Solution of the Transmission Problem

The solution of the following transmission problem for the Laplace equation is constructed: Δu ₊=0 in G ⁺, Δu ₋=0 in G ⁻, u ₊−u ₋=f in ∂ G ⁺, n⋅(∇ u ₊−a ∇ u ₋)+b τ⋅(∇ u ₊−∇ u ₋)+h ₊ u ₊+h ₋ u ₋=g in ∂ G ⁺.     

Global existence and L ∞ estimates of solutions for a quasilinear parabolic system

In this paper, we study the global existence, L^∞ estimates and decay estimates of solutions for the quasilinear parabolic system u_t = div (|∇ u|^m ∇ u) + f(u, v), v_t = div (|∇ v|^m ∇ v) + g(u,v) with zero Dirichlet boundary condition in a bounded domain Ω ⊂ R^N. In particular, we find a critical value for the existence and nonexistence of global solutions to the equation u_t = div (|∇ u|^m ∇ u) + λ |u|^{α - 1} u.     

Systems of equations driven by stable processes

Let Z^j_t, j = 1, . . . , d, be independent one-dimensional symmetric stable processes of index α ∈ (0,2). We consider the system of stochastic differential equations where the matrix A(x)=(A_ij(x))1≤ i, j ≤ d is continuous and bounded in x and nondegenerate for each x. We prove existence and uniqueness of a weak solution to this system. The approach of this paper uses the martingale problem method. For this, we establish some estimates for pseudodifferential operators with singular state-dependent symbols. Let λ₂ > λ₁ > 0. We show that for any two vectors a, b∈ ℝ_d with |a|, |b| ∈ (λ₁, λ₂) and p sufficiently large, the L^p-norm of the operator whose Fourier multiplier is (|u · a|^α - |u · b|^α) / ∑_j=1^d |u_i|^α is bounded by a constant multiple of |a−b|^θ for some θ > 0, where u=(u₁ , . . . , u_d) ∈ ℝ^d. We deduce from this the L^p-boundedness of pseudodifferential operators with symbols of the form ψ(x,u)=|u · a(x)|^α / ∑_j=1^d |u_i|^α, where u=(u₁,...,u_d) and a is a continuous function on ℝ^d with |a(x)|∈ (λ₁, λ₂) for all x∈ ℝ^d. Research partially supported by NSF grant DMS-0244737. Research partially supported by NSF grant DMS-0303310.     

Continuation of Time Bounds for a Regularized Boussinesq System

We study the periodic solution of a perturbed regularized Boussinesq system (Bona et al., J. Nonlinear Sci. 12:283–318, 2002, Bona et al., Nonlinearity 17:925–952, 2004), namely the system η _t +u _x +β(−η _xxt +u _xxx )+α((ηu) _x +ηη _x +uu _x )=0,u _t +η _x +β(η _xxx −u _xxt )+α((ηu) _x +ηη _x +uu _x )=0, with 0<α,β≤1. We prove that the solution, starting from an initial datum of size ε, remains smaller than ε for a time scale of order (ε ⁻¹ α ⁻¹ β)², whereas the natural time is of order ε ⁻¹ α ⁻¹ β.     

A Paley-Wiener theorem and Wiener-Hopf-type integral equations in Clifford analysis

Using the properties of the monogenic extension of the Fourier transform, we state a Paley-Wiener-type theorem for monogenic functions. Based on an multiplier algebra related to boundary values of monogenic functions we consider integral equations of Wiener-Hopf-typeK±u _±=f on ℝ ⁿ , whereK ∈S′ andu _± are boundary values of monogenic functions in ℝ₊ ⁿ⁺¹ and ℝ_{_} ⁿ⁺¹ respectivly.     

Boundary Harnack principle for second order elliptic equations with unbounded drift

We prove the boundary Harnack principle for ratios of solutions u/v of non-divergence second order elliptic equations Lu = a _ij D _ij u + b _i D _i u = 0 in a bounded domain Ω ⊂  \mathbb R {\mathbb R} ⁿ . We assume that b _i ∈ L ⁿ (Ω) and Ω is a twisted H?lder domain of order α ∈ (1/2, 1]. Based on this result, we derive the H?lder regularity of u/v for uniform domains. Bibliography: 27 titles.     

Asymptotic behavior of solutions to a wave equation with a non linear dissipative term in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this note we investigate the asymptotic behavior of solutions to the wave equation:u"-Δu+g(u')=0 in ℝⁿxℝ₊.     

Global existence,asymptotic behaviour,and global non‐existence of solutions for damped non‐linear wave equations of Kirchhoff type in the whole space

We study the global existence, asymptotic behaviour, and global non‐existence (blow‐up) of solutions for the damped non‐linear wave equation of Kirchhoff type in the whole space: u_tt+u_t=(a+b∥∇u∥^2γ)Δu+∣u∣^αu in ℝ^N×ℝ₊ for a, b⩾0, a+b>0, γ⩾1, and α>0, with initial data u(x, 0)=u₀(x) and u_t(x, 0)=u₁(x). Copyright © 2000 John Wiley & Sons, Ltd.     

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

In three spaces, we obtain exact classical solutions of the boundary-value periodic problem u _tt−a ² u _xx=g(x,t), u(0,t)=u(π,t)=0, u(x,t+T)=u(x,t)=0, x,t∈ĝ Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1537–1544, November, 1998.     

