Entire Solutions with Merging Fronts to Reaction–Diffusion Equations

We deal with a reaction–diffusion equation u _t = u _xx + f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ₁(x + c ₁ t) (c ₁ < 0) and ψ₂(x + c ₂ t) (c ₂ > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all . We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ₁(x + c ₁ t) and ψ₂(x + c ₂ t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c ₁, we show the existence of an entire solution which behaves as ψ₁( − x + c ₁ t) in and φ(x + ct) in for t≈ − ∞.     

The Evolution of Stress Intensity Factors and the Propagation of Cracks in Elastic Media

When a crack Γ _s propagates in an elastic medium the stress intensity factors evolve with the tip x(s) of Γ _s . In this paper we derive formulae which describe the evolution of these stress intensity factors for a homogeneous isotropic elastic medium under plane strain conditions. Denoting by ψ=ψ(x,s) the stress potential (ψ is biharmonic and has zero traction along the crack Γ _s ) and by κ(s) the curvature of the crack at the tip x(s), we prove that the stress intensity factors A ₁(s), A ₂(s), as functions of s, satisfy:

Regularity of the Inverse of a Planar Sobolev Homeomorphism

Let be a domain. Suppose that f ∈ W^1,1_loc(Ω,R²) is a homeomorphism such that Df(x) vanishes almost everywhere in the zero set of J_f. We show that f^-1 ∈ W^1,1_loc(f(Ω),R²) and that Df⁻¹(y) vanishes almost everywhere in the zero set of Sharp conditions to quarantee that f⁻¹ ∈ W^1,q(f(Ω),R²) for some 1<q≤2 are also given.     

Existence of Exact Solutions for the Eversion of Elastic Cylinders

The existence of eversion deformations of an elastic cylinder into another right circular cylinder is studied. It is found that such deformations are associated with strain-energy functions of separable form W(λ₁,λ₂,λ₃) =φ(λ₁)+φ(λ₂)+φ (λ₃), and thus can serve as a test for materials of this kind. Under certain constitutive assumptions, there always exists a cylindrical eversion deformation that satisfies exact pointwise traction free boundary conditions over the entire surface of the cylinder. For such solutions the cavity must remain open upon eversion. This revised version was published online in August 2006 with corrections to the Cover Date.     

Reynolds' analogy for the thermal convection driven by nonisothermal stretching surfaces

In the present paper the steady boundary-layer flows induced by permeable stretching surfaces with variable temperature distribution are investigated under the aspect of Reynolds' analogy r = St _x /C _f(x). It is shown that for certain stretching velocities and wall temperature distributions, “Reynolds' function”r, i.e. the ratio of the local Stanton number St _x and the skin friction coefficient C _f(x) equals −1/2 for any value of the Prandtl number Pr and of the dimensionless suction/injection velocity f _w. In all of these cases, the dimensionless temperature field ϑ is connected to the dimensionless downstream velocity f ^′ by the simple relationship ϑ=(f ^′)^Pr. It is also shown that in the general case, Reynolds' function r may possess several singularities in f _w. The largest of them represents a critical value, so that for f _w<f _w,crit the solutions of the energy equation (although they still satisfy all the boundary conditions) become nonphysical.     

On asymptotic properties of solutions of functional differential equations with linearly transformed argument

We establish new properties of solutions of the functional differential equation x′(t) = ax(t) + bx(t − r) + cx′(t − r) + px(qt) + hx′(qt) + f ₁(x(t), x(t − r), x′(t − r), x(qt), x′(qt)) in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 1, pp. 144–160, January–March, 2007.     

On the fundamental linear fractional order differential equation

This paper deals with the rational function approximation of the irrational transfer function G(s) = \fracX(s)E(s) = \frac1[(t₀s)^2m + 2z(t₀s)^m + 1]G(s) = \frac{X(s)}{E(s)} = \frac{1}{[(\tau _{0}s)^{2m} + 2\zeta (\tau _{0}s)^{m} + 1]} of the fundamental linear fractional order differential equation (t₀)^2m\fracd^2mx(t)dt^2m + 2z(t₀)^m\fracd^mx(t)dt^m + x(t) = e(t)(\tau_{0})^{2m}\frac{d^{2m}x(t)}{dt^{2m}} + 2\zeta(\tau_{0})^{m}\frac{d^{m}x(t)}{dt^{m}} + x(t) = e(t), for 0<m<1 and 0<ζ<1. An approximation method by a rational function, in a given frequency band, is presented and the impulse and the step responses of this fractional order system are derived. Illustrative examples are also presented to show the exactitude and the usefulness of the approximation method.     

