On the mean curvature of spacelike surfaces in certain three-dimensional Robertson–Walker spacetimes and Calabi–Bernstein’s type problems

Several uniqueness results for the spacelike slices in certain Robertson–Walker spacetimes are proved under boundedness assumptions either on the mean curvature function of the spacelike surface or on the restriction of the time coordinate on the surface when the mean curvature is a constant. In the nonparametric case, a uniqueness result and a nonexistence one are proved for bounded entire solutions of some constant mean curvature spacelike differential equations.     

Critical exponent of Fujita type for semilinear wave equations in Friedmann–Lemaître–Robertson–Walker spacetime

We consider the nonlinear massless wave equation belonging to some family of the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. We prove the global in time small data solutions for supercritical powers in the case of decelerating expansion universe.     

Asymptotic behaviour of the relativistic Boltzmann equation in the Robertson–Walker spacetime

In this paper, we study the relativistic Boltzmann equation in the spatially flat Robertson–Walker spacetime. For a certain class of scattering kernels, global existence of classical solutions is proved. We use the standard method of Illner and Shinbrot for the global existence and apply the splitting technique of Guo and Strain for the regularity of solutions. The main interest of this paper is to study the evolution of matter distribution, rather than the evolution of spacetime. We obtain the asymptotic behaviour of solutions and will understand how the expansion of the universe affects the evolution of matter distribution.     

On the geometry of maximal spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime

In this paper, we establish new characterizations of totally geodesic spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime, which is supposed to obey the null convergence condition. As applications, we get nonparametric results concerning to entire maximal vertical graphs in a such ambient spacetime. Proceeding, we obtain a lower estimate of the index of relative nullity of complete r-maximal spacelike hypersurfaces immersed in Robertson–Walker spacetimes of constant sectional curvature. In particular, we prove a sort of weak extension of the classical Calabi–Bernstein theorem.     

Constant mean curvature tori as stationary solutions to the Davey–Stewartson equation

A well known result of Da Rios and Levi-Civita says that a closed planar curve is elastic if and only if it is stationary under the localized induction (or smoke ring) equation, where stationary means that the evolution under the localized induction equation is by rigid motions. We prove an analogous result for surfaces: an immersion of a torus into the conformal 3-sphere has constant mean curvature with respect to a space form subgeometry if and only if it is stationary under the Davey–Stewartson flow.     

On Bernstein-type properties of complete spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime

We apply the well know Omori?CYau generalized maximum principle (Omori in J Math Soc Jpn 19:205?C214, 1967; Yau in Commun Pure Appl Math 28:201?C228, 1975), as well as a suitable extension of it that was established in a joint work with Caminha (Caminha and de Lima in Gen Relat Grav 41:173?C189, 2009), in order to investigate Bernstein-type properties of complete spacelike hypersurfaces immersed in a generalized Robertson?CWalker spacetime, which is supposed to obey a standard convergence condition.     

Global solutions for the generalized Camassa–Holm equation

In this paper, we study the Cauchy problem of the generalized Camassa–Holm equation. Firstly, we prove the existence of the global strong solutions provide the initial data satisfying a certain sign condition. Then, we obtain the existence and the uniqueness of the global weak solutions under the same sign condition of the initial data.     

Global weak solutions for the Dullin–Gottwald–Holm equation

We prove the existence and uniqueness of global weak solutions to the Dullin–Gottwald–Holm equation provided the initial data satisfies certain conditions.     

Construction of weak solutions of a certain stochastic Navier–Stokes equation

We prove the existence of weak solutions of stochastic Navier–Stokes equation on a two-dimensional torus, which appears in a certain variational problem. Our equation does not satisfy the coercivity condition. We construct its weak solutions due to an approximation by a sequence of solutions of equations with enlarged viscosity terms and then by showing an a priori estimate for them.     

Dynamical bifurcation of the damped Kuramoto–Sivashinsky equation

In this paper, we study the primary instability of the damped Kuramoto–Sivashinsky equation under a periodic boundary condition. We prove that it bifurcates from the trivial solution to an attractor which determines the long time dynamics of the system. Using the attractor bifurcation theorem and the center manifold theory, we describe the bifurcated attractor in detail.     

On the solutions of the Dullin–Gottwald–Holm equation in Besov spaces

This paper is concerned with the Cauchy problem for the Dullin–Gottwald–Holm equation. First, the local well-posedness for this system in Besov spaces is established. Second, the blow-up criterion for solutions to the equation is derived. Then, the existence and uniqueness of global solutions to the equation are investigated. Finally, the sharp estimate from below and lower semicontinuity for the existence time of solutions to this equation are presented.     

Global existence of solutions to the Einstein–Maxwell–Massive scalar field system in 3 + 1 formulation on Bianchi spacetimes

Global existence to the coupled Einstein–Maxwell–Massive Scalar Field system which rules the dynamics of a kind of charged pure matter in the presence of a massive scalar field is proved, in Bianchi I–VIII spacetimes. The result is established in the case of a cosmological constant bounded from below by a strictly negative constant depending only on the potential of the massive scalar field, for strictly negative initial data of the mean curvature.     

Continuity of strong solutions of the Reaction–Diffusion equation in initial data

This paper is concerned with the continuity of strong solutions of the Reaction–Diffusion equation with respect to initial conditions. It is proved that the solutions of the equation are continuous in H¹$H^1$ with respect to initial data provided the nonlinearity satisfies a dissipative condition with polynomial growth order p≥2$p\ge 2$.     

Existence of weak solutions of the g-Kelvin–Voight equation

The 2D $g$-Navier–Stokes equations have the form $\frac{?u}{?t}?\nu \Delta u+u.?u+?p=f\text{in }\Omega$ with the continuity equation $?.\left(gu\right)=0\text{in }\Omega$ in a bounded domain $\Omega ?{\mathbb{R}}^2$ where $g=g\left(x_{1,}{x}_2\right)$ is a smooth real valued function defined on $\Omega$. We use the method described by Roh [J. Roh, $g$-Navier Stokes equations, Ph.D. Thesis, University of Minnesota, 2001] for the derivation of $g$-Kelvin–Voight equations represented by $\frac{?u}{?t}?\nu {\Delta }_{g}u+\frac{\nu }{g}(?g??)u?\alpha {\Delta }_g{u}_t+\frac{\alpha }{g}(?g??)u_t+u??u+?p=f\left(x\right)\text{ in }\Omega$$?.\left(gu\right)=0\text{in }\Omega$ We discuss the existence and uniqueness of weak solutions of $g$-Kelvin–Voight equations by the use of the well known Feado–Galerkin method.