An Almost Sure Limit Theorem for Super-Brownian Motion

We establish an almost sure scaling limit theorem for super-Brownian motion on ℝ ^d associated with the semi-linear equation u_t=\frac12Du+bu-au²u_{t}=\frac{1}{2}\Delta u+\beta u-\alpha u^{2} , where α and β are positive constants. In this case, the spectral theoretical assumptions required in Chen et al. (J. Funct. Anal. 254:1988–2019, 2008) are not satisfied. An example is given to show that the main results also hold for some sub-domains in ℝ ^d .     

Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

Summary We consider four models of partial differential equations obtained by applying a generalization of the method of normal forms to two-component reaction-diffusion systems with small diffusionu _t=εDu _xx+(A+εA ₁)u+F(u),u ∈ ℝ². These equations (quasinormal forms) describe the behaviour of solutions of the original equation forε → 0. One of the quasinormal forms is the well-known complex Ginzburg-Landau equation. The properties of attractors of the other three equations are considered. Two of these equations have an interesting feature that may be called asensitivity to small parameters: they contain a new parameterϑ(ε)=−(aε ^−1/2 mod 1) that influences the behaviour of solutions, but changes infinitely many times whenε → 0. This does not create problems in numerical analysis of quasinormal forms, but makes numerical study of the original problem involvingε almost impossible.     

A mathematical analysis for a diffusive epidemic model with criss-cross dynamics

Abstract. In this paper, an initial boundary value problem with homogeneous Neumann bound-ary condition is studied for a reaction diffusion system which models the spread of infectious dis-eases within two population groups by means of serf and criss-cross infection mechanism, Exis-tence, uniqueness and houndedness of the nonnegative global solution     

On the existence and nonexistence of global solutions for the porous medium equation with strongly nonlinear sources in a cone

In this paper, we study the initial-boundary value problem of the porous medium equation u _t  = Δu ^m  + V(x)u ^p in a cone D = (0, ∞) × Ω, where V(x) ~ (1 + |x|) ^σ . Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω and let l denote the positive root of l ² + (n − 2)l = ω ₁. We prove that if m ≤ p ≤ m + (2 + σ)/(n + l), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if p > m + (2 + σ)/n, then the problem has global solutions for some u ₀ ≥ 0.     

On the growth of mass for a viscous Hamilton-Jacobi equation

We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationu _t = Δu + |Δu| ^p ,t>0,x ∈ ℝ ^N , wherep≥1 andu(0,.)=u ₀≥0,u ₀≢0,u ₀∈L ¹. DenotingI _∞=lim _t→∞‖u(t)‖₁≤∞, we show that the asymptotic behavior of the mass can be classified along three cases as follows:

Asymptotiques d’un systeme dynamique conservatif associe a une equation elliptique non lineaire

LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar ^P-B. We prove that any non-negative solutionu ofu _tt+Δ_gu=f(u) onM x ℝ⁺ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ _g Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C ²(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that for every λ the map (u(0,·),u _t(0,·))→(u(t,·), u _t(t,·)) defines a dynamical system onW ^1/2(M)⊂C(M)×L ²(M) which possesses a compact maximal attractor.      

Norms of Moore-Penrose Inverses of Fredholm Toeplitz Operators

In this paper we estimate the norm of the Moore-Penrose inverse T(a)⁺ of a Fredholm Toeplitz operator T(a) with a matrix-valued symbol a∈L_{N × N}^∞ defined on the complex unit circle. In particular, we show that in the ”generic case” the strict inequality ||T(a)⁺|| > ||a⁻¹||_∞ holds. Moreover, we discuss the asymptotic behavior of ||T(t^ra)⁺|| for . The results are illustrated by numerical experiments.     

Complement on elliptic and hyperbolic inversions

In performing the inversions we analize the case when putu ₁=ax ₁+iby ₁ orax ₁+εby ₁ (i ²=−1,ε ² = 1)x ₁ andy ₁ being orthogonal coordinates of a variable pointM ₁ belonging to a plane whilea andb are positive number which are fixed.     

On a problem of Cilleruelo and Nathanson

Let ℤ denote the set of all integers and ℕ the set of all positive integers. Let A be a set of integers. For every integer u, we denote by d _A (u) and s _A (u) the number of solutions of u=a − a′ with a,a′ ∈ A and u=a+a′ with a,a′ ∈ A and a≤a′, respectively.     

On the behavior of solutions of a semilinear parabolic or elliptic equation satisfying a nonlinear boundary condition in a cylindrical domain

One considers a semilinear parabolic equation u _t = Lu − a(x)f(u) or an elliptic equation u _tt + Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ₊ with the nonlinear boundary condition , where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝⁿ; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems for t → ∞. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007.     

Large solutions for equations involving the p-Laplacian and singular weights

In this paper we consider the boundary blow-up problem Δ_pu = a(x)u^q in a smooth bounded domain Ω of \mathbbR^N{\mathbb{R}}^N, with u = +∞ on ∂Ω. Here D_pu = div(|?u|^p-2?u)\Delta_{p}u = {\rm div}(|\nabla u|^{p-2}\nabla u) is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary behavior of positive solutions.