Abstract Riccati equations in an L1 space of finite measure and applications to transport theory

We consider operator-valued Riccati initial-value problems of the form R′(t) + TR(t) + R(t)T = TA(t) + TB(t)R(t) + R(t)TC(t) + R(t)TD(t)R(t), R(0) = R₀. Here A to D and R₀ have values as non-negative bounded linear operators in L¹ (μ), where μ is a finite measure, and T is a closed non-negative operator in L¹ (μ) satisfying additional technical conditions. For such problems the notion of strongly mild solutions is defined, and local existence and uniqueness theorems for such solutions are established. The results of the analysis are applied to the reflection kernels with both isotropically scattering homogeneous and anisotropically scattering inhomogeneous medium.     

Oscillations of Higher Order Neutral Differential Equations with Variable Coefficients

We obtain sufficient conditions for the oscillation of all solutions of the higher order neutral differential equation dⁿ/d^m[y(t) + P(t) y(t - μ)] + Q(t) y(t ?σ) = 0, t ≧ t₀ where n ≧ 1, P ? C[t₀, ∞), R ], Q ? C[t₀, ∞), R ] and τ, μ ? R ⁺. Our results extend and improve several known results in the literature.     

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.     

The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data

Parabolic partial differential equations with overspecified data play a crucial role in applied mathematics and engineering, as they appear in various engineering models. In this work, the radial basis functions method is used for finding an unknown parameter p(t) in the inverse linear parabolic partial differential equation u_t = u_xx + p(t)u + φ, in [0,1] × (0,T], where u is unknown while the initial condition and boundary conditions are given. Also an additional condition ∫₀¹k(x)u(x,t)dx = E(t), 0 ≤ t ≤ T, for known functions E(t), k(x), is given as the integral overspecification over the spatial domain. The main approach is using the radial basis functions method. In this technique the exact solution is found without any mesh generation on the domain of the problem. We also discuss on the case that the overspecified condition is in the form ∫₀^s(t) u(x,t)dx = E(t), 0 < t ≤ T, 0 < s(t) < 1, where s and E are known functions. Some illustrative examples are presented to show efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Anti‐periodic solutions for evolution equations with mappings in the class (S+)

In this paper, we study the existence of anti‐periodic solutions for the first order evolution equation in a Hilbert space H, where G : H → ? is an even function such that ?G is a mapping of class (S₊) and f : ? → ? satisfies f(t + T) = –f(t) for any t ∈ ? with f(·) ∈ L²(0, T; H). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Fujita-type global existence—global non-existence theorem for a system of reaction diffusion equations with differing diffusivities

In this paper we condiser non-negative solutions of the initial value problem in ?^N for the system where 0 ? δ ? 1 and pq > 0. We prove the following conditions. Suppose min(p,q)≥1 but pq1.

• (a) If δ = 0 then u=v=0 is the only non-negative global solution of the system.
• (b) If δ>0, non-negative non-globle solutions always exist for suitable initial values.
• (c) If 0<?1 and max(α, β) ≥ N/2, where qα = β + 1, pβ = α + 1, then the conclusion of (a) holds.
• (d) If N > 2, 0 < δ ? 1 and max (α β) < (N - 2)/2, then global, non-trivial non-negative solutions exist which belong to L^∞(?^N×[0, ∞]) and satisfy 0 < u(X, t) ? c∣x∣^?2α and 0 < v(X, t) ? c ∣x∣^?2bT for large ∣x∣ for all t > 0, where c depends only upon the initial data.
• (e) Suppose 0 > δ 1 and max (α, β) < N/2. If N> = 1,2 or N > 2 and max (p, q)? N/(N-2), then global, non-trivial solutions exist which, after makinng the standard ‘hot spot’ change of variables, belong to the weighted Hilbert space H¹ (K) where K(x) ? exp(¼∣x∣²). They decay like e^{[max(α,β)-(N/2)+ε]t} for every ε > 0. These solutions are classical solutions for t > 0.
• (f) If max (α, β) < N/2, then threre are global non-tivial solutions which satisfy, in the hot spot variables where where 0 < ε = ε(u₀, v₀) < (N/2)?;max(α, β). Suppose min(p, q) ? 1.
• (g) If pq ≥ 1, all non-negative solutions are global. Suppose min(p, q) < 1.
• (h) If pg > 1 and δ = 0, than all non-trivial non-negative maximal solutions are non-global.
• (i) If 0 < δ ? 1, pq > 1 and max(α,β)≥ N/2 all non-trivial non-negative maximal solutions are non-global.
• (j) If 0 < δ ≥ 1, pq > 1 and max(α,β) < N/2, there are both global and non-negative solutions.

We also indicate some extensions of these results to moe general systems and to othere geometries.     

