaking, such a kind of inequality says that the total energy of solutions can be controlled by the energy…
We establish near-optimal quantitative uniqueness of continuation for solutions of evolution equations vanishing on the lateral boundary. These results were obtained simply by combining existing observability inequalities and energy…
Self-similar solutions of the coherent diffusion equation are derived and measured. The set of real similarity solutions is generalized by the introduction of a nonuniform phase surface, based on the elegant Gaussian modes of optical…
We study the unique continuation property for the neutron transport equation and for a simplified model of the Fokker-Planck equation in a bounded domain with absorbing boundary condition. An observation estimate is derived. It depends on…
In this paper, we establish a globally quantitative estimate of unique continuation at one time point for solutions of parabolic equations with Neumann boundary conditions in bounded domains. Our proof is mainly based on Carleman commutator…
We prove in this Note an observation estimate at one point in time for the Kolmogorov equation in the whole space. Such estimate implies the observability and the null controllability for the Kolmogorov equation with a control region which…
Stochastic diffusion equations are crucial for modeling a range of physical phenomena influenced by uncertainties. We introduce the generalized finite difference method for solving these equations. Then, we examine its consistency,…
We consider solvability of the generalized reaction-diffusion equation with both space- and time-dependent diffusion and reaction terms by means of the similarity method. By introducing the similarity variable, the reaction-diffusion…
A Carleman estimate and the unique continuation property of solutions for a multi-terms time fractional diffusion equation up to order $\alpha\,\,(0<\alpha<2)$ and general time dependent second order strongly elliptic time elliptic operator…
In this paper, we study quantitative spatial analytic bounds and unique continuation inequalities of solutions for fractional heat equations with an analytic lower order term on the whole space. At first, we show that the solution has a…
In these lectures we present some useful techniques to study quantitative properties of solutions of elliptic PDEs. Our aim is to outline a proof of a recent result on propagation of smallness. The ideas are also useful in the study of the…
A well-known boundary observability inequality for the elasticity system establishes that the energy of the system can be estimated from the solution on a sufficiently large part of the boundary for a sufficiently large time. This…
We study the wave equation in an interval with two linearly moving endpoints. We give the exact solution by a series formula, then we show that the energy of the solution decay at the rate $1/t$. We also establish observability results, at…
By mean of generalized Fourier series and Parseval's equality in weighted $L^{2}$--spaces, we derive a sharp energy estimate for the wave equation in a bounded interval with a moving endpoint. Then, we show the observability, in a sharp…
We revisit the problem of uniqueness for the Ricci flow and give a short, direct proof, based on the consideration of a simple energy quantity, of Hamilton/Chen-Zhu's theorem on the uniqueness of complete solutions of uniformly bounded…
We propose a new fully-discretized finite difference scheme for a quantum diffusion equation, in both one and two dimensions. This is the first fully-discretized scheme with proven positivity-preserving and energy stable properties using…
We investigate the well-posedness of the fast diffusion equation (FDE) in a wide class of noncompact Riemannian manifolds. Existence and uniqueness of solutions for globally integrable initial data was established in [5]. However, in the…