Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for <Emphasis Type="Italic">p</Emphasis> ≥ 1

We prove that the mixed problem for the Klein–Gordon–Fock equation u _tt (x, t) ? u _xx (x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q _T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L _p [0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L _p (Q _T ) for p ≥ 1. We construct the solution in explicit analytic form.     

Estimates and regularization for solutions of some ill-posed problems of elliptic and parabolic type

In this paper, we examine, in a systematic fashion, some ill-posed problems arising in the theory of heat conduction. In abstract terms, letH be a Hilbert space andA: D (A)?H→H be an unbounded normal operator, we consider the boundary value problemü(t)=Au(t), 0<t<∞,u(0)=u ₀∈D(A), \(\mathop {\lim }\limits_{t \to 0} \left\| {u\left( t \right)} \right\| = 0\) . The problem of recoveringu ₀ whenu(T) is known for someT>0 is not well-posed. Suppose we are given approximationsx ₁,x ₂,…,x _N tou(T ₁),…,u(T _N) with 0<T, <…<T _N and positive weightsP _i,i=1,…,n, \(\sum\limits_{i = 1}^N {P_i = 1} \) such that \(Q_2 \left( {u_0 } \right) = \sum\limits_{i = 1}^N {P_i } \left\| {u\left( {T_i } \right) - x_i } \right\|^2 \leqslant \varepsilon ^2 \) . If ‖u _t(0)‖≤E for some a priori constantE, we construct a regularized solution ν(t) such that \(Q\left( {\nu \left( 0 \right)} \right) \leqslant \varepsilon ^2 \) while \(\left\| {u\left( 0 \right) - \nu \left( 0 \right)} \right\| = 0\left( {ln \left( {E/\varepsilon } \right)} \right)^{ - 1} \) and \(\left\| {u\left( t \right) - \nu \left( t \right)} \right\| = 0\left( {\varepsilon ^{\beta \left( t \right)} } \right)\) where 0<β(t)<1 and the constant in the order symbol depends uponE. The function β(t) is larger thant/m whent _k andk is the largest integer such that \((\sum\limits_{k = 1}^N {P_i (T_i )} )< (\sum\limits_{k = 1}^N {P_i (T_i )} = m\) , which β(t)=t/m on [T _k, m] and β(t)=1 on [m, ∞). Similar results are obtained if the measurement is made in the maximum norm, i.e.,Q _∞(u ₀)=max{‖u(T _i)?x _i‖, 1≤i≤N}.     

Necessary and sufficient conditions for quadratic stabilizability of an uncertain system

Consider an uncertain system (Σ) described by the equationx(t)=A(r(t))x(t)+B(s(t))u(t), wherex(t) ∈R ⁿ is the state,u(t) ∈R ^m is the control,r(t) ∈ ? ?R ^p represents the model parameter uncertainty, ands(t) ∈L ?R ^l represents the input connection parameter uncertainty. The matrix functionsA(·),B(·) are assumed to be continuous and the restraint sets ?,L are assumed to be compact. Within this framework, a notion of quadratic stabilizability is defined. It is important to note that this type of stabilization is robust in the following sense: The Lyapunov function and the control are constructed using only the bounds ?,L. Much of the previous literature has concentrated on a fundamental question: Under what conditions onA(·),B(·), ?,L can quadratic stabilizability be assured? In dealing with this question, previous authors have shown that, if (Σ) satisfies certain matching conditions, then quadratic stabilizability is indeed assured (e.g., Refs. 1–2). Given the fact that matching is only a sufficient condition for quadratic stabilizability, the objective here is to characterize the class of systems for which quadratic stabilizability can be guaranteed.     

On an over-determined problem of free boundary of a degenerate parabolic equation

This work is concerned with the inverse problem of determining initial value of the Cauchy problem for a nonlinear diffusion process with an additional condition on free boundary. Considering the flow of water through a homogeneous isotropic rigid porous medium, we have such desire: for every given positive constants K and T ₀, to decide the initial value u ₀ such that the solution u(x, t) satisfies $\mathop {\sup }\limits_{x \in H_u (T_0 )} |x| \geqslant K$ , where H _u(T ₀) = {x, ?^N: u(x, T ₀) > 0}. In this paper, we first establish a priori estimate u _t ? C(t)u and a more precise Poincaré type inequality $\left\| \phi \right\|_{L^2 (B_\varrho )}^2 \leqslant \varrho \left\| {\nabla \phi } \right\|_{L^2 (B_\varrho )}^2 $ , and then, we give a positive constant C ₀ and assert the main results are true if only $\left\| {u_0 } \right\|_{L^2 (\mathbb{R}^N )} \geqslant C_0 $ .     

Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I

We consider the blow-up of the solution to a semilinear heat equation with nonlinear boundary condition. We establish conditions on nonlinearities sufficient to guarantee that u(x, t) exists for all time t > 0 as well as conditions on data forcing the solution u(x, t) to blow up at some finite time t*. Moreover, an upper bound for t* is derived. Under somewhat more restrictive conditions, lower bounds for t* are also derived.     