Removable singularities of weak solutions to the navier-stokes equations

Consider the Navier-Stokes equations in Ω×(0,T), where Ω is a domain in R³. We show that there is an absolute constant ε₀ such that ever, y weak solution u with the property that Sup_tε(a,b)|u(t)|_L(D)≤ε0 is necessarily of class C^∞ in the space-time variables on any compact suhset of D × (a,b) , where D?? and 0 a<b<T. As an application. we prove that if the weak solution u behaves around (x_o, t_o) εΩ×(o,T) 1ike u(x, t) = o(|x - xo|^-1) as x→x ₀ uniforlnly in t in some neighbourliood of t_o, then (x_o,t_o) is actually a removable singularity of u.     

The distribution of Brownian motion in Rn at a natural stopping time

For a measure μ on Rⁿ let ((B_t, P^μ) be Brownian motion in Rⁿ with initial distribution μ. Let D be an open subset of Rⁿ with exit time ζ ≡ inf {t > 0: B_t ? D}. In the case where D is a Green region with Green function G and μ is a measure in D such that G_μ is not identically infinite on any component of D, we have given necessary and sufficient conditions for a measure ν in D to be of the form ν(dx) = P^μ(B_T ? dx, T <ζ), where T is some natural stopping time for (B_t), and we have applied this characterization to show that a measure ν in D satisfies G_ν ? G_μ iff ν is of the form ν(dx) = P^α(B_T ? dx, T <ζ) + β(dx), where T is some natural stopping time for (B_t) and α and β are measures in D such that α + β = μ and β lives on a polar set. We have proved analogous results in the case where D = R² and μ is a finite measure on R² such that ∫ log⁺ ∥x∥ du(x) < ∞, and applied this to give a characterization of the stopping times T for Brownian motion in R² such that (log⁺ ∥B_T∧t∥)_0<t<∞ is P^μ-uniformly integrable.     

The mean value of the zeta-function on <Emphasis Type="Italic">σ</Emphasis>=1

Let R(T): = ò₁^T|z(1+it)|² dt - z(2)T+plogTR(T):= \int_{1}^{T}|\zeta (1+it)|^{2}\,\mathrm{d}t - \zeta (2)T+\pi\log T. We derive a precise explicit expression for R(t) which is used to derive asymptotic formulas for ò₁^TR(t) dt\int_{1}^{T}R(t)\,\mathrm{d}t and ò₁^TR²(t) dt\int_{1}^{T}R^{2}(t)\,\mathrm{d}t. These results improve on earlier upper bounds of Balasubramanian, Ramachandra and the author for the integrals in question.     

Single point blow-up for a semilinear initial value problem

The initial value problem on [?R, R] is considered: u_t(t, x) = u_xx(t, x) + u(t, x)^γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥_∞ → ∞ as t → T. For the specific initial value ? = kψ, where ψ ? 0, ψ_xx + ψ^γ = 0, ψ(±R) = 0, k is sufficiently large, it is shown that if x ≠ 0, then lim_{t → T}u(t, x) and lim_{t → T}u_x(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0.     

The amount of non-uniqueness for factored equations with Euler-Poisson-Darboux factors

Of concern are factored Euler–Poisson–Darboux equations of the type ∏^N_j=1(d²/dt²+(ρ/t)d/dt+A_j)u(t)=0, where, for example, A_j=−c_jΔ, Δ being the Dirichlet Laplacian acting on L²(Ω), Ω⊂ℝⁿ, and 0<c₁<…<c_N. More generally −A_j can be the square of the generator of a (C₀) group on a Banach space. When the constant ρ is negative, the initial value problem for the factored EPD equation is ill-posed. Nevertheless, we determine how many initial conditions are necessary to guarantee uniqueness of a solution. This number jumps up as ρ crosses a negative integer from right to left. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Convergence of the Dirichlet solutions of the very fast diffusion equation

For any −1<m<0, positive functions f, g and u₀≥0, we prove that under some mild conditions on f, g and u₀ as R→∞ the solution u^R of the Dirichlet problem u_t=(u^m/m)_xx in (−R,R)×(0,∞), u(R,t)=(f(t)|m|R)^1/m, u(−R,t)=(g(t)|m|R)^1/m for all t>0, u(x,0)=u₀(x) in (−R,R), converges uniformly on every compact subset of R×(0,T) to the solution of the equation u_t=(u^m/m)_xx in R×(0,T), u(x,0)=u₀(x) in R, which satisfies some mass loss formula on (0,T) where T is the maximal time such that the solution u is positive. We also prove that the solution constructed is equal to the solution constructed in Hui (2007) [15] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.     

