Solvability of convolution type integro-differential equations on ℝ

The paper looks for the solutions of integro-differential equations of the form

$ - \frac{{d\varphi }}{{dx}} + A\varphi (x) = g(x) + B\int_\mathbb{R} {k(x - t)\lambda (t)\varphi (t)dt, x \in \mathbb{R}} $

in the class of functions which are absolutely continuous and of slow growth on ?. It is assumed that A and B are nonnegative parameters, 0 ≤ g ∈ L ₁ (?), 0 ≤ k ∈ L ₁ (?), ∫_? k(x) dx = 1 and 0 ≤ λ(x) ≤ 1 is a measurable function in ?. The equation is solved by a special factorization of the corresponding integro-differential operator in combination with appropriately modified standard methods of the theory of convolution type integral equations.

     

Global solvability of monge-ampère type equations

We study the global solvability of Monge-Ampère equations of mixed type by "blowing up" the problem onto the torus embedded at the singular point of the equations     

Estimates of Wiman–Valiron type for evolution equations

We establish estimates of Wiman–Valiron type for solutions of evolution equations with a pseudodifferential operator of the Hörmander class in a Hilbert space. Estimates of this type characterize the behavior of the solution of the problem as t→∞ or t → 0 depending on the decay or growth rate of the Fourier coefficients of the initial data.     

On some functional equations of Goląb-Schinzel type

Summary LetE be a real Hausdorff topological vector space. We consider the following binary law * on ·E:(, ) * (, ) = (, ^k + ) for(, ), (, ) × E where is a nonnegative real number,k andl are integers.In order to find all subgroupoids of ( ·E, *) which depend faithfully on a set of parameters, we have to solve the following functional equation:f(f(y) ^k x +f(x) ^l y) =f(x)f(y) (x, y E). (1)In this paper, all solutionsf: of (1) which are in the Baire class I and have the Darboux property are obtained. We obtain also all continuous solutionsf: E of (1). The subgroupoids of (^* ·E, *) which dapend faithfully and continuously on a set of parameters are then determined in different cases. We also deduce from this that the only subsemigroup ofL _n ¹ of the form {(F(x ₂,x ₃, ,x _n ),x ₂,x ₃, ,x _n ); (x ₂, ,x _n ) ^{n – 1} }, where the mappingF: ^{n – 1 *} has some regularity property, is {1} × ^{n – 1} .We may noitice that the Gob-Schinzel functional equation is a particular case of equation (1)(k = 0, l = 1, = 1). So we can say that (1) is of Gob—Schinzel type. More generally, whenE is a real algebra, we shall say that a functional equation is of Gob—Schinzel type if it is of the form:f(f(y) ^k x +f(x) ^l y) =F(x,y,f(x),f(y),f(xy)) wherek andl are integers andF is a given function in five variables. In this category of functional equations, we study here the equation:f(f(y) ^k x +f(x) ^l y) =f(xy) (x, y f: ). (4)This paper extends the results obtained by N. Brillouët and J. Dhombres in [3] and completes some results obtained by P. Urban in his Ph.D. thesis [11] (this work has not yet been published).Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

Periodic solutions of fully nonlinear autonomous equations of Benjamin–Ono type

We prove the existence of time-periodic, small amplitude solutions of autonomous quasi-linear or fully nonlinear completely resonant pseudo-PDEs of Benjamin–Ono type in Sobolev class. The result holds for frequencies in a Cantor set that has asymptotically full measure as the amplitude goes to zero.     

Existence and regularity of positive solutions of elliptic equations of Schr?dinger type

We prove the existence of positive solutions with optimal local regularity of the homogeneous equation of Schr?dinger type $$ - {\rm{div}}(A\nabla u) - \sigma u = 0{\rm{ in }}\Omega $$ for an arbitrary open ?? ? ? ⁿ under only a form-boundedness assumption on ?? ?? D??(??) and ellipticity assumption on A ?? L ^??(??) ^n×n . We demonstrate that there is a two-way correspondence between form boundedness and existence of positive solutions of this equation as well as weak solutions of the equation with quadratic nonlinearity in the gradient $$ - {\rm{div}}(A\nabla u) = (A\nabla v) \cdot \nabla v + \sigma {\rm{ in }}\Omega $$ As a consequence, we obtain necessary and sufficient conditions for both formboundedness (with a sharp upper form bound) and positivity of the quadratic form of the Schr?dinger type operator H = ?div(A?·)-?? with arbitrary distributional potential ?? ?? D??(??), and give examples clarifying the relationship between these two properties.     

