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.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

Solution of Initial Value Problems with Monogenic Initial Functions in Banach Spaces with L p -Norm

This paper deals with the initial value problem of the type

A wavelike functional equation of Pexider type

Summary. Let (G, +) and (H, +) be abelian groups such that the equation 2u = v 2u = v is solvable in both G and H. It is shown that if f₁, f₂, f₃, f₄, : G ×G ? H f_1, f_2, f_3, f_4, : G \times G \longrightarrow H satisfy the functional equation f₁(x + t, y + s) + f₂(x - t, y - s) = f₃(x + s, y - t) + f₄(x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , then f₁, f₂, f₃, and f₄ are given by f₁ = w + h, f₂ = w - h, f₃ = w + k, f₄ = w - k where w : G ×G ? H w : G \times G \longrightarrow H is an arbitrary solution of f (x + t, y + s) + f (x - t, y - s) = f (x + s, y - t) + f (x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , and h, k : G ×G ? H h, k : G \times G \longrightarrow H are arbitrary solutions of D_y,t³g(x,y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and D_x,t³g(x,y) = 0 \Delta_{x,t}^{3}g(x,y) = 0 for all x, y, s, t ? G x, y, s, t \in G .     

A Threshold Result for a Non-local Parabolic Equation

In this paper, we consider the Cauchy problem: (ECP) u_t−Δu+p(x)u=u(x,t)∫u²(y,t)/∣x−y∣dy; x∈ℝ³, t>0, u(x, 0)=u₀(x)⩾0 x∈ℝ³, (0.2) The stationary problem for (ECP) is the famous Choquard–Pekar problem, and it has a unique positive solution ū(x) as long as p(x) is radial, continuous in ℝ³, p(x)⩾ā>0, and lim_{∣x∣→∞}p(x)=p¯>0. In this paper, we prove that if the initial data 0⩽u₀(x)⩽(≢)ū(x), then the corresponding solution u(x, t) exists globally and it tends to the zero steady-state solution as t→∞, if u₀(x)⩾(≢)ū(x), then the solution u(x,t) blows up in finite time. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Quenching phenomena for a non-local diffusion equation with a singular absorption

In this paper we study the quenching problem for the non-local diffusion equation

Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions

In this paper, we consider the following nonlinear fractional three-point boundary-value problem:

The Evolution of the Weyl and Maxwell Fields in Curved Space—Times

The covariant Weyl (spin s = 1/2) and Maxwell (s = 1) equations in certain local charts (u, φ) of a space-time (M, g) are considered. It is shown that the condition g⁰⁰(x) > 0 for all x ε u is necessary and sufficient to rewrite them in a unified manner as evolution equations δ_tφ = L_(s)φ. Here L_(s) is a linear first order differential operator on the pre—Hilbert space (C (U_t, ^2s+1). (…)), where U_t ? IR³ is the image of the coordinate map of the spacelike hyper-surface t = const, and (φ, C) = ?U_t ? *Q d⁽³⁾x with a suitable Hermitian n × n- matrix Q = Q(t,x). The total energy of the spinor field ? with respect to U_t is then simply given by E = 〈?,?〉. In this way inequalities for the energy change rate with respect to time, δ_t|?|² = 2Re (?, L_(s)?) are obtained. As an application, the Kerr—Newman black hole is studied, yielding quantitative estimates for the energy change rate. These estimates especially confirm the energy conservation of the Weyl field and the well—known superradiance of electromagnetic waves.     

Smooth solutions to some differential-difference equations of neutral type

The paper is devoted to the scalar linear differential-difference equation of neutral type

The Cauchy problem for a linear parabolic partial differential equation

Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: (p(x)u_x)_x ? q(x)u = p(x)u_t, 0 < x < 1,0 < t? T; $\text{u(0, t) = ?}{}_1\text{(t), 0 < t ? T}$; $\text{u(1,t) = ?}{}_2\text{(t), 0 < t ? T}$; p(0) u_x(0, t) = g(t), 0 < t₀ ? t ? T, is ill-posed in the sense of Hadamard. Complex variable and Dirichlet series techniques are used to establish Hölder continuous dependence of the solution upon the data under the additional assumption of a known uniform bound for ¦ u(x, t)¦ when 0 ? x ? 1 and 0 ? t ? T. Numerical results are obtained for the problem where the data ?₁, ?₂ and g are known only approximately.     

