Mathematics

This paper is devoted to Bresse systems, a robust model for circular beams, given by a set of three coupled wave equations. The main objective is to establish the existence of global attractors for dynamics of semilinear problems with localized damping. In order to deal with localized damping a unique continuation property (UCP) is needed. Therefore we also provide a suitable UCP for Bresse systems. Our strategy is to set the problem in a Riemannian geometry framework and see the system as a single equation with different Riemann metrics. Then we perform Carleman-type estimates to get our result.

Accurate asymptotic solutions are presented for axisymmetric deformation of thin layers constrained by either two rigid plates or two rigid spheres. Those solutions are developed using Saint-Venant's principle and the layer thinness as the only assumptions. The solutions are valid in the entire range of Poisson's ratios, and allow one to distinguish among compressible, nearly incompressible, and incompressible layers. That classification involves both material and geometric parameters.

In this paper, we study the mixed problem for new class of nonlinear reaction-diffusion PDEs with the nonlocal nonlinearity with variable exponents. Here we obtain results on solvability and behavior of solutions both when these are yet dissipative and when these get become non-dissipative problems. These problems are possess special properties: these can be to remain dissipative all time or can get become non dissipative after finite time. It is shown that if the studied problems get become non-dissipative can have an infinite number of different unstable solutions with varying speeds and also an infinite number of different states of spatio-temporal (diffusion) chaos that are generated by cascades of bifurcations governed by the corresponding steady states. The behavior of these solutions is analyzed in detail and it is explained how space-time chaos can arise. Since these problems desctibe the dynamics of cancer, we explain the behavior of the dynamics of cancer using obtained here results.

We study quasilinear elliptic equations of the type -\Delta_{p} u = \sigma u^{q} + \mu \; \; \text{in} \;\; \bf{R}^n in the case 0<q< p-1 , where \mu and \sigma are nonnegative measurable functions, or locally finite measures, and \Delta_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u) is the p -Laplacian. Similar equations with more general local and nonlocal operators in place of \Delta_{p} are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions u : u(x) \approx (\bf{W}_p \sigma(x))^{\frac{p-q}{p-q-1}} + \bf{K}_{p,q} \sigma(x) + \bf{W}_p \mu (x), \quad x \in \bf{R}^n, where \bf{W}_p and \bf{K}_{p, q} are, respectively, the Wolff potential and the intrinsic Wolff potential, with the constants of equivalence depending only on p , q and n . The contributions of \mu and \sigma in these pointwise estimates are totally separated, which is a new phenomenon even when p=2 . In the homogeneous case \mu=0 , such estimates were obtained earlier by a different method only for minimal positive solutions.

We consider the nonlinear Schrödinger equation (NLS) on a torus of arbitrary dimension. The equation is studied in presence of an external potential field whose time-dependent amplitude is taken as control. Assuming that the potential satisfies a saturation property, we show that the NLS equation is approximately controllable between any pair of eigenstates in arbitrarily small time. The proof is obtained by developing a multiplicative version of a geometric control approach introduced by Agrachev and Sarychev. We give an application of this result to the study of the large time behavior of the NLS equation with random potential. More precisely, we assume that the amplitude of the potential is a random process whose law is 1 -periodic in time and non-degenerate. Combining the controllability with a stopping time argument and the Markov property, we show that the trajectories of the random equation are almost surely unbounded in regular Sobolev spaces.

We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a strong enough energy estimate, we prove that its low super-actions are almost preserved for very long times. Roughly speaking, it means that, to exchange energy, modes have to oscillate at the same frequency. Contrary to the previous existing results, we do not require the solutions to be especially smooth. They only have to live in the energy space. We apply our result to nonlinear Klein-Gordon equations in dimension d = 1 and nonlinear Schr{ö}dinger equations in dimension d ??2.

In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein - de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the linear damping term with critical decay rate. We prove in this work that the results obtained in a previous work, where the damping coefficient takes two particular values 0 or 2 , can be extended for any positive damping coefficient. In the blow-up case, the upper bound of the exponent of the nonlinear term is given, and the lifespan estimate of the global existence time is derived as well.

A particular type of dyadic model for the magnetohydrodynamics (MHD) with forward energy cascade is studied. The model includes intermittency dimension δ in the nonlinear scales. It is shown that when δ is small, positive solution with large initial data for either the dyadic MHD model or the dyadic Hall MHD develops blow-up in finite time.

We describe the asymptotic behavior of positive solutions u ϵ of the equation ?�Δu+au=3 u 5?��?in Ω??R 3 with a homogeneous Dirichlet boundary condition. The function a is assumed to be critical in the sense of Hebey and Vaugon and the functions u ϵ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation ?�Δu+(a+ϵV)u=3 u 5 in Ω .

Motivated by the astrophysical problems of star formations from molecular clouds, we make the first step on the possible behaviors of certain molecular clouds. This article (1) establishes the diffuse boundary problem of Euler-Poisson system for describing the evolution of molecular clouds; (2) proves the local existence, uniqueness and continuation principle of the classical solution to the diffuse boundary problem; (3) proves the classical solution (without any symmetry condition) to the diffuse problem blows up at finite time if there is no the first class of global solution and the data is admissible (large scale, irregularly-shaped, expanding and rotational molecular clouds); (4) proves certain singularities can be removed from the boundary if the data is strongly admissible. This result partially answers Makino's conjecture [69] on the finite blowup of any tame solution without symmetries and gives the possibilities of star formations, fragmentation and possibilities of formations of shocks and physical vacuum boundary in perfect fluids with Newtonian self-gravity.