On radially symmetric solutions of the equation ?Δu + u = |u| p-1 u. An ODE approach

Questions of the existence in a ball of radially symmetric solutions of the equation indicated in the title with the Dirichlet zero boundary conditions are studied in many publications and generally speaking, there was obtained more or less complete answer on these questions. It is known now that if the dimension of the space d????3 and 1 <?p?<?(d?+ 2)/(d ? 2) or if d?=?2 and p?>?1, then for any integer l??? 0 this problem in a ball or in the entire space ${x \in \mathbb {R}^d}$ has a radially symmetric solution with precisely l zeros as a function of r?=?|x|. If d??? 3 and p????(d?+?2)/(d ? 2), then the problem in the entire space has no nontrivial solution. For the first time, this problem was studied by a variant of the variational method. However, it is known to the specialists in the field that it is also interesting to obtain the same results by using methods of the qualitative theory of ODEs. In the present article, we shall give a simple proof of the result above in this way. An earlier proof of this result of the other authors is essentially more complicated than our one.     

On the solution set of an equation of the type f(t, Φ(u(t)) = 0

We investigate the solution set Γ of an equation of the type f(t, Φ(u(t)) = 0, where Φ is a linear homeomorphism from a topological vector space X onto L ¹(T) and f: T×R → R is a Carathéodory function. More precisely, we characterize the property of Γ of intersecting each closed hyperplane of X.     

Some properties of solutions ofu″+a(t) f (u)=0

Monatshefte für Mathematik -     

On the asymptotic behaviour of a pantograph–type difference equation

In this paper we study the asymptotic behaviour of solutions of the pantograph-type differnce equation, and obtain aymptotic estimates, which can imply asymptotic stability or stability of solutions     

A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds

We consider Kerr spacetimes with parameters a and M such that |a|≪M, Kerr-Newman spacetimes with parameters |Q|≪M, |a|≪M, and more generally, stationary axisymmetric black hole exterior spacetimes (M,g)(\mathcal{M},g) which are sufficiently close to a Schwarzschild metric with parameter M>0 and whose Killing fields span the null generator of the event horizon. We show uniform boundedness on the exterior for solutions to the wave equation □ _g ψ=0. The most fundamental statement is at the level of energy: We show that given a suitable foliation Σ _τ , then there exists a constant C depending only on the parameter M and the choice of the foliation such that for all solutions ψ, a suitable energy flux through Σ _τ is bounded by C times the initial energy flux through Σ₀. This energy flux is positive definite and does not degenerate at the horizon, i.e. it agrees with the energy as measured by a local observer. It is shown that a similar boundedness statement holds for all higher order energies, again without degeneration at the horizon. This leads in particular to the pointwise uniform boundedness of ψ, in terms of a higher order initial energy on Σ₀. Note that in view of the very general assumptions, the separability properties of the wave equation or geodesic flow on the Kerr background are not used. In fact, the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high-frequency geometric optics regime.     

Asymptotic behavior for t→∞ of the solutions of the equation ψxx+u(x,t)ψ+(λ/4)ψ=0 with a potential u,satisfying the Korteweg-de Vries equation. II

This is the third paper in a series of papers of the authors, devoted to a rigorous investigation of the asymptotic behavior of the solutions of the KdV equation as t. The immediate purpose of the paper is the investigation of the solution of the Schrödinger equation in the neighborhood of the singular point x=3t for a special class of potentials, introduced in the previous papers. As it will be proved, in the final analysis this class of potentials describes the asymptotic behavior of the solutions, decreasing for x, of the KdV equation as t. The solution , far from the singular point, has been investigated earlier. In the paper we investigate a series which gives the solution of the Schrödinger equation and we consider the asymptotic properties of this series.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 8–32, 1984.     

On solutions of a common generalization of the Go?a?b-Schinzel equation and of the addition formulae

Under some additional assumptions we determine solutions of the equation

f(x+M(f(x))y)=f(x)○f(y),     

On the equation −Δu + e u − 1 = 0 with measures as boundary data

If Ω is a bounded domain in ${\mathbb{R}^N}$ , we study conditions on a Radon measure μ on ?Ω for solving the equation ?Δu + e ^u ? 1 = 0 in Ω with u = μ on ?Ω. The conditions are expressed in terms of Orlicz capacities.     

On a relationship between solutions of the third Painlevé equation

The direct and inverse Bäcklund transformations for the third Painlevé equation in the case O is used to obtain a nonlinear functional relationship connecting the solutions of this equation for different values of the parameters that occur in it.Belarus State University of Information Technology and Electronics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 364–366, March, 1995.     

Stability of Runge–Kutta methods in the numerical solution of equation u′(t)=au(t)+a0u([t])

This paper is concerned with the stability analysis of the Runge–Kutta methods for the equation u′(t)=au(t)+a₀u([t]). The stability regions for the Runge–Kutta methods are determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given.     

Rigorous justification of the asymptotic solutions of “sliding wave” type

In the paper we give a rigorous justification of the asymptotic expansion of Green's function for the diffraction problem on a smooth convex contour in the shadow zone. We assume that one of the source and observation points is on the boundary and the other one outside . We consider the case of the Dirichlet problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 48–64, 1983.     

On a nonlinear wave equation involving the term −∂∂x(μ(x,t,u,6ux62)ux): Linear approximation and asymptotic expansion of solution in many small parameters

In this paper, we consider the following nonlinear Kirchhoff wave equation (1)$\{\begin{array}{c}{u}_{tt}?\frac{?}{?x}\left(\mu (x,t,u,{6u_{x}6}^2)u_x\right)=f(x,t,u,u_x,u_t)\text{,}0<x<1,0<t<T\text{,}\hfill \\ u(0,t)=g_0\left(t\right)\text{,}u(1,t)=g_1\left(t\right)\text{,}\hfill \\ u(x,0)={\stackrel{?}{u}}_0\left(x\right)\text{,}{u}_t(x,0)={\stackrel{?}{u}}_1\left(x\right)\text{,}\hfill \end{array}$ where ${\stackrel{?}{u}}_0$, ${\stackrel{?}{u}}_1$, $\mu$, $f$, $g_0$, $g_1$ are given functions and ${6u_{x}6}^2={\int }_0^1{u}_x^2(x,t)dx$. First, combining the linearization method for nonlinear term, the Faedo–Galerkin method and the weak compact method, a unique weak solution of problem (1) is obtained. Next, by using Taylor’s expansion of the function $\mu (x,t,y,z)$ around the point $(x,t,y_0,z_0)$ up to order $N+1$, we establish an asymptotic expansion of high order in many small parameters of solution.     

On optimal Hölder regularity of solutions to the equation Δu + b ⋅ ∇u = 0 in two dimensions

We show that for an $L^2$ drift b in two dimensions, if the Hardy norm of $\text{div}b$ is small, then the weak solutions to $\mathrm{\Delta }u+b?\mathrm{?}u=0$ have the same optimal Hölder regularity as in the case of divergence-free drift, that is, $u\in C_{\text{loc}}^{\alpha }$ for all $\alpha \in (0,1)$.