Existence and uniqueness of periodic solutions for a class of nonlinear n ‐th order differential equations with delays

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T ‐periodic solutions for a class of nonlinear n ‐th order differential equations with delays of the form x⁽ⁿ⁾(t) + f (x^{(n‐ 1)}(t)) + g (t, x (t ‐ τ (t))) = p (t).     

The nonlinear heat equation with absorption: Effects of variable coefficients

We consider the nonnegative solutions to the nonlinear degenerate parabolic equation u_t = (D(x, t)u^{m − 1}u_x)_x − b(x, t)u^p with m > 1, 0 < p < 1, and positive D(x, t), b(x, t). After obtaining the uniqueness and Hölder regularity results, we investigate the dependence of such phenomena as extinction in finite time and instantaneous shrinking of the support on the behaviour of D(x, t) and b(x, t).     

Oscillation criteria for nonlinear second order differential equations with damping

Summary Using the integral average method, we give some new oscillation criteria for the second order differential equation with damped term (a(t)Ψ(x(t))K(x'(t)))'+p(t)K(x'(t))+q(t)f(x(t))=0, t<span style='font-size:10.0pt; font-family:"Lucida Sans Unicode"'>≧t₀. These results improve and generalize the oscillation criteria in<span lang=EN-US style='font-size:10.0pt;mso-ansi-language:EN-US'>[1], because they eliminate both the differentiability of p(t) and the sign of p(t), q(t). As a consequence, improvements of Sobol's type oscillation criteria are obtained.     

Blocking Sets and Derivable Partial Spreads

We prove that a GF(q)-linear Rédei blocking set of size q ^t + q ^t–1 + ··· + q + 1 of PG(2,q ^t) defines a derivable partial spread of PG(2t – 1, q). Using such a relationship, we are able to prove that there are at least two inequivalent Rédei minimal blocking sets of size q ^t + q ^t–1 + ··· + q + 1 in PG(2,q ^t), if t 4.     

Asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces

This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u _t −Δu+b(x)·∇(u|u| ^{q −1})=f(x, t) in ℝ ⁿ ×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u ₀ is supposed to be in an appropriate L ^p space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant     

The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation

A detailed analysis is given to the solution of the cubic Schrödinger equation iq_t + q_xx + 2|q|²q = 0 under the boundary conditions as |x|→∞. The inverse-scattering technique is used, and the asymptotic state is a series of solitons. However, there is no soliton whose amplitude is stationary in time. Each soliton has a definite velocity and “pulsates” in time with a definite period. The interaction of two solitons is considered, and a possible extension to the perturbed periodic wave [q(x + T,t) = q(x,t) as |x|→∞] is discussed.     

Extinction and non-extinction for a polytropic filtration equation with a nonlocal source

In this article, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive polytropic filtration equation u _t ?=?div(|?u ^m | ^p?2?u ^m )?+?a∫_Ω u ^q (y,?t)dy with a, q, m?>?0, p?>?1, m(p???1)?R ^N (N?>?2). More precisely speaking, it is shown that if q?>?m(p???1), any non-negative solution with small initial data vanishes in finite time, and if 0?q?m(p???1), there exists a solution which is positive in Ω for all t?>?0. For the critical case q?=?m(p???1), whether the solutions vanish in finite time or not depends on the comparison between a and μ, where μ?=?∫?_Ωφ ^p?1(x)dx and φ is the unique positive solution of the elliptic problem ?div(|?φ| ^p?2?φ)?=?1, x?∈?Ω; φ(x)?=?0, x?∈??Ω.     

On the Existence of Periodic Solutions of Nonlinear Difference Equations

We obtain new sufficient conditions for the existence and uniqueness of an N-periodic solution (N is a positive integer) of a nonlinear difference equation with continuous argument of the form x(t + 1) = f(x(t)) and investigate the properties of this solution.     

A study of (<Emphasis Type="Italic">x</Emphasis>(<Emphasis Type="Italic">q</Emphasis> + 1), <Emphasis Type="Italic">x</Emphasis>; 2, <Emphasis Type="Italic">q</Emphasis>)-minihypers

In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x ≤ q ² − q, not decomposable in the sum of another minihyper and a line, a (j(q + 1), j; 2, q)-minihyper, where j = q ² − q − x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44:169–196, 2007), giving further results on these minihypers.     

