On Kneser solutions of higher order nonlinear ordinary differential equations

The equationx ⁽ⁿ⁾(t)=(−1) ⁿ │x(t)│ ^k withk>1 is considered. In the casen≦4 it is proved that solutions defined in a neighbourhood of infinity coincide withC(t−t₀)^−n/(k−1), whereC is a constant depending only onn andk. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times (t−t ₀)^−n/(k−1). It is shown that they do not necessarily coincide withC(t−t₀)^−n/(k−1). This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics. Dedicated to Professor Vladimir Maz'ya on the occasion of his 60th birthday. The author was supported by the Swedish Natural Science Research Council (NFR) grant M-AA/MA 10879-304.     

An abstract nonlinear Volterra equation

The existence, uniqueness, regularity and dependence upon data of solutions of the abstract Volterra equation u(t)+∫ ₀ ^t a(t-s)A(u(s))ds∈f(t), t≧0 are studied in a real Banach space. The nonlinear operatorA is assumed to bem-accretive and the assumptions on the kernela do not exclude the possibility that lim _t→0+ a(t)=+∞.     

Evolution equations with a free boundary condition

In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary) with a free boundary condition. That is, the following initial boundary value problem ∂₁,u −Δu = Γ(u)(∇u, ∇u) [tT T_{u u}N, on M × [0, ∞), u(t, x) ∈ Σ, for x ∈ ∂M, t > 0, ∂u/t6n(t, x) ⊥_u T_u(t,x) Σ, for x ∈ ∂M, t > 0, u(o,x) = u_o(x), on M, where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M. Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties, even for the case N = ℝⁿ, in which the nonlinear equation reduces to ∂_tu-Δu = 0. We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical conditions on N and Σ which are weaker than K_N <-0 and Σ is totally geodesic in N.     

Nonlocal problems for the equations of Kelvin-Voight fluids and their ɛ-approximations in classes of smooth functions

Existence theorems are proved for the solutions of the first and second initial boundary-value problems for the equations of Kelvin-Voight fluids and for the penalized equations of Kelvin-Voight fluids in the smoothness classes W _∞ ^r (ℝ⁺;W ₂ ^2+k (Ω)), W ₂ ^r (ℝ⁺;W ₂ ^2+k (Ω)) and S ₂ ^r (ℝ⁺;W ₂ ^2+k (Ω)) (r=1,2; k=0,1,2, …) under the condition that the right-hand sides f(x,t) belong to the classes W _∞ ^r-1 (ℝ⁺;W ₂ ^k (Ω)), W ₂ ^r-1 (ℝ⁺;W ₂ ^k (Ω)) and S ₂ ^r-1 (ℝ⁺;W ₂ ^k (Ω)), respectively, and for the solutions of the first and second T-periodic boundary-value problems for the same equations in the smoothness classes W _∞ ^r−1 (ℝ; W ₂ ^2+k (Ω)) and W ₂ ^r−1 (0, T; W ₂ ^2+k (Ω)) (r=1,2, k=0,1,2…) under the condition that f(x,t) are T-periodic and belong to the spaces W _∞ ^r−1 (ℝ⁺; W ₂ ^k (Ω)) and W ₂ ^r−1 (0,T; W ₂ ^k (Ω)), respectively. It is shown that as ɛ→0, the smooth solutions {v^ɛ} of the perturbed initial boundary-value and T-periodic boundary-value problems for the penalized equations of Kelvin-Voight fluids converge to the corresponding smooth solutions (v,p) of the initial boundary-value and T-periodic boundary-value problems for the equations of Kelvin-Voight fluids. Bibliography: 27 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 214–242. Translated by T. N. Surkova.     

Multivalued perturbations ofm-accretive differential inclusions

Given anm-accretive operatorA in a Banach spaceX and an upper semicontinuous multivalued mapF: [0,a]×X→2 ^X , we consider the initial value problemu′∈−Au+F(t,u) on [0,a],u(0)=x ₀. We concentrate on the case when the semigroup generated by—A is only equicontinuous and obtain existence of integral solutions if, in particular,X* is uniformly convex andF satisfies β(F(t,B))≤k(t)β(B) for all boundedB⊂X wherek∈L ¹([0,a]) and β denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compactR _δ-set in this situation. In general, the extra condition onX* is essential as we show by an example in whichX is not uniformly smooth and the set of all solutions is not compact, but it can be omited ifA is single-valued and continuous or—A generates aC _o-semigroup of bounded linear operators. In the simpler case when—A generates a compact semigroup, we give a short proof of existence of solutions, again ifX* is uniformly (or strictly) convex. In this situation we also provide a counter-example in ℝ⁴ in which no integral solution exists. The author gratefully acknowledges financial support by DAAD within the scope of the French-German project PROCOPE.     