Neumann boundary condition for a non-autonomous Hamilton-Jacobi equation in a quarter plane

We consider Hamilton-Jacobi equation u _t +H(u, u _x ) = 0 in the quarter plane and study initial boundary value problems with Neumann boundary condition on the line x = 0. We assume that p → H(u, p) is convex, positively homogeneous of degree one. In general, this problem need not admit a continuous viscosity solution when s → H(s, p) is non increasing. In this paper, explicit formula for a viscosity solution of the initial boundary value problem is given for the cases s → H(s, p) is non decreasing as well as s → H(s, p) is non increasing.     

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.     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

Asymptotically autonomous parabolic evolution equations

We study the long-term behaviour of the parabolic evolution equation $\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \]$\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \] If A(t) A(t) converges to a sectorial operator A with s(A)?i \Bbb R = ? \sigma(A)\cap i \Bbb R =\emptyset as t?￥ t\to\infty , then the evolution family solving the homogeneous problem has exponential dichotomy. If also f(t)? f_￥ f(t)\to f_\infty , then the solution u converges to the 'stationary solution at infinity', i.e., lim_t?￥u(t) = -A\sp-1f_￥=:u_￥,        lim_t?￥u￠(t)=0,        lim_t?￥A(t)u(t)=Au_￥. \lim_{t\to\infty}u(t)= -A\sp{-1}f_\infty=:u_\infty, \qquad \lim_{t\to\infty}u'(t)=0, \qquad \lim_{t\to\infty}A(t)u(t)=Au_\infty. .     

Some reproducing identities for families of mean values

Gini, Lehmer, Beckenbach, and others studied the meanG _s (a, b) = (a ^s +b ^s )/(a ^{s 1} +b ^s-1 ) We proveTheorem 1 The identity (ina, b)G _s (G _t ,G _u ) =G _v holds if and only if (s, t, u, v) is (s, t, t, t) (the trivial solution) or one of (1, 1 –k, 1 +k, 1), (1/2, 1 –k, k, 1/2), or (0,–k, k, 0) (the exotic solutions,k is any real number)Theorem 2 IfP _s (a, b) is the power mean [(a ^s +b ^s )/2]^1/s , thenP _s (P _t ,P _u ) =P _v has only the trivial solution (s, t, u, v) = (s, t, t, t) and the exotic solution (0,t, –t, 0) The family of meansG _s (respP _s ) includes the classical arithmetic, geometric, and harmonic means     

Boundary value problems for complete quasi-parabolic differential equations of odd order with variable domains of operators

We prove the well-posed solvability in the strong sense of the boundary value Problems

$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$

where the unbounded operators A _s (t), s > 0, in a Hilbert space H have domains D(A _s (t)) depending on t, are subordinate to the powers A ^1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A ₀(t) and the symmetric operators A _s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that

$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$

, where the smoothing operators are defined by

$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$

.

     

On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type

Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q₁(t)| y(t-s₁)|^a sgn y(t-s₁) +q₂(t)| y(t-s₂)|^b sgn y(t-s₂)=0,    t 3 t₀,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ₁ and σ₂ are nonnegative constants, α > 0, β > 0, and a, p, q ₁, q₂ ? C([t₀, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.     

On a smooth solution of a nonlinear periodic boundary-value problem

We establish conditions for the existence of a smooth solution of a quasilinear hyperbolic equationu _tt - u_xx = ƒ(x, t, u, u, u _x),u (0,t) = u (π,t) = 0,u (x, t+ T) = u (x, t), (x, t) ∈ [0, π] ×R, and prove a theorem on the existence and uniqueness of a solution. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1574–1576, November, 1999.     

The dimension of the determinantal scheme vs (t,t,2) of the catalecticant matrix

Let T be a Gorenstein sequence of a graded artinian Gorenstein ring k[x ₀,x ₁,x ₂]/I We develop a dimension formula for PGor(T) in terms of the alignment character. Based on our formula, we find a very large component of V_s (t,t,2) when s=r_t-1+1 and t is large enough. This answers Diesel’s conjecture negatively. Further we show that V_s (t,t,2) is irreducible of dimension 3s-1 for s ≤t+1,t ≥ 2 and dim V_s (t,t,2)= 3s-1 for small s. Finally an algorithm to calculate dim V_s (t,t,2) is constructed, and we find the values of dim V_s (t,t,2) for t ≤ 16.     

Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

The following system considered in this paper: