Critical exponents for large time asymptotics for fractal Hamilton-Jacobi-KPZ equations

Nonlinear and nonlocal evolution equations of the form u_t = ℒ︁u ± |∇u |^q, where ℒ︁ is a pseudodifferential operator representing the infinitesimal generator of a Lévy stochastic process, have been derived (see, [6]) as models for growing interfaces in the case when the continuous Brownian diffusion surface transport is augmented by a random hopping mechanism. The goal of this note is to report properties of solutions to this equation resulting from the interplay between the strengths of the “diffusive” linear and “hyperbolic” nonlinear terms, posed in the whole space R ^N , and supplemented with nonnegative, bounded, and sufficiently regular initial conditions. The full text of the paper, including complete proofs and other results, will appear in the Transactions of the American Mathematical Society. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global solutions for operator Riccati equations with unbounded coefficients: A non-linear semigroup approach

Let X be a Banach space of real-valued functions on [0, 1] and let ?(X) be the space of bounded linear operators on X. We are interested in solutions R:(0, ∞) → ?(X) for the operator Riccati equation where T is an unbounded multiplication operator in X and the B_i(t)'s are bounded linear integral operators on X. This equation arises in transport theory as the result of an invariant embedding of the Boltzmann equation. Solutions which are of physical interest are those that take on values in the space of bounded linear operators on L¹(0, 1). Conditions on X, R(0), T, and the coefficients are found such that the theory of non-linear semigroups may be used to prove global existence of strong solutions in ?(X) that also satisfy R(t) ? ?(L¹(0,1)) for all t ≥ 0.     

Spectral analysis of transport equations with bounce‐back boundary conditions

We investigate the spectral properties of the time‐dependent linear transport equation with bounce‐back boundary conditions. A fine analysis of the spectrum of the streaming operator is given and the explicit expression of the strongly continuous streaming semigroup is derived. Next, making use of a recent result from Sbihi (J. Evol. Equations 2007; 7 :689–711), we prove, via a compactness argument, that the essential spectrum of the transport semigroup and that of the streaming semigroup coincide on all L^p‐spaces with 1<p<∞. Application to the linear Boltzmann equation for granular gases is provided. Copyright © 2008 John Wiley & Sons, Ltd.     

Fixed points,Volterra equations,and Becker’s resolvent

Summary In a recent paper we derived a stability criterion for a Volterra equation which is based on the contraction mapping principle. It turns out that this criterion has significantly wider application. In particular, when we use Becker’s form of the resolvent it readily establishes critical resolvent properties which have been very illusive when investigated by other techniques. First, it enables us to show that the resolvent is L¹. Next, it allows us to show that the resolvent satisfies a uniform bound and that it tends to zero. These properties are then used to prove boundedness of solutions of a nonlinear problem, establish the existence of periodic solutions of a linear problem, and to investigate asymptotic stability properties. We also apply the results to a Liénard equation with distributed delay and possibly negative damping so that relaxation oscillations may occur.     

Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations

We study the asymptotic behavior of linear evolution equations of the type t_∂g=Dg+Lg−λg, where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator Dg+Lg. In the case Dg=−x_∂g, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg=−x_∂(xg), it is known that λ=1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation t_∂f=Lf.By means of entropy–entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural L² space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part.     

Asymptotics of the solutions to theN-particle Kolmogorov-Feller equations and the asymptotics of the solution to the Boltzmann equation in the region of large deviations

We construct a representation in which the asymptotics of the solution to the Kolmogorov-Feller equation in the Fock space Γ(L ₁(ℝ ⁿ )) is of a form similar to the WKB asymptotic expansion; namely, the Boltzmann equation inL ₁(ℝ ⁿ ) plays the role of the Hamilton equations, the linearized Boltzmann equation extended to Γ(L ₁(ℝ ⁿ )) plays the role of the transport equation, and the Hamilton-Jacobi equation follows from the conservation of the total probability for the solutions of the Boltzmann equation. We also construct the asymptotics of the solution to the Boltzmann equation with small transfer of momentum; this asymptotics is given by the tunnel canonical operator corresponding to the self-consistent characteristic equation. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 694–709, November, 1995. The author is deeply grateful to Prof. A. M. Chebotarev, whose assistance has made the writing of this paper possible. This work was financially supported by the International Science Foundation under grants Nos. MFO000 and MFO300.     

