Asymptotic symmetry of solutions of nonlinear partial differential equations

For a large class of partial differential equations on exterior domains or on ?^N we show that any solution tending to a limit from one side as x goes to infinity satisfies the property of “asymptotic spherical symmetry”. The main examples are semilinear elliptic equations, quasilinear degenerate elliptic equations, and first-order Hamilton-Jacobi equations.     

Asymptotic behavior of solutions of pseudo-parabolic partial differential equations

Summary Consider a solution to a second-order pseudo-parabolic equation with sufficiently smooth time-independent coefficients in a cylindrical domain. If it vanishes on the cylindrical surface for all times and if its restriction to a fixed instant belongs toC _2+a , then its pointwise values decay exponentially as t→∞ while its Dirichlet norm grows expontially as t→−∞. Similar conclusion still hold for solutions to non-homogeneous equations under non-homogeneous boundary conditions provided the free term and the boundary data posses these asymptotic behaviors. Work of the second named author was partially supported by N.S.F. Grant No. GP-19590. Entrata in Redazione il 29 gennaio 1971.     

Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations

In this work, the asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses is investigated. By impulsive differential inequality and Riccati transformation, sufficient conditions of asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses are obtained. An example is also inserted to illustrate the impulsive effect.     

Asymptotic solutions of nonlinear differential equations

An asymptotic expression for solutions of nonlinear differential equations is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 676–678, May, 1991.     

Asymptotic behavior of solutions of nonlinear neutral differential equations with impulses

This paper is concerned with a nonlinear neutral differential equations with impulses of the form

(*)     

Asymptotic behavior of solutions of second-order nonlinear delay differential equations with impulses

This paper is concerned with second-order nonlinear delay differential equations with impulses of the form

     

Asymptotic behavior of nonoscillatory solutions of nonlinear differential equations with retarded arguments

For nonlinear retarded differential equations $\text{y}{}^{\mathrm{2n}}\text{(t)?}\text{∑}\text{i=1}\text{m}\text{f}{}_{\mathrm{i}}\text{(t,y(t),y(g}{}_{\mathrm{i}}\text{(t)))=0}$ and $\text{y}{}^{\mathrm{n}}\text{(t)?}\text{∑}\text{i=1}\text{m}\text{P}{}_{\mathrm{i}}\text{(t)F}{}_{\mathrm{i}}\text{(y(g}{}_{\mathrm{i}}\text{(t)))=h(t),}$ the sufficient conditions are given on f_i, p_i, F_i, and h under which every bounded nonoscillatory solution of (${}^1$) or (${}^7$) tends to zero as t → ∞.     

Asymptotic behavior of nonoscillatory solutions of nonlinear differential equations with forcing term

Sunto In questo lavoro si studiano il comportamento asintotico delle soluzioni non oscillatorie di una classe di equazioni differenziali di ordine superiore al secondo. In particolare, si dánno condizioni sufficienti perchè tutte le soluzioni, non oscillatorie e limitate, dell’equazione considerata, abbiano per limite lo zero quando la variabile indipendente tende all’infinito.

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Entrata in Redazione il 6 febbraio 1976.     

Asymptotic behavior of solutions of forced nonlinear neutral delay differential equations with impulses

Sufficient conditions for asymptotic behavior of the solutions of nonlinear forced neutral delay differential equations with impulses are found. The results given in [2,4,6,7] are generalized and improved.     

Asymptotic behavior of unbounded solutions of essentially nonlinear second-order differential equations. II

We establish asymptotic representations for one class of unbounded solutions of second-order differential equations whose right-hand sides contain a sum of terms with nonlinearities of a more general form than nonlinearities of the Emden-Fowler type. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 901–921, July, 2006.     

Asymptotic behavior of solutions of nonlinear Volterra equations

An energy decay rate is obtained for solutions of wave type equations in a bounded region in Rⁿ whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface.     

Traveling wave solutions of nonlinear partial differential equations

We propose a simple algebraic method for generating classes of traveling wave solutions for a variety of partial differential equations of current interest in nonlinear science. This procedure applies equally well to equations which may or may not be integrable. We illustrate the method with two distinct classes of models, one with solutions including compactons in a class of models inspired by the Rosenau–Hyman, Rosenau–Pikovsky and Rosenau–Hyman–Staley equations, and the other with solutions including peakons in a system which generalizes the Camassa–Holm, Degasperis–Procesi and Dullin–Gotwald–Holm equations. In both cases, we obtain new classes of solutions not studied before.     

Quasi-periodic solutions of nonlinear elliptic partial differential equations

In a recent paper [9] the KAM theory has been extended to non-linear partial differential equations, to construct quasi-periodic solutions. In this article this theory is illustrated with three typical examples: an elliptic partial differential equation, an ordinary differential equation and a difference equation related to monotone twist mappings.