Maxwell equations in complex form,squaring procedure and separating the variables

The Riemann–Silberstein–Majorana–Oppenheimer complex approach to the Maxwell electrodynamics is investigated within the matrix formalism. Within the squaring procedure we construct four types of formal solutions of the Maxwell equations on the base of scalar D’Alembert solutions. General problem of separating physical electromagnetic solutions in the linear space λ₀Ψ⁰ + λ₁Ψ¹ + λ₂Ψ² + λ₃Ψ³ is investigated, the Maxwell equations reduce to a new form including parameters λ _a . Several particular cases, plane waves and cylindrical waves, are considered in detail. Possible extension of the technique to a curved space–time models is discussed.     

Oscillating multipliers for some eigenfunction expansions

Let P be a non-negative, self-adjoint differential operator of degree d on ℝⁿ. Assume that the associated Bochner-Riesz kernel s _R ^δ satisfies the estimate, |s _R ^δ (x, y)| ≤ C R^n/d(1+R^1/d|x - y|^-αδ+β)for some fixed constants a>0 and β. We study L^p boundedness of operators of the form m(P), m coming from the symbol class S _p ^−α . We prove that m(P) is bounded on L^P if . We also study multipliers associated to the Hermite operator H on ℝⁿ and the special Hermite operator L on ℂⁿ given by the symbols . As a special case we obtain L^p boundedness of solutions to the Wave equation associated to H and L.     

Multivalued nonlinear equations on the half line: A fixed point approach

New fixed point theorems of the authors are used to establish the existence of one (or more) C[0, ∞) solutions to the nonlinear integral inclusion y(t)∈ ∫₀^∞ K(t, s) F(s, y(s))ds fort ∈ [0,∞).     

Asymptotic properties of solutions of the emden-fowler equation at infinity

We prove a number of theorems on asymptotic properties of solutions of the equation y″+x ^a y ^σ = 0, σ < 0. First, we prove the absence of solutions on (x ₁, +∞) for some values of the parameters a and σ; after that, we obtain asymptotic formulas for solutions defined on (x ₀, +∞).     

On the theory of the Beltrami equation

We study ring homeomorphisms and, on this basis, obtain a series of theorems on the existence of the so-called ring solutions for degenerate Beltrami equations. A general statement on the existence of solutions for the Beltrami equations that extends earlier results is formulated. In particular, we give new existence criteria for homeomorphic solutions f of the class W _loc ^1,1 with f ⁻¹ ∈ W _loc ^1,2 in terms of tangential dilatations and functions of finite mean oscillation. The ring solutions also satisfy additional capacity inequalities. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1571–1583, November, 2006.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. II. Quasilinear problem

In three spaces, we find exact classical solutions of the boundary-value periodic problem u_tt - a²u_xx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator and whose right-hand side is a nonlinear operator. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998.     

Oscillation of forced neutral differential equations with positive and negative coefficients

In this paper the forced neutral difterential equation with positive and negative coefficients d/dt [x(t)-R(t)x(t-r)] P(t)x(t-x)-Q(t)x(t-σ)=f(t),t≥t0,is considered,where f∈L^1(t0,∞)交集C([t0,∞],R^ )and r,x,σ∈(0,∞),The sufficient conditions to oscillate for all solutions of this equation are studied.     

Asymptotic Complexity in Filtration Equations

We show that the solutions of nonlinear diffusion equations of the form u _t = ΔΦ(u) appearing in filtration theory may present complicated asymptotics as t → ∞ whenever we alternate infinitely many times in a suitable manner the behavior of the nonlinearity Φ. Oscillatory behaviour is demonstrated for finite-mass solutions defined in the whole space when they are renormalized at each time t > 0 with respect to their own second moment, as proposed in [Tos05, CDT05]; they are measured in the L ¹ norm and also in the Euclidean Wasserstein distance W ₂. This complicated asymptotic pattern formation can be constructed in such a way that even a chaotic behavior may arise depending on the form of Φ. In the opposite direction, we prove that the assumption that the asymptotic normalized profile does not depend on time implies that Φ must be a power-law function on the appropriate range of values. In other words, the simplest asymptotic behavior implies a homogeneous nonlinearity.     

Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, u^￠_n(t) = ?_{j ? \mathbbZ^d} J_n-ju_j(t)-u_n(t)u^{\prime}_{n}(t) = \sum_{{j\in}{{{\mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t) with t ≥ 0 and n ? \mathbbZ^dn \in {\mathbb{Z}}^{d}. We assume that J is nonnegative and verifies ?_{n ? \mathbbZ^d}J_n = 1\sum_{{n \in {\mathbb{Z}}}^{d}}J_{n}= 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.     

On some properties of solutions of second-order linear functional differential equations

The properties of solutions of the equationu″(t) =p ₁(t)u(τ₁(t)) +p ₂(t)u′(τ₂(t)) are investigated wherep _i :a, + ∞[→R (i=1,2) are locally summable functions τ₁ :a, + ∞[→R is a measurable function, and τ₂ :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ _i (t) ≥t (i = 1,2),p ₁(t)≥0,p ₂ ² (t) ≤ (4 - ɛ)τ ₂ ^′ (t)p ₁(t), ɛ =const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .     

Self-similar extremal processes

Given an extremal process X: [0,∞)→[0,∞)^d with lower curve C and associated point process N={(t_k, X_k):k≥0}, t_k distinct and X_k independent, given a sequence ζ _n =(τ _n , ξ _n ), n≥1, of time-space changes (max-automorphisms of [0,∞)^d+1), we study the limit behavior of the sequence of extremal processes Y_n(t)=ξ _n ^-1 ○ X ○ τ_n(t)=C_n(t) V max {ξ _n ^-1 ○ X_k: t_k ≤ τ_n(t){ ⇒ Y under a regularity condition on the norming sequence ζ_n and asymptotic negligibility of the max-increments of Y_n. The limit class consists of self-similar (with respect to a group η_α=(σ_α, L_α), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property L_α ○ Y(t) ₌ ^d Y ○ α_α(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

Entropy and reduced distance for Ricci expanders

Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂_gij/∂t = − 2R_ij and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expanding entropy W₊ is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnack-like quantity for the conjugate heat equation, and leads to functionals μ₊ and v₊. The forward reduced volume θ₊ is monotone in general and constant exactly on expanders. A natural conjecture asserts that g(t)/t converges as t → ∞ to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known a-priori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expanders; these include vol(g)/t^n/2 (Hamilton) and -λ (Perelman), as well as our new quantities. In general, we show that, if vol(g) grows like t^n/2(maximal volume growth) then W⁺, θ⁺ and -λ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjecture.     

Oscillation of nonlinear vector differential equations

Summary Oscillation criteria are obtained for vector partial differential equations of the type Δv+b(x, v)v=0, x∈G, v∈E^m, where G is an exterior domain in E_n, and b is a continuous nonnegative valued function in G × E^m. A solution v: G→E^m is called h-oscillatory in G whenever the scalar product [v(x), h] (|h|=1) has zeros x in G with |x| arbitrarily large. It is shown that the spherical mean of [v(x), h] over a hypersphere of radius r in Eⁿ satisfies a nonlinear ordinary differential inequality. As a consequence, the main theorems give sufficient conditions on b(x, t), depending upon the dimension n, for all solutions v to be h-oscillatory in G. Entrata in Redazione il 26 giugno 1975.     

A priori estimate for continuation of the solution of the heat equation in the space variable

Summary Let u(x, t) satisfy the heat equation in a region D in the x−t plane and vanish initially. Let |u|<M₀ in D and |u _i ² |<∈ in D* ⊂ D. Then, in D−D*, |u|< , where B=B(x, t) satisfies O<B<1 and B(x, t) and M₁ are given explicity by simple formulas. The a priori bound is applied to an error analysis of a numerical procedure for continuing solutions of the heat equation in the space variable. Work was performed under the auspices of the U. S. Atomic Energy Commission.