Oscillation criteria for fourth‐order differential equations with middle term

Oscillation criteria for self‐adjoint fourth‐order differential equations were established for various conditions on the coefficients r(x) > 0, q(x) and p(x). However, most of these results deal with the case when lim_{x → ∞}∫^x₁q(s) ds < +∞. In this note we give a new oscillation criterion in the case when this condition is not fulfilled, in particular when q(x)↗ + ∞ (even with exponential growth).     

Spectral theory and L∞-time decay estimates for Klein–Gordon equations on two half axes with transmission: The tunnel effect

Consider two copies N₁, N₂ of the interval [0, ∞]. Consider Klein-Gordon equations with (different) constant coefficients on ? × N_j ( = time × space). Assume the coincidence of the values of the solution at the boundary points of the N_j for all times and a transmission condition relating its first (one-sided) space derivatives at these points. Under a symmetry condition, we extend the spatial part of the equation and the transmission conditions to a self-adjoint operator (by Friedrichs extension) and reformulate our problem in terms of an abstract wave equation in a suitable Hilbert space. We derive an expansion of the solution in generalized eigenfunctions of this self-adjoint extension and show, that the L^∞-norms (in space) of the solution and its first k space derivatives at the time t decay for t → ∞ at least as const. t^¼, if the initial conditions satisfy a compatibility condition of order k derived in this paper. The loss of decay rate in comparison with the full line case (const. t^?½, cf. [28]) is caused by the tunnel effect. Further we show that an abstract wave equation in a Hilbert space with a Friedrichs extension as spatial part can always be derived from a stationarity principle for an associated action-type functional. This yields a physical legitimation of our model by the principle of stationary action and moreover a criterion for the physical interpretability of all models created by the linear interaction concept [4, 6, 8, 10], in particular for the coupling of media of different dimension (alternative to [13, 16] for similar models).     

Oscillatory solutions of nonhomogeneous linear differential equations

The complex oscillation of nonhomogeneous linear differential equations with transcendental coefficients is discussed. Results concerning the equation f ^(k)+a _k−1 f ^(k−1)+...+a ₀ f=F where a ₀,...,a _k−i and Fare entire functions, possessing an oscillatory solution subspace in which all solutions (with at most one exception) have infinite exponent of convergence of zeros are obtained. All solutions of the equation are also characterized when the coefficients a ₀,a ₁,...,a _k−1 are polynomials and F=h exp (p ₀), where p ₀ is a polynomial and h is an entire function. Author supported by Max-Planck-Gesellschaft and by NSFC.     

Optimal bounds for bilinear forms associated with linear equations

The construction of upper and lower bounds to the bilinear quantity g₀, f, where f is the solution of an operator equation Tf = f₀, requires either an approximation for f or one for T^?1. In this paper the question of “best” approximation of T^?1 by an operator of the form B = βI, where β is a real constant, is investigated for linear operators that are either self-adjoint or can be related by suitable manipulations to others that are. Particular attention is paid to a special operator, previously studied by Robinson, of importance in predicting the dynamic polarisabilities of quantum-mechanical systems.     

Oscillation and non-oscillation for second-order linear difference equations

We prove new oscillation and non-oscillation theorems for the second-order linear difference equation Δ²x_n−1 + p_nx_n = 0, where is a real sequence with p_n 0. These results are extensions of earlier results of Zhang and Zhou [Comput. Math. Appl. 39 (2000) 1–7].     

A priori estimates for second order elliptic equations and their applications

We study the equation (λ+H)u=f whereH is a self-adjoint operator associated with the Dirichlet form inL ²(IR ^d ,pdx). A priori estimates of the first and the second order derivatives of solutions are obtained under minimal restrictions on the coefficients of the operator and measure. As a consequence we give a criterion of the essential self-adjointness of the operatorH ↾C ₀ ^∞ (IR ^d ) with non-smooth coefficients. Recipient of a Dov Biegun Postdoctoral Fellowship.     

