Global Properties Of Solutions Research Articles

Asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid

UDC 517.9 We consider the Cauchy problem for an evolution equation modeling bidirectional surface waves in a convecting fluid. We study the existence, uniqueness, and asymptotic properties of global solutions to the initial value problem associated withthis equation in . We obtain some polynomial decay estimates of the energy.

Global Structure of Spherically Symmetric Solutions of Einstein’s Equations with an Electromagnetic Field

We classify all global spherically symmetric solutions of Einstein’s equations with an electromagnetic field and a cosmological constant. The classification comprises 11 topologically inequivalent solutions. The spacetime is assumed to be a warped product of two surfaces. The study of global properties of solutions is carried out by the method of conformal blocks, which consists in analyzing the zeros and poles of a conformal factor contained in the spacetime metric.

Existence, uniqueness and global behavior of the solutions to some nonlinear vector equations in a finite dimensional Hilbert space

The initial value problem and global properties of solutions are studied for the vector equation: (‖u′‖lu′)′+‖A12u‖βAu+g(u′)=0 in a finite dimensional Hilbert space under suitable assumptions on g.

Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source

In this paper, a class of pseudo-parabolic equations with an exponential source is concerned. First, by the elliptic regularity theory, we establish the local existence and uniqueness of solutions. Sequently, the global existence and blow up of solutions with lower initial energy is considered via the potential wells method. Finally, we find a sufficient condition under which the solution blows up without any limit of initial energy by constructing a new functional which falls between the energy functional and Nehari functional. During this process, the properties of global solutions are also studied.

Global behavior of the solutions to a class of nonlinear, singular second order ODE

In this paper the initial value problem and global properties of solutions are studied for the scalar second order ODE: (|u′|lu′)′+c|u′|αu′+d|u|βu=0, where α,β,l,c,d are positive constants. In particular, existence, uniqueness and regularity as well as optimal decay rates of solutions to 0 are obtained depending on the various parameters, and the oscillatory or non-oscillatory behavior is elucidated.

On a representation of linear differential equations

Here we present a certain representation of linear differential equations that enables us to describe global transformations, a construction of global canonical forms, distribution of zeros of solutions and other global properties of solutions of the equations studied.

On the Log-Laplace Equation for Nonlocal Operators Generating Sub-Markovian Semigroups

We consider the semilinear Cauchy problem for a class of pseudo-differential operators generating sub-Markovian semigroups. Solutions of such problems with negative definite nonlinearity play an important role in constructing branching measure-valued processes. We establish local existence and uniqueness of solutions in the context of the Dirichlet space associated to the problem. Comparison and global properties of solutions are also studied.

Asymptotic Properties of Global Solutions of Systems of Functional Differential Equations of Neutral Type with Nonlinear Deviations of an Argument

For a system of functional differential equations of the neutral type with nonlinear deviations of an argument dependent on an unknown function, we establish sufficient conditions for the existence of a solution continuously differentiable and bounded for t ∈ ℝ and study its properties.

Global features of nonlinear renormalization-group equations

The analysis of selected nonlinear problems in the renormalization group is found to show striking contrasts between the usual local linearized fixed-point analysis and the properties of global solutions of nonlinear equations derived from an approximation of the Wegner-Houghton differential formulation. The competition between various fixed points that is incorporated in general global solutions can upset the asymptotically valid critical behavior deduced from the local analysis. In general, the critical-point exponents of such a solution will not satisfy equalities, but rather the corresponding inequalities. However, these nonscaling solutions have extraneous singularities that are not related to the thermodynamic singularities of the system. If singularities of this type are excluded, then the global solution has the same critical-point exponents as the local solution derived by linearizing around the stablest fixed point. It is shown that in this case the critical surface in the Hamiltonian space is closely related to the surface of order-2 critical points in a thermodynamic field space. The boundaries of this surface are correspondingly related to the critical points of higher order in this thermodynamic space. The nonlinear global solution predicts multiple power scaling behavior from a single scaling equation deduced from the renormalization group. Previously such behavior was obtained by postulating the simultaneous validity of two of more "linear" scaling hypotheses.