Differential-algebraic equations with a small nonlinear term

In the present paper, we generalize some known results of the theory of differentialalgebraic equations to a more complicated case under the assumption that the nonlinear term is small. We give an asymptotic estimate of the behavior of the solution with respect to the parameter μ.     

Doubly nonlinear parabolic equations with singular lower order term

In this paper we are concerned with positive solutions of the doubly nonlinear parabolic equation u_t=div(u^m−1|∇u|^p−2∇u)+Vu^m+p−2 in a cylinder $\text{Ω×(0,T)}$, with initial condition u(·,0)=u₀(·)⩾0 and vanishing on the parabolic boundary $\text{∂Ω×(0,T)}$. Here $\text{Ω⊂}\text{R}{}^{\mathrm{N}}$ (resp. $\text{H}{}^{\mathrm{n}}$) is a bounded domain with smooth boundary, $\text{V∈L}{}_{\mathrm{<mtext>loc</mtext>}}{}^1\text{(Ω)}$, $\text{m∈}\text{R}$, 1<p<N and m+p−2>0. The critical exponents $\text{q}{}^1$ are found and the nonexistence results are proved for $\text{q}{}^1\text{⩽m+p<3}$.     

Oscillation of higher order nonlinear dynamic equations with a nonlinear neutral term

This study introduces the oscillation of higher order nonlinear dynamic equations with a nonlinear neutral term. The findings are obtained via utilising an integral criterion as well as a comparison theorem with the oscillatory properties of a first-order dynamic equation. We propose novel oscillation criteria that improve, extend and simplify existing ones in the literature. The results are associated with several numerical examples.     

Elliptic solutions of nonlinear integrable equations and related topics

The theory of elliptic solitons for the Kadomtsev-Petviashvili (KP) equation and the dynamics of the corresponding Calogero-Moser system is integrated. It is found that all the elliptic solutions for the KP equation manifest themselves in terms of Riemann theta functions which are associated with algebraic curves admitting a realization in the form of a covering of the initial elliptic curve with some special properties. These curves are given in the paper by explicit formulae. We further give applications of the elliptic Baker-Akhiezer function to generalized elliptic genera of manifolds and to algebraic 2-valued formal groups.Dedicated to the memory of J.-L. Verdier     

Oscillation theorems for differential equations with a nonlinear damping term

In this paper, we are concerned with a class of nonlinear second-order differential equations with a nonlinear damping term. Passage to more general class of equations allows us to remove a restrictive condition usually imposed on the nonlinearity, and, as a consequence, our results apply to wider classes of nonlinear differential equations. Two illustrative examples are considered.     

The Cauchy-Goursat problem for wave equations with nonlinear dissipative term

The Cauchy-Goursat problem for wave equations with nonlinear dissipative term is studied. The existence, uniqueness, and blow-up of global solutions of this problem are considered. The local solvability of this problem is also discussed.     

Oscillation theorems for second-order nonlinear differential equations with damped term

Several oscillation criteria are given for the second-order damped nonlinear differential equation (a(t)[y′(t)]^σ′i +p(t)[y′(t)]^σ +q(t)f(y(t)) = 0, where σ > 0 is any quotient of odd integers, a ϵ C(R, (0, ∞)), p(t) and q(t) are allowed to change sign on [t_o, ∞), and f ϵ Cl (R, R) such that xf (x) > 0 for x≠0. Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results.     

A uniqueness result for Kirchhoff equations with non-Lipschitz nonlinear term

We consider the second order Cauchy problem

     

On oscillation of nonlinear second-order differential equations with damping term

The paper is concerned with oscillation of a novel class of nonlinear differential equations with a damping term. First it is demonstrated how known oscillation results for another intensively studied class of equations can be translated to the one in question, and vice versa. Advantages and drawbacks of such translation are carefully examined. Then an oscillation criterion for the new class of equations is established. The principal result of the paper is compared to those reported in the literature, and an illustrative example to which known oscillation criteria fail to apply is provided.     

Instability of the standing waves for nonlinear Klein-Gordon equations with damping term

This paper is concerned with the nonlinear Klein-Gordon equations with damping term. In terms of the variational argument, the sharp conditions for blowing up and global existence are derived out by applying the potential well argument and using the concavity method. Further, the instability of the standing waves is shown.     

The first Darboux problem for wave equations with nonlinear dissipative term

The first Darboux problem for wave equations with nonlinear dissipative term is considered. The uniqueness, local and global existence and blow-up of solutions of the problem mentioned are investigated. The paper’s originality is the coalescence of two standard methods: a priori estimate of solutions in the class of continuous functions is given by energetic methods; basing on this result a priori estimate in the class of continuously differentiable functions using classical method of characteristics is obtained.