On the solution of a certain class of nonlinear singular integro-differential equations with Carleman shift

The solution of a class of nonlinear singular integro-differential equations with Carleman shift is obtained in the space H_μ⁽ⁿ⁾(Γ). The approach followed depends essentially on solution of linear singular integro-differential equation with shift. A criterion for the Noetherity of a correspondence singular integral functional operator of second order with Carleman shift preserving orientation is obtained and the index formula is given.     

Polynomial quasisolutions of linear second-order differential-difference equations

We consider the scalar linear second-order differential-difference equation with delay {fx159-01}. This equation is investigated by the method of polynomial quasisolutions based on the representation of an unknown function in the form of a polynomial {ie159-01}. Upon the substitution of this polynomial in the original equation, the residual Δ(t) = O(t ^N−1) appears. An exact analytic representation of this residual is obtained. We show the close connection between a linear differential-difference equation with variable coefficients and a model equation with constant coefficients, the structure of whose solution is determined by the roots of the characteristic quasipolynomial. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 140–152, January, 2008.     

A Numerical solution to the linear and nonlinear Fredholm integral equations using Legendre wavelet functions

Two methods for solving the Fredholm integral equation of the second kind in linear case, i.e. f (x) – λ ∫_a^b K (x,y)f (y)dy = g (x), and nonlinear case, i.e., f (x) = g (x) + λ ∫_a^b K (x,y)F (f (y))dy, are proposed. In order to solve the linear equation, the kernel K (x,y) as well as the functions f and g are initially approximated through Legendre wavelet functions. This leads to a system of linear equations its solution culminates in a solution to the Fredholm integral equation. In nonlinear case only K (x,y) is approximated by Legendre wavelet base functions. This leads to a separable kernel and makes it possible to employ a number of earlier methods in solving nonlinear Fredholm integral equation with separable kernels. Another feature of the proposed method is that it finds the solution as a function instead of specific solution points, what is done by the majority of the existing methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear equations and sets of integers

We prove two results concerning solvability of a linear equation in sets of integers. In particular, it is shown that for every k∈ℕ, there is a noninvariant linear equation in k variables such that if A⫅{1,…,N} has no solution to the equation then |A|\leqq 2^-ck/(logk)2N|A|\leqq 2^{-ck/{(\log k)}^{2}}N, for some absolute constant c>0, provided that N is large enough.     

On the boundary behavior of solutions of the Beltrami equations

We show that any homeomorphic solution of the Beltrami equation v from the Sobolev class W _loc^1,1 is a so-called lower Q-homeomorphism with Q(z) = K _μ(z), where K _μ(z) is the dilatation ratio of this equation. On this basis, we develop the theory of boundary behavior and removing of singularities of these solutions.     

Asymptotic forms of solutions of certain linear ordinary differential equations

We analyze the asymptotic behavior as x → ∞ of the product integral Π_x0^xe^A(s)ds, where A(s) is a perturbation of a diagonal matrix function by an integrable function on [x₀,∞). Our results give information concerning the asymptotic behavior of solutions of certain linear ordinary differential equations, e.g., the second order equation y″ = a(x)y.     

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W_p^k(ℝ^s) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W_p^k(ℝ^s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L_p (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ∈L_p(ℝ^s) (φ∈C(ℝ^s) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,L_p(ℝ^s)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167) in the univariate setting to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C20, 41A25, 39B12. Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant G121210654.     

Space-time means and solutions to a class of nonlinear parabolic equations

Cauchy problem and initial boundary value problem for nonlinear parabolic equation inCB([0,T):L ^p ) orL ^q (0,T; L ^p ) type space are considered. Similar to wave equation and dispersive wave equation, the space-time means for linear parabolic equation are shown and a series of nonlinear estimates for some nonlinear functions are obtained by space-time means. By Banach fixed point principle and usual iterative technique a local mild solution of Cauchy problem or IBV problem is constructed for a class of nonlinear parabolic equations inCB([0,T);L ^p orL ^q (0,T; L ^p ) with ϕ(x)∈L ^r . In critical nonlinear case it is also proved thatT can be taken as infinity provided that ||ϕ(x)||_r is sufficiently small, where (p,q,r) is an admissible triple. Project supported by the National Natural Science Foundation of China (Grant No. 19601005).