Conditions for blow-up of solutions of some nonlinear Volterra integral equations

In this paper we give necessary and sufficient conditions for blow-up of solutions for a particular class of nonlinear Volterra equations. We also give some examples.     

The well-posedness of the integral equations for thin wire antennas

In this paper the author shows that the Pocklington and Hallenintegral equations for the current induced on a thin wire byan incident harmonic electromagnetic field are well-posed. Itis also shown that the solutions must lie in certain Sobolevspaces of absolutely continuous functions     

Some properties of solutions to a family of integral equations arising in the models of living systems

We consider the well-posedness problem of nonlinear integral and differential equations with delay which arises in the elaboration of mathematical models of living systems. The questions of existence, uniqueness, and nonnegativity of solutions to these systems on an infinite semiaxis are studied as well as continuous dependence of solutions on the initial data on finite time segments.     

Symmetry of solutions to some systems of integral equations

In this paper, we study some systems of integral equations, including those related to Hardy-Littlewood-Sobolev (HLS) inequalities. We prove that, under some integrability conditions, the positive regular solutions to the systems are radially symmetric and monotone about some point. In particular, we established the radial symmetry of the solutions to the Euler-Lagrange equations associated with the classical and weighted Hardy-Littlewood-Sobolev inequality.

     

Separation of variables for some systems with a fourth-order integral of motion

We construct separation variables for Yehia’s integrable deformations of the Kovalevskaya top and the Chaplygin system on a sphere. In the general case, the corresponding quadratures are given by the Abel-Jacobi map on a two-dimensional submanifold of the Jacobian of a genus-three algebraic curve, which is not hyperelliptic.     

Local well-posedness for higher order nonlinear dispersive systems

We study nonlinear dispersive systems of the form

Global well-posedness of coupled parabolic systems

The initial boundary value problem of a class of reaction-diffusion systems(coupled parabolic systems)with nonlinear coupled source terms is considered in order to classify the initial data for the global existence,finite time blowup and long time decay of the solution.The whole study is conducted by considering three cases according to initial energy:the low initial energy case,critical initial energy case and high initial energy case.For the low initial energy case and critical initial energy case the sufficient initial conditions of global existence,long time decay and finite time blowup are given to show a sharp-like condition.In addition,for the high initial energy case the possibility of both global existence and finite time blowup is proved first,and then some sufficient initial conditions of finite time blowup and global existence are obtained,respectively.     

Conditions for the well-posedness of nonlocal problems for second-order linear differential equations

For second-order linear differential equations, we obtain sharp sufficient conditions for the well-posedness of nonlocal problems with functional and multipoint boundary conditions.     

On well-posedness for some dispersive perturbations of Burgers' equation

We show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin–Ono equation

${?}_{t}u?D_x^{\alpha }{?}_{x}u={?}_x(u^2),0<\alpha \le 1,$

is locally well-posed in $H^s(\mathbb{R})$ when $s>s_{\alpha }:=\frac{3}{2}?\frac{5\alpha }{4}$. As a consequence, we obtain global well-posedness in the energy space $H^{\frac{\alpha }{2}}(\mathbb{R})$ as soon as $\frac{\alpha }{2}>s_{\alpha }$, i.e. $\alpha >\frac{6}{7}$.     

A critical look at some point-process models for repairable systems

The use of a modeldriven approach to the analysis of repairablesystems is considered and shown to be useful as a way of understandingthe characteristics of such a system. However, considerablestatistical problems arise from the use of a set of standardmodel-building elements. In particular, identification problemsarise in many of the models. The argument is illustrated byexamples from software reliability and mechanical reliability.The conclusion is that, in many cases, the exploratory data-analysisapproach is as effective as the use of more sophisticated models.     

On some systems of nonlinear integral Hammerstein-type equations on the semiaxis

We prove the existence for a one-parameter family of solutions of a system of nonlinear integral Hammerstein-type equations on the positive semiaxis and study the asymptotic behavior of the obtained solutions at infinity.     

On some global well-posedness and asymptotic results for quasilinear parabolic equations

We prove that the quasilinear parabolic initial-boundary value problem (1.1) below is globally well-posed in a class of high order Sobolev solutions, and that these solutions possess compact, regular attractors ast+.     

Remark on the well-posedness of the problem of minimization of integral functionals

In this note, we study the well-posedness of the problem of constrained minimization of an integral functional and show that global well-posedness is equivalent to pointwise well-posedness almost everywhere. Our result extends an earlier one by Zolezzi.This research was supported by NSF Grant No. DMS 88-02688.     

Liouville type theorems for integral equations and integral systems

In this paper, we establish some Liouville type theorems for positive solutions of some integral equations and integral systems in R ^N . The main technique we use is the method of moving planes in an integral form.     

Global well-posedness for nonlocal fractional Keller-Segel systems in critical Besov spaces

In this paper, we study Keller-Segel systems with fractional diffusion and a nonlocal term. We establish the global existence, uniqueness and stability of solutions for systems with small initial data in critical Besov spaces. Our main tools are the L^p−L^q estimates for in Besov spaces and the perturbation of linearization.     

On the well-posedness of the prediction-control problem for certain systems of equations

We consider the inverse problem for equations of Sobolev type and their applications to linearized Navier-Stokes systems and phase-field systems. We obtain conditions for the well-defined solvability of these systems.     

Investigation of solutions to one family of mathematical models of living systems

We consider a family of integral equations used as models of some living systems. We prove that an integral equation is reducible to the equivalent Cauchy problem for a non-autonomous differential equation with point or distributed delay dependently on the choice of the survival function of system elements. We also study the issues of the existence, uniqueness, nonnegativity, and continuability of solutions. We describe all stationary solutions and obtain sufficient conditions for their asymptotic stability. We have found sufficient conditions for the existence of a limit of solutions on infinity and present an example of equations where the rate of generation of elements of living systems is described by a unimodal function (namely, the Hill function).