Determination of the Unknown Cauchy Data in a Linear Parabolic Problem from the Measured Data at the Final Time

The problem of determining the initial value u(x, 0) = μ ₀(x) in the parabolic equation u _t = (k(x)u _x (x, t)) _x + F(x, t) from the final overdetermination μ _T (x) = u(x, T) is formulated. It is proved that the Fréchet derivative of the cost functional ${{J(\mu_0) = \|\mu_T(x) - u(x, T)\|_0^2}}$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. The existence of a quasisolution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method.     

Commuting and centralizing generalized derivations on lie ideals in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, and L a noncentral Lie ideal of R. If F and G are generalized derivations of R and k ≥1 a fixed integer such that [F(x), x] _k x ? x[G(x), x] _k = 0 for any x ∈ L, then one of the following holds:

1. either there exists an a ∈ U and an α ∈ C such that F(x) = xa and G(x) = (a + α)x for all x ∈ R
2. or R satisfies the standard identity s ₄(x ₁, …, x ₄) and one of the following conclusions occurs

1. there exist a, b, c, q ∈ U, such that a ?b + c ?q ∈ C and F(x) = ax + xb, G(x) = cx + xq for all x ∈ R
2. there exist a, b, c ∈ U and a derivation d of U such that F(x) = ax+d(x) andG(x) = bx+xc?d(x) for all x ∈ R, with a + b ? c ∈ C.

     

Extremes of Gaussian processes with smooth random expectation and smooth random variance

Let ξ(t), t ∈ [0, T],T > 0, be a Gaussian stationary process with expectation 0 and variance 1, and let η(t) and μ(t) be other sufficiently smooth random processes independent of ξ(t). In this paper, we obtain an asymptotic exact result for P(sup _t∈[0,T](η(t)ξ(t) + μ(t)) > u) as u→∞.     

Cauchy problem for the Korteweg-de Vries equation in the case of a nonsmooth unbounded initial function

In the strip П = (?1, 0) × ?, we establish the existence of solutions of the Cauchy problem for the Korteweg-de Vries equation u _t + u _xxx + uu _x = 0 with initial condition either 1) u(?1, x) = ?xθ(x), or 2) u(?1, x) = ?xθ(?x), where θ is the Heaviside function. The solutions constructed in this paper are infinitely smooth for t ∈ (?1, 0) and rapidly decreasing as x → +∞. For the case of the first initial condition, we also establish uniqueness in a certain class. Similar special solutions of the KdV equation arise in the study of the asymptotic behavior with respect to small dispersion of the solutions of certain model problems in a neighborhood of lines of weak discontinuity.     

Grid approximation of singularly perturbed parabolic convection-diffusion equations subject to a piecewise smooth initial condition

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation on an interval is considered. The higher order derivative in the equation is multiplied by a parameter ? that can take arbitrary values in the half-open interval (0, 1]. The first derivative of the initial function has a discontinuity of the first kind at the point x ₀. For small values of ?, a boundary layer with the typical width of ? appears in a neighborhood of the part of the boundary through which the convective flow leaves the domain; in a neighborhood of the characteristic of the reduced equation outgoing from the point (x ₀, 0), a transient (moving in time) layer with the typical width of ?^1/2 appears. Using the method of special grids that condense in a neighborhood of the boundary layer and the method of additive separation of the singularity of the transient layer, special difference schemes are designed that make it possible to approximate the solution of the boundary value problem ?-uniformly on the entire set $\bar G$ , approximate the diffusion flow (i.e., the product ?(?/?x)u(x, t)) on the set $\bar G^ * = \bar G\backslash \{ (x_0 ,0)\} $ , and approximate the derivative (?/?x)u(x, t) on the same set outside the m-neighborhood of the boundary layer. The approximation of the derivatives ?²(?²/?x ²)u(x, t) and (?/?t)u(x, t) on the set $\bar G^ * $ is also examined.     

Fixed point theory and product,sub-, super-spaces

In this paper, we definen-segmentwise metric spaces and then we prove the following results:

1. (i)|Let (X, d) be ann-segmentwise metric space. ThenX ⁿ has the fixed point property with respect to uniformly continuous bounded functions if and only if, for any continuous functionF: C ^*(X) → C^*(X) and for anyn-tuple of distinct points x₁, x₂, ?, x_n ∈X, there exists anh ∈C ^*(X) such that $$F(h)(x_1 ) = h(x_1 ),i = 1,2,...,n;$$ whereC ^*(X) has either the uniform topology or the subspace product (Tychonoff) topology \((C^ * (X) \subseteq X^X )\) .
2. LetX _i (i = 1, 2, ?) be countably compact Hausdorff spaces such thatX ₁ × ? × X_n has the fixed point property for alln ∈N Then the product spaceX ₁ × X₂ × ? has the fixed point property. We shall also discuss several problems in the Fixed Point Theory and give examples if necessary. Among these examples, we have:
3. There exists a connected metric spaceX which can be decomposed as a disjoint union of a closed setA and an open setB such thatA andB have the fixed point property andX does not have.
4. There exists a locally compact metrizable spaceX which has the fixed point property but its one-point compactificationX ⁺ does not have the fixed point property.

Other relevant results and examples will be presented in this paper.     

