Liouville theorems and blow up behaviour in semilinear reaction diffusion systems
Liouville theorems for a class of linear second-order operators with nonnegative characteristic form.
Liouville type theorems for some conformally invariant fully nonlinear equations
This is a report on some joint work with Aobing Li on Liouville type theorems for some conformally invariant fully nonlinear equations.
Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations
We prove a Liouville type theorem for sign-changing radial solutions of a subcritical semilinear heat equation . We use this theorem to derive a priori bounds, decay estimates, and initial and final blow-up rates for radial solutions of rather general semilinear parabolic equations whose nonlinearities have a subcritical polynomial growth. Further consequences on the existence of steady states and time-periodic solutions are also shown.
Local and global existence and behaviour for of solutions of the Navier-Stokes equations
Local asymptotic symmetry of singular solutions to nonlinear elliptic equations.
Local attractivity in nonautonomous semilinear evolution equations
We study the local attractivity of mild solutions of equations in the form u’(t) = A(t)u(t) + f (t, u(t)), where A(t) are (possible) unbounded linear operators in a Banach space and where f is a (possible) nonlinear mapping. Under conditions of exponential stability of the linear part, we establish the local attractivity of various kinds of mild solutions. To obtain these results we provide several results on the Nemytskii operators on the space of the functions which converge to zero at infinity...
Local Decay in time of solutions to Schrödinger's equation with a dilation-analytic interaction.
Local energy decay for several evolution equations on asymptotically euclidean manifolds
Let be a long range metric perturbation of the Euclidean Laplacian on , . We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to . The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group where has a suitable development at zero (resp. infinity).
Local energy decay for waves governed by a system of nonlinear Schrödinger equations in a nonuniform medium.
Local Energy Decay in Even Dimensions for the Wave Equation with a Time-Periodic Non-Trapping Metric and Applications to Strichartz Estimates
2000 Mathematics Subject Classification: 35B40, 35L15.We obtain local energy decay as well as global Strichartz estimates for the solutions u of the wave equation ∂t2 u-divx(a(t,x)∇xu) = 0, t ∈ R, x ∈ Rn, with time-periodic non-trapping metric a(t,x) equal to 1 outside a compact set with respect to x. We suppose that the cut-off resolvent Rχ(θ) = χ(U(T, 0)− e−iθ)−1χ, where U(T, 0) is the monodromy operator and T the period of a(t,x), admits an holomorphic continuation to {θ ∈ C : Im(θ) ≥ 0}, for...
Local energy decay of solutions to the wave equation for nontrapping metrics
Local well-posedness for the incompressible Euler equations in the critical Besov spaces
In this paper we establish the existence and uniqueness of the local solutions to the incompressible Euler equations in , , with any given initial data belonging to the critical Besov spaces . Moreover, a blowup criterion is given in terms of the vorticity field....
Localization and Blow-Up of Thermal Waves in Nonlinear Heat Conduction with Peaking.
Localization effects for eigenfunctions near to the edge of a thin domain
Localization effects for eigenfunctions near to the edge of a thin domain
It is proved that the first eigenfunction of the mixed boundary-value problem for the Laplacian in a thin domain is localized either at the whole lateral surface of the domain, or at a point of , while the eigenfunction decays exponentially inside . Other effects, attributed to the high-frequency range of the spectrum, are discussed for eigenfunctions of the mixed boundary-value and Neumann problems, too.
Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem
We consider the problemwhere is a smooth and bounded domain,
Logarithmic decay of the energy for an hyperbolic-parabolic coupled system
This paper is devoted to the study of a coupled system which consists of a wave equation and a heat equation coupled through a transmission condition along a steady interface. This system is a linearized model for fluid-structure interaction introduced by Rauch, Zhang and Zuazua for a simple transmission condition and by Zhang and Zuazua for a natural transmission condition. Using an abstract theorem of Burq and a new Carleman estimate proved near the interface, we complete the results obtained...
Logarithmic decay of the energy for an hyperbolic-parabolic coupled system
This paper is devoted to the study of a coupled system which consists of a wave equation and a heat equation coupled through a transmission condition along a steady interface. This system is a linearized model for fluid-structure interaction introduced by Rauch, Zhang and Zuazua for a simple transmission condition and by Zhang and Zuazua for a natural transmission condition. Using an abstract theorem of Burq and a new Carleman estimate proved near the interface, we complete the results obtained...
Logarithmically improved blow-up criterion for smooth solutions to the Leray--magnetohydrodynamic equations
In this paper, the Cauchy problem for the Leray--MHD model is investigated. We obtain the logarithmically improved blow-up criterion of smooth solutions for the Leray--MHD model in terms of the magnetic field only in the framework of homogeneous Besov space with negative index.