E α -Ulam type stability of fractional order ordinary differential equations

In this paper, the concepts of $\mathbb{E}_{\alpha}$ -Ulam-Hyers stability, generalized $\mathbb{E}_{\alpha}$ -Ulam-Hyers stability, $\mathbb{E}_{\alpha}$ -Ulam-Hyers-Rassias stability and generalized $\mathbb{E}_{\alpha}$ -Ulam-Hyers-Rassias stability for fractional order ordinary differential equations are raised. Without loss of generality, $\mathbb{E}_{\alpha}$ -Ulam-Hyers-Rassias stability result is derived by using a singular integral inequality of Gronwall type. Two examples are also provided to illustrate our results.     

Effect of noise on front propagation in reaction-diffusion equations of KPP type

We consider reaction-diffusion equations of KPP type in one spatial dimension, perturbed by a Fisher-Wright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed Fisher-KPP equations

$\partial_tu=\partial_x^2u+u(1-u)+\epsilon\sqrt{u(1-u)}\dot{W},$

and

$\partial_tu=\partial_x^2u+u(1-u)+\epsilon\sqrt{u}\dot{W},$

where \(\dot{W}=\dot{W}(t,x)\) is a space-time white noise. We prove the Brunet-Derrida conjecture that the speed of traveling fronts for small ε is

$2-\pi^2|{\log}\,\epsilon^2|^{-2}+O((\log|{\log}\,\epsilon|)|{\log}\,\epsilon|^{-3}).$

     

Global regularity for a class of Monge-Ampère type equations

In this paper we are concerned with the global regularity of solutions to the Dirichlet problem for a class of Monge-Ampère type equations.By employing the concept of(a,η) type domain,we emphasize that the boundary regularity depends on the convexity of the domain in nature.The key idea of our proof is to provide more effective global H?lder estimates of convex solutions to the problem based on carefully choosing auxiliary functions and constructing sub-solutions.     

Measures not charging semipolars and equations of schrödinger type

SupposeX is a Borel right process andm is a -finite excessive measure forX. Given a positive measure not chargingm-semipolars we associate an exact multiplicative functionalM(). No finiteness assumptions are made on . Given two such measures and ,M()=M() if and only if and agree on all finely open measurable sets. The equation (q–L)u+u=f whereL is the generator of (a subprocess of)X may be solved for appropriatef by means of the Feynman-Kac formula based onM(). Both uniqueness and existence are considered.Supported in part by NSF Grant DMS 92-24990.     

Asymptotic formulae of Liouville–Green type for higher even-order equations

Asymptotic formulae of Liouville–Green type for general linear ordinary differential equations of an arbitrary even-order 2m are investigated. A theorem on asymptotic behaviour at the infinity of 2m linearly independent solutions is proved. It is shown that numerous results known in the literature are contained in this theorem as particular cases.     

Estimates of the Nash–Aronson type for degenerating parabolic equations

We consider second-order parabolic equations describing diffusion with degeneration and diffusion on singular and combined structures. We give a united definition of a solution of the Cauchy problem for such equations by means of semigroup theory in the space L ₂ with a suitable measure. We establish some weight estimates for solutions of Cauchy problems. Estimates of Nash–Aronson type for the fundamental solution follow from them. We plan to apply these estimates to known asymptotic diffusion problems, namely, to the stabilization of solutions and to the “central limit theorem.”     

Fractional equations of Volterra type involving a Riemann–Liouville derivative

In this paper, we will discuss the existence of solutions of fractional equations of Volterra type with the Riemann–Liouville derivative. Existence results are obtained by using a Banach fixed point theorem with weighted norms and by a monotone iterative method too. An example illustrates the results.     

Asymptotic behavior of positive solutions to emden–fowler type equations

We study the asymptotic behavior of positive solutions to nonlinear elliptic equations of Emden–Fowler type with absorption term. For operators with variable coefficients we obtain conditions on coefficients under which the solutions have the same asymptotics as solutions to the model equation Δu = −x| ^p |u| ^σ−1 u. For positive solutions we obtain lower order terms of the asymptotic expansion at infinity. Bibliography: 10 titles.     

Control of space-time chaos in a system of equations of the FitzHugh–Nagumo type

We perform an analytic and numerical study of a system of partial differential equations that describes the propagation of nerve impulses in the heart muscle. We show that, for fixed parameter values, the system has infinitely many distinct stable wave solutions running along the spatial axis at arbitrary velocities and infinitely many distinct modes of space-time chaos, where the bifurcation parameter is the velocity of running wave propagation along the spatial axis, which does not explicitly occur in the original system of equations. We suggest an algorithm for controlling the space-time chaos in the system, which permits one to stabilize any of its unstable periodic running waves.     

Hölder estimates of solutions to difference partial differential equations of elliptic-parabolic type

For solutions to difference partial differential equations of elliptic-parabolic type, there is achieved Hölder estimates independent of the time discrete mesh.