Locating the peaks of solutions via the maximum principle II: A local version of the method of moving planes

Let Ω be a bounded, smooth domain in ?²ⁿ, n ≥ 2. The well‐known Moser‐Trudinger inequality ensures the nonlinear functional J_ρ(u) is bounded from below if and only if ρ ≤ ρ_2n := 2²ⁿn!(n ? 1)!ω_2n, where in , and ω_2n is the area of the unit sphere ??^{2n ? 1} in ?²ⁿ. In this paper, we prove the inf_u∈X J_ρ(u) is always attained for ρ ≤ ρ_2n. The existence of minimizers of J_ρ at the critical value ρ = ρ_2n is a delicate problem. The proof depends on the blowup analysis for a sequence of bubbling solutions. Here we develop a local version of the method of moving planes to exclude the boundary bubbling. The existence of minimizers for J_ρ at the critical value ρ = ρ_2n is in contrast to the case of two dimensions. © 2003 Wiley Periodicals, Inc.     

Long time classical solutions of a nonlinear wave equation

The existence and uniqueness of long time classical solutions of the Cauchy problem u_{t t}+μu_t = div(a(u)▽u), where a(u) = 1+u and μ ≥ 0, are studied for the case of two space dimensions. Let the initial data u(0,.) = φ and u_t(0,.) = ψ be supported compactly on R². Then for every T > 0, such a solution exists on [0,T] whenever (φ,ψ) is small enough in H⁴ (R²) x H³(R²). A result on the asymptotic relation between the maximal T and the size of the initial data is given.     

Nonlinear degenerate parabolic equations with time dependent singular coefficients

We are concerned with the nonexistence of positive solutions of the nonlinear parabolic partial differential equations in a cylinder Ω × (0, T) with initial condition u(., 0) = u₀(.) ? 0 and vanishing on the boundary ?Ω × (0, T), given by where $\Omega \in \mathbf {R}^NWe are concerned with the nonexistence of positive solutions of the nonlinear parabolic partial differential equations in a cylinder Ω × (0, T) with initial condition u(., 0) = u₀(.) ? 0 and vanishing on the boundary ?Ω × (0, T), given by where $\Omega \in \mathbf {R}^N$ (resp. a Carnot Carathéodory metric ball in $\mathbf {R}^{2N+1})$ with smooth boundary and the time dependent singular potential function V(x, t) ∈ L¹_loc(Ω × (0, T)), $\alpha , \beta \in \mathbf {R}$, 1 < p < N, p ? 1 + α + β > 0. We find the best lower bounds for p + β and provide proofs for the nonexistence of positive solutions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces

Let V and V* be a real reflexive Banach space and its dual space, respectively. This paper is devoted to the abstract Cauchy problem for doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations of the form: ?_V y^t (u￠(t)) + ?_V j(u(t)) + B(t, u(t)) ' f(t){\partial_V \psi^t (u{^\prime}(t)) + \partial_V \varphi(u(t)) + B(t, u(t)) \ni f(t)} in V*, 0 < t < T, u(0) = u ₀, where ?_V y^t, ?_V j: V ? 2^V*{\partial_V \psi^t, \partial_V \varphi : V \to 2^{V^*}} denote the subdifferential operators of proper, lower semicontinuous and convex functions y^t, j: V ? (-￥, +￥]{\psi^t, \varphi : V \to (-\infty, +\infty]}, respectively, for each t ? [0,T]{t \in [0,T]}, and f : (0, T) → V* and u₀ ? V{u_0 \in V} are given data. Moreover, let B be a (possibly) multi-valued operator from (0, T) × V into V*. We present sufficient conditions for the local (in time) existence of strong solutions to the Cauchy problem as well as for the global existence. Our framework can cover evolution equations whose solutions might blow up in finite time and whose unperturbed equations (i.e., B o 0{B \equiv 0}) might not be uniquely solved in a doubly nonlinear setting. Our proof relies on a couple of approximations for the equation and a fixed point argument with a multi-valued mapping. Moreover, the preceding abstract theory is applied to doubly nonlinear parabolic equations.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Hilbert boundary value problem in the weighted spaces <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">ρ</Emphasis>)

The paper studies a Hilbert boundary value problem in L ¹(ρ), where ρ(t) = |1–t|^α and α is a real number. For α > ?1, it is proved that the homogeneous problem has n + κ linearly independent solutions when n + κ ≥ 0, where a(t) is the coefficient of the problem, besides, κ ind a(t) and n = [α] + 1 if α is not an integer, and n = α if α is an integer. Conditions under which the problem is solvable are found for the case when α > ?1 and n+κ < 0. For α ≤ ?1 the number of linearly independent solutions of the homogeneous problem depends on the behavior of the function a(t) at the point t = 1.     

Ricci flow on surfaces with cusps

We consider the normalized Ricci flow ? _t g = (ρ ? R)g with initial condition a complete metric g ₀ on an open surface M where M is conformal to a punctured compact Riemann surface and g ₀ has ends which are asymptotic to hyperbolic cusps. We prove that when χ(M) < 0 and ρ < 0, the flow g(t) converges exponentially to the unique complete metric of constant Gauss curvature ρ/2 in the conformal class.