Existence of Semilinear Differential Equations with Nonlocal Initial Conditions

In this paper we give the existence of mild solutions for semilinear Cauchy problems u′（t） = Au（t） ＋f（t, u（t））, t ∈ I, a.e. with nonlocal initial condition u（O） = g（u） ＋uo when the map g loses compactness in Banach spaces.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

Adaptive numerical schemes for a parabolic problem with blow-up

In this paper we present adaptive procedures for the numericalstudy of positive solutions of the following problem: u_t = u_xx (x, t) (0, 1) x [0, T), u_x(0, t) = 0 t [0, T), u_x(1, t) = u^p(1, t) t [0, T), u(x, 0) = u₀(x) x (0, 1), with p > 1. We describe two methods. The first one refinesthe mesh in the region where the solution becomes bigger ina precise way that allows us to recover the blow-up rate andthe blow-up set of the continuous problem. The second one combinesthe ideas used in the first one with moving mesh methods andmoves the last points when necessary. This scheme also recoversthe blow-up rate and set. Finally, we present numerical experimentsto illustrate the behaviour of both methods.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

Boundary noncrossings of additive Wiener fields∗

Let {W _i (t), t ∈ ?₊}, i = 1, 2, be two Wiener processes, and let W ₃ = {W ₃(t), t ∈ ? ₊ ² } be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary noncrossing probability P _f = P{W ₁(t ₁) + W ₂(t ₂) + W ₃(t) + f(t) ≤ u(t), t ∈ ? ₊ ² }, where f, u : ? ₊ ² → ? are two general measurable functions. We further show that, for large trend functions γf > 0, asymptotically, as γ → ∞, P _γf is equivalent to \( {P}_{\gamma}\underset{\bar{\mkern6mu}}{{}_f} \) , where \( \underset{\bar{\mkern6mu}}{f} \) is the projection of f onto some closed convex set of the reproducing kernel Hilbert space of the field W(t) = W ₁(t ₁) + W ₂(t ₂) + W ₃(t). It turns out that our approach is also applicable for the additive Brownian pillow.     

Algorithms for the vector maximization problem

We consider a convex setB inR ⁿ described as the intersection of halfspacesa _i ^T x ≦b _i (i ∈ I) and a set of linear objective functionsf _j =c _j ^T x (j ∈ J). The index setsI andJ are allowed to be infinite in one of the algorithms. We give the definition of theefficient points ofB (also called functionally efficient or Pareto optimal points) and present the mathematical theory which is needed in the algorithms. In the last section of the paper, we present algorithms that solve the following problems:

1. To decide if a given point inB is efficient.
2. To find an efficient point inB.
3. To decide if a given efficient point is the only one that exists, and if not, find other ones.
4. The solutions of the above problems do not depend on the absolute magnitudes of thec _j. They only describe the relative importance of the different activitiesx _i. Therefore we also consider $$\begin{gathered} \max G^T x \hfill \\ x efficient \hfill \\ \end{gathered} $$ for some vectorG.

     

Right inverses for linear, constant coefficient partial differential operators on distributions over open half spaces

Results of Hörmander on evolution operators together with a characterization of the present authors [Ann. Inst. Fourier, Grenoble 40, 619–655 (1990)] are used to prove the following: Let P ∈ ?[z₁,...,z _n ] and denote by P _m its principal part. If P ? P_m is dominated by P _m then the following assertions for the partial differential operators P(D) and P _m(D) are equivalent for N ∈ S ^n?1:

1. P(D) and/or P_m D)admit a continuous linear right inverse on C ^∞(H ₊(N)).
2. P(D) admits a continuous linear right inverse on C ^∞ (? ⁿ ) and a fundamental solution E ∈ C ^∞(?ⁿ) satisfying Supp $E \subset \overline {H - (N)} $

where H ₊(N) := {x ∈ ? ⁿ :±(x,N) τ; 0}.     

A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line

In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation

$i{\partial _t}u + {\partial_{xx}u - i |u{|^2}{\partial _x}u = 0}$

on the half line (?∞, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which are obtained from the initial data u₀(x) = u(x, 0) and the boundary data g₀(t) = u(0, t), g₁(t) = u_x(0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.

     

On conditional extreme values of random vectors with polar representation

An effective approach for studying the asymptotics of bivariate random vectors is to search for the limits of conditional probabilities where the conditioning variable becomes large. In this context, elliptical and related distributions have been extensively investigated. A quite general model was presented by Fougères and Soulier (Limit conditional distributions for bivariate vectors with polar representation in Stochastic Models, 2010), who derived a conditional limit theorem for random vectors (X, Y) with a polar representation R · (u(T), v(T)), where R, T are stochastically independent and R is in the Gumbel max-domain of attraction. We reformulate their assumptions, such that they have a simpler structure, display more clearly the geometry of the curves (u(t), v(t)) and allow us to deduce interesting generalizations into two directions:

u has several global maxima instead of only one,

the curve (u(t), v(t)) is no longer differentiable, but forms a “cusp”.

The latter generalization yields results where only random norming leads to a non-degenerate limit statement. Ideas and results are elucidated by several figures.