Nonlinear filtering and large deviations:a pde-control theoretic approach

We consider the asymptotic nonlinear filtering problem dx=f(x)dt + ?^1/2 dw,dy=h(x) dt + ? dv and obtain lim_?→0 ? log q ²(x,t) = -W(x,t) for unnormalized conditional densities q ²(x,t) using PDE methods. HereW(x,t) is the value function for a deterministic optimal control problem arising in Mortensen's deterministic estimation, and is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. ijab has also studied this filtering problem, and we extend his large deviation result for certain unnormalized conditional measures. The resulting variational problem corresponds to the above control problem     

The Generalized Dirichlet to Neumann Map for the KdV Equation on the Half-Line

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if q _t and q _xxx have the same sign (KdVI) or two boundary conditions if q _t and q _xxx have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if {q(x,0),q(0,t)} and {q(x,0),q(0,t),q _x (0,t)} are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values {q _x (0,t),q _xx (0,t)} and {q _xx (0,t)}, respectively. We show that this can be achieved without solving for q(x,t) by analysing a certain “global relation” which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function Φ ^(t)(t,k), where Φ ^(t) satisfies the t-part of the associated Lax pair evaluated at x=0. The analysis of the global relation requires the construction of the so-called Gelfand–Levitan–Marchenko triangular representation for Φ ^(t). In spite of the efforts of several investigators, this problem has remained open. In this paper, we construct the representation for Φ ^(t) for the first time and then, by employing this representation, we solve explicitly the global relation for the unknown boundary values in terms of the given initial and boundary conditions and the function Φ ^(t). This yields the unknown boundary values in terms of a nonlinear Volterra integral equation. We also discuss the implications of this result for the analysis of the long t-asymptotics, as well as for the numerical integration of the KdV equation.     

Existence of the mild solution for some fractional differential equations with nonlocal conditions

We are concerned in this paper with the existence of mild solutions to the Cauchy Problem for the fractional differential equation with nonlocal conditions: D ^q x(t)=Ax(t)+t ⁿ f(t,x(t),Bx(t)), t∈[0,T], n∈ℤ⁺, x(0)+g(x)=x ₀, where 0<q<1, A is the infinitesimal generator of a C ₀-semigroup of bounded linear operators on a Banach space X.     

Reconstructing the past from imprecise knowledge of the present: Effective non‐uniqueness in solving parabolic equations backward in time

Identifying sources of ground water pollution and deblurring astronomical galaxy images are two important applications generating growing interest in the numerical computation of parabolic equations backward in time. However, while backward uniqueness typically prevails in parabolic equations, the precise data needed for the existence of a particular backward solution is seldom available. This paper discusses previously unexplored non‐uniqueness issues, originating from trying to reconstruct a particular solution from imprecise data. Explicit 1D examples of linear and nonlinear parabolic equations are presented, in which there is strong computational evidence for the existence of distinct solutions w^red(x,t) and w^green(x,t), on 0 ≤ t ≤ 1. These solutions have the property that the traces w^red(x,1) and w^green(x,1) at time t = 1 are close enough to be visually indistinguishable, while the corresponding initial values w^red(x,0) and w^green(x,0) are vastly different, well‐behaved, physically plausible functions, with comparable L² norms. This implies effective non‐uniqueness in the recovery of w^red(x,0) from approximate data for w^red(x,1). In all these examples, the Van Cittert iterative procedure is used as a tool to discover unsuspected, valid, additional solutions w^green(x,0). This methodology can generate numerous other examples and indicates that multidimensional problems are likely to be a rich source of striking non‐uniqueness phenomena. Published 2012. This article is a US Government work and is in the public domain in the USA.     

Boundary blow-up for differential equations with indefinite weight

We obtain results of existence and multiplicity of solutions for the second-order equation x″+q(t)g(x)=0, with x(t) defined for all t∈]0,1[ and such that x(t)→+∞ as t→0⁺ and t→1⁻. We assume g having superlinear growth at infinity and q(t) possibly changing sign on [0,1].     

Singular solutions of some nonlinear parabolic equations

We consider the existence and uniqueness of singular solutions for equations of the formu ₁=div(|Du|^p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫_|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u ^q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. Dedicated to Professor Shmuel Agmon     

Stable Determination of X-Ray Transforms of Time Dependent Potentials from Partial Boundary Data

We consider compact smooth Riemmanian manifolds with boundary of dimension greater than or equal to two. For the initial-boundary value problem for the wave equation with a lower order term q(t, x), we can recover the X-ray transform of time dependent potentials q(t, x) from the dynamical Dirichlet-to-Neumann map in a stable way. We derive conditional Hölder stability estimates for the X-ray transform of q(t, x). The essential technique involved is the Gaussian beam Ansatz, and the proofs are done with the minimal assumptions on the geometry for the Ansatz to be well-defined.     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Instability of Certain Nonlinear Stochastic Differential Equations

We prove existence and uniqueness of the solution X^ε_t of the SDE, X^ε_t = εB_t + ∫^t₀u^{q −1} _ε(s, X^ε_t) ds, where X^ε_t is a one-dimensional process and u_ε(t, x) the density of X^ε_t (ε > 0, q > 1). We show that the closure of (X^ε_t; 0 ≤ t ≤ 1) with respect to Hölder norm, when ε goes to 0, is a.s. equal to an explicit family of continuous functions. We obtain similar results, considering SDE′s where the drift coefficient is equal to ± sgn(x) u(t, x).     

Anti-periodic solutions for a class of nonlinear th-order differential equations with delays

In this paper, we use the Leray–Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear nth-order differential equations with delays of the form

x⁽ⁿ⁾(t)+f(t,x⁽ⁿ⁻¹⁾(t))+g(t,x(t−τ(t)))=e(t).