A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities

A priori bounds for solutions of a wide class of quasilinear degenerate elliptic inequalities are proved. As an outcome we deduce sharp Liouville theorems. Our investigation includes inequalities associated to p-Laplacian and the mean curvature operators in Carnot groups setting. No hypotheses on the solutions at infinity are assumed. General results on the sign of solutions for quasilinear coercive/noncoercive inequalities are considered. Related applications to population biology and chemical reaction theory are also studied.     

Asymptotic Behavior of Solutions for a Class of Fractional Integro-differential Equations

In this paper, we study the asymptotic behavior of solutions for a general class of fractional integro-differential equations. We consider the Caputo fractional derivative. Reasonable sufficient conditions under which the solutions behave like power functions at infinity are established. For this purpose, we use and generalize some well-known integral inequalities. It was found that the solutions behave like the solutions of the associated linear differential equation with zero right hand side. Our findings are supported by examples.     

On the minimal solution for some variational inequalities

Applying an asymptotic method, the existence of the minimal solution to some variational elliptic inequalities defined on bounded or unbounded domains is established. The minimal solution is obtained as limit of solutions to some classical variational inequalities defined on domains becoming unbounded when some parameter tends to infinity. The considered quasilinear operators are only monotone (not strictly) and noncoercive. Some related comparison principles are also investigated.     

Existence of periodic solutions for first order differential equations

In this paper we study the periodic boundary value problem for first order differential equations by combining techniques of the theory of differential inequalities, namely the method of upper and lower solutions, and the alternative method for nonlinear problems at resonance. The results obtained are in terms of the behavior of the nonlinear part at infinity.     

Steady Plane Flow of Viscoelastic Fluid Past an Obstacle

We consider the steady plane flow of certain classes of viscoelastic fluids in exterior domains with a non-zero velocity prescribed at infinity. We study existence as well as asymptotic behaviour of solutions near infinity and show that for sufficiently small data the solution decays near infinity as fast as the fundamental solution to the Oseen problem.     

Asymptotic Behavior of Solutions to a Class of Semilinear Parabolic Equations with Boundary Degeneracy

This paper concerns the asymptotic behavior of solutions to one-dimensional semilinear parabolic equations with boundary degeneracy both in bounded and unbounded intervals. For the problem in a bounded interval, it is shown that there exist both nontrivial global solutions for small initial data and blowing-up solutions for large one if the degeneracy is not strong. Whereas in the case that the degeneracy is strong enough, the nontrivial solution must blow up in a finite time. For the problem in an unbounded interval, blowing-up theorems of Fujita type are established. It is shown that the critical Fujita exponent depends on the degeneracy of the equation and the asymptotic behavior of the diffusion coefficient at infinity, and it may be equal to one or infinity. Furthermore, the critical case is proved to belong to the blowing-up case.     

Zero-Hopf Bifurcation at the Origin and Infinity for a Class of Generalized Lorenz System

In this paper, the zero-Hopf bifurcations are studied for a generalized Lorenz system. Firstly, by using the averaging theory and normal form theory, we provide sufficient conditions for the existence of small amplitude periodic solutions that bifurcate from zero-Hopf equilibria under appropriate parameter perturbations. Secondly, based on the Poincar{\''e} compactification, the dynamic behavior of the generalized Lorenz system at infinity is described, and the zero-Hopf bifurcation at infinity is investigated. Additionally, for the above theoretical results, some related illustrations are given by means of the numerical simulation.     

Nonexistence of nonnegative solutions of elliptic systems of divergence type

In this paper we deal with noncoercive elliptic systems of divergence type, that include both the p-Laplacian and the mean curvature operator and whose right-hand sides depend also on a gradient factor. We prove that any nonnegative entire (weak) solution is necessarily constant. The main argument of our proofs is based on previous estimates, given in Filippucci (2009) [12] for elliptic inequalities. Actually, the main technique for proving the central estimate has been developed by Mitidieri and Pohozaev (2001) [23] and relies on the method of test functions. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.     

Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems

In this paper we introduce the theory of dominant solutions at infinity for nonoscillatory discrete symplectic systems without any controllability assumption. Such solutions represent an opposite concept to recessive solutions at infinity, which were recently developed for such systems by the authors. Our main results include: (i) the existence of dominant solutions at infinity for all ranks in a given range depending on the order of abnormality of the system, (ii) construction of dominant solutions at infinity with eventually the same image, (iii) classification of dominant and recessive solutions at infinity with eventually the same image, (iv) limit characterization of recessive solutions at infinity in terms of dominant solutions at infinity and vice versa, and (v) Reid’s construction of the minimal recessive solution at infinity. These results are based on a new theory of genera of conjoined bases for symplectic systems developed for this purpose in this paper.     