On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D outside C. (1) With f (z) assumed to be C n (n < ∞-times continuously differentiable) z ∈ D + and in a neighborhood of C, f (z) and its derivatives f (n) (z) are proved uniformly continuous in the closed domain D + = [D + + C]. (2) Cauchy’s integral formulas and their derivatives z ∈ D + (or z ∈ D ) are proved to converge uniformly in D + (or in D = [D +C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[ f (z)]) are shown extended to hold for the complement function F(z), defined to be C n z ∈ D and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four general- ized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f (z) in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f (z) in domain D , based on the continuous numerical value of f (z) z ∈ D + = [D + + C], is presented for resolution as a conjecture.     

Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr ²(t)>, of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G ^*(ω), in the Fourier frequency domain, and the stress relaxation modulus, G _r (t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of <Δr ²(t)> known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing <Δr ²(t)> as a local power law. If the logarithmic slope of <Δr ²(t)> can be accurately determined, these estimates generally perform well at the frequency extremes. Received: 8 September 2000/Accepted: 9 March 2000     

Exact analytic solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium

 The problem of the self-similar boundary flow of a “Darcy-Boussinesq fluid” on a vertical plate with temperature distribution T _w(x) = T _∞+A·x ^λ and lateral mass flux v _w(x) = a·x ^(λ−1)/2, embedded in a saturated porous medium is revisited. For the parameter values λ = 1,−1/3 and −1/2 exact analytic solutions are written down and the characteristics of the corresponding boundary layers are discussed as functions of the suction/ injection parameter in detail. The results are compared with the numerical findings of previous authors. Received on 8 March 1999     

A unified approach to closed-form solutions of moving heat-source problems

A closed-form model for the computation of temperature distribution in an infinitely extended isotropic body with a time-dependent moving-heat sources is discussed. The temperature solutions are presented for the sources of the forms: (i) ₀₁(t)=₀ exp(−λt), (ii) ₀₂(t) =₀(t/t ^*)exp(−λt), and ₀₃(t)=₀[1+a cos(ωt)], where λ and ω are real parameters and t ^* characterizes the limiting time. The reduced (or dimensionless) temperature solutions are presented in terms of the generalized representation of an incomplete gamma function Γ(α,x;b) and its decomposition C _Γ and S _Γ. The solutions are presented for moving, -point, -line, and -plane heat sources. It is also demonstrated that the present analysis covers the classical temperature solutions of a constant strength source under quasi-steady state situations. Received on 13 June 1997     

On a Crystalline Variational Problem, Part I:¶First Variation and Global L∞ Regularity

Let φ:ℝ ⁿ → [0,+∞[ be a given positively one-homogeneous convex function, and let ?_φ≔{φ≤ 1 }. Pursuing our interest in motion by crystalline mean curvature in three dimensions, we introduce and study the class ?_φ (ℝ ⁿ ) of “smooth” boundaries in the relative geometry induced by the ambient Banach space (ℝ ⁿ , φ). It can be seen that, even when ?_φ is a polytope, ?_φ(ℝ ⁿ ) cannot be reduced to the class of polyhedral boundaries (locally resembling ∂?_φ). Curved portions must be necessarily included and this fact (as well as the nonsmoothness of ∂?_φ) is the source of several technical difficulties related to the geometry of Lipschitz manifolds. Given a boundary δE in the class ?_φ(ℝ ⁿ ), we rigorously compute the first variation of the corresponding anisotropic perimeter, which leads to a variational problem on vector fields defined on δE. It turns out that the minimizers have a uniquely determined (intrinsic) tangential divergence on δE. We define such a divergence to be the φ-mean curvature κ_φ of δE; the function κ_φ is expected to be the initial velocity of δE, whenever δE is considered as the initial datum for the corresponding anisotropic mean curvature flow. We prove that κ_φ is bounded on δE and that its sublevel sets are characterized through a variational inequality.     

Existence of the minimal positive solution of some nonlinear elliptic systems when the nonlinearity is the sum of a sublinear and a superlinear term

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｃｏｎｓｉｄｅｒｔｈｅｅｌｌｉｐｔｉｃｓｙｓｔｅｍ(1λ)　-Δｕ=ｆ(λ,ｘ,ｕ)-ｖ　　(ｉｎΩ),-Δｖ=δｕ-γｖ(ｉｎΩ),ｕ=ｖ=0(ｏｎΩ),ｗｈｅｒｅΩｉｓａｓｍｏｏｔｈｂｏｕｎｄｅｄｄｏｍａｉｎｉｎＲＮ(Ｎ≥2)ａｎｄλｉｓａｒｅａｌｐａｒａｍｅｔｅｒ.Ｔｈｅｓｏｌｕｔｉｏｎｓ(ｕ,ｖ)ｏｆｔｈｉｓｓｙｓｔｅｍｒｅｐｒｅｓｅｎｔｓｔｅａｄｙｓｔａｔｅｓｏｌｕｔｉｏｎｓｏｆｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｓｙｓｔｅｍｓｄｅｒｉｖｅｄｆｒｏｍｓｅｖｅｒａｌａｐ…     

Sharp Estimates of Bounded Solutions to Some Second-order Forced Dissipative Equations

Résumé A l’aide d’inégalités différentielles, on établit une estimation proche de l’optimalité pour la norme dans de l’unique solution bornée de u′′ + cu′ + Au = f(t) lorsque A = A ^* ≥ λ I est un opérateur borné ou non sur un espace de Hilbert réel H, V = D(A ^1/2) et λ, c sont des constantes positives, tandis que . By using differential inequalities, a close-to-optimal bound of the unique bounded solution of u′′ + cu′ + Au = f(t) is obtained whenever A = A ^* ≥ λ I is a bounded or unbounded linear operator on a real Hilbert space H, V = D(A ^1/2) and λ, c are positive constants, while .

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Symmetry of Mountain Pass Solutions of Some Vector Field Equations

We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u ₁(x),...,u _N (x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be , a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes [(1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14].     

Darboux problem for a differential equation with fractional derivative

We find conditions for the unique solvability of the problem u _xy (x, y) = f(x, y, u(x, y), (D ₀ ^r u)(x, y)), u(x, 0) = u(0, y) = 0, x ∈ [0, a], y ∈ [0, b], where (D ₀ ^r u)(x, y) is the mixed Riemann-Liouville derivative of order r = (r ₁, r ₂), 0 < r ₁, r ₂ < 1, in the class of functions that have the continuous derivatives u _xy (x, y) and (D ₀ ^r u)(x, y). We propose a numerical method for solving this problem and prove the convergence of the method. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 456–467, October–December, 2005.     

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity

For elliptic equations ε²Δu − V(x) u + f(u) = 0, x ∈ R ^N , N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal. An erratum to this article can be found at     

Parameter Dependence of Stable Manifolds for Delay Equations with Polynomial Dichotomies

We establish the existence of Lipschitz stable invariant manifolds for semiflows generated by a delay equation x′ = L(t)x _t + f (t, x _t , λ), assuming that the linear equation x′ = L(t)x _t admits a polynomial dichotomy and that f is a sufficiently small Lipschitz perturbation. Moreover, we show that the stable invariant manifolds are Lipschitz in the parameter λ. We also consider the general case of nonuniform polynomial dichotomies.     

A Variational Approach to the Fredholm Alternative for the p-Laplacian near the First Eigenvalue

We are concerned with the existence of a weak solution to the degenerate quasi-linear Dirichlet boundary value problem

Hydromagnetic stagnation point flow of a viscous fluid over?a?stretching or shrinking sheet

We establish the existence and uniqueness results over the semi-infinite interval [0,∞) for a class of nonlinear third order ordinary differential equations of the form