A boundary blow-up for sub-linear elliptic problems with a nonlinear gradient term.
A boundary Harnack principle for infinity-Laplacian and some related results.
A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations
First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt. 46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman...
A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations
First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt.46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman...
A class of weighted function spaces, and intermediate Caccioppoli-Schauder estimates
A coerciveness inequality for a class of nonelliptic operators and its applications
A Commuting Vectorfields Approach to Strichartz type Inequalities and Applications to Quasilinear Wave Equations
A counterexample to an endpoint bilinear Strichartz inequality.
A discrete maximum principle [Book]
A functional method applied to operator equations.
A fundamental estimate and an inequality of Caccioppoli's type for nonlinear parabolic systems of higher order
A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems – Analysis, assessments and applications to parameter estimation
We address the issue of parameter variations in POD approximations of time-dependent problems, without any specific restriction on the form of parameter dependence. Considering a parabolic model problem, we propose a POD construction strategy allowing us to obtain some a priori error estimates controlled by the POD remainder – in the construction procedure – and some parameter-wise interpolation errors for the model solutions. We provide a thorough numerical assessment of this strategy with the...
A generalization of the Keller-Segel system to higher dimensions from a structural viewpoint
We consider initial boundary problems of a two-chemical substances chemotaxis system. In the four-dimensional setting, it was shown that solutions exist globally in time and remain bounded if the total mass is less than , whereas the solution emanating from some initial data of large magnitude may blows up. This result can be regarded as a generalization of the well-known problem in the Keller–Segel system to higher dimensions. We will compare mathematical structures of the Keller–Segel system...
A generalized maximum principle and estimates of max vrai for nonlinear parabolic boundary value problems
A gradient estimate for solutions of the heat equation
A gradient estimate for solutions of the heat equation. II
The author obtains an estimate for the spatial gradient of solutions of the heat equation, subject to a homogeneous Neumann boundary condition, in terms of the gradient of the initial data. The proof is accomplished via the maximum principle; the main assumption is that the sufficiently smooth boundary be convex.
-harmonic equations and the Dirac operator.
A Harnack Inequality for the Equation ..., when |b| ... Kn + 1.
A lower bound for the gradient of -harmonic functions.
A mixed boundary value problem for the Laplace equation in an angle (1st part)