Transmission problem in thermoelasticity with symmetry

In this paper we show the existence, uniqueness and regularityof the solutions to the thermoelastic transmission problem.Moreover, when the solutions are symmetrical we show that theenergy decays exponentially as time goes to infinity, no matterhow small is the size of the thermoelastic part.     

Global topological perturbations of nonlinear elliptic eigenvalue problems

This paper is concerned with bifurcation from infinity for nonlinear elliptic equations, which are not necessarily linearizable at infinity. The methods employed are global perturbation techniques by means of which one obtains access to continua of positive solutions bifurcating from infinity via continua bifurcating from trivial solutions.     

Reduction from a Semi-Infinite Interval to a Finite Interval of a Nonlinear Boundary Value Problem for a System of Second-Order Equations with a Small Parameter

We consider a boundary value problem over a semi-infinite interval for a nonlinear autonomous system of second-order ordinary differential equations with a small parameter at the leading derivatives. We impose certain constraints on the Jacobian under which a solution to the problem exists and is unique. To transfer the boundary condition from infinity, we use the well-known approach that rests on distinguishing the variety of solutions satisfying the limit condition at infinity. To solve an auxiliary Cauchy problem, we apply expansions of a solution in the parameter.     

Asymptotic solutions for mixed-type equations with a small deviation

An asymptotic matrix solution is formulated for a class of mixed-type linear vector equations with a single variable deviation which is small at infinity. This matrix solution describes the asymptotic behavior of all exponentially bounded solutions. A sufficient condition is obtained for there to be no other solutions.     

INTERPOLATION INEQUALITIES AND DECAY ESTIMATES FOR DERIVATIVES

Under the only assumption of the cone property for a given domain Rn, it is proved that inter polation inequalities for intermediate derivatives of functions in the Sobolev spaces Wm,p or even in some weighted Sobolve spaces still hold. That is, the usual additional restrictions that is bounded or has the uniform cone property are both removed. The main tools used are polynomial inequalities, by which it is also ob- tained pointwise version inter polation inequalities for smooth and analytic functions. Such pointwise version in - equalities give explicit decay estimates for derivatives at infinity in unbounded domains which have the cone property. As an application of the decay ertimates, a previous result on radial basis function approximation of smooth functions is extended to the derivative-simultaneous approximation.     

A transformation to avoid solutions at infinity for polynomial systems

In finding all solutions to polynomial systems, the existence of solutions at infinity makes the problem more difficult, particularly when a continuation method is being used as the solution technique. Systems with solutions at infinity do arise in applications; for example, in computer graphics and geometric modeling. In this paper, a simple transformation of the system is given which eliminates solutions at infinity.     

Value Functions and Transversality Conditions for Infinite-Horizon Optimal Control Problems

This paper investigates a relationship between the maximum principle with an infinite horizon and dynamic programming and sheds new light upon the role of the transversality condition at infinity as necessary and sufficient conditions for optimality with or without convexity assumptions. We first derive the nonsmooth maximum principle and the adjoint inclusion for the value function as necessary conditions for optimality. We then present sufficiency theorems that are consistent with the strengthened maximum principle, employing the adjoint inequalities for the Hamiltonian and the value function. Synthesizing these results, necessary and sufficient conditions for optimality are provided for the convex case. In particular, the role of the transversality conditions at infinity is clarified.     

The Separation of Zeros of Solutions of Higher Order Linear Differential Equations With Entire Coefficients

Suppose that a homogeneous linear differential equation has entire coefficients of finite order and a fundamental set of solutions each having zeros with finite exponent of convergence. Upper bounds are given for the number of zeros of these solutions in small discs in a neighbourhood of infinity.     

The existence of periodic solutions of nonlinear wave equations on Sn

We are concerned with the existence of time periodic solutions of nonlinear wave equations on n-dimensional spheres. The existence of nontrivial periodic solution in case 0 is a solution is proved. the existence of multiple, especially infinitely many, time periodic solutions is established for several classes f nonlinear terms which satisfy some symmetry properties such as time translation invariance or oddness. Furhermore, this paper also studies the effect of perturbations which are not small and which destroy the symmetry, and shows how multiple solutions persist despite the nonsymmetric perturbations if the growth of nonlinear term at infinity is suitable controlled.     

Generalized solitary waves in the gravity-capillary Whitham equation

We study the existence of traveling wave solutions to a unidirectional shallow water model, which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between 0 and 1/3) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave.