Fractional Abstract Cauchy Problems

This paper is concerned with fractional abstract Cauchy problems with order \({\alpha\in(1,2)}\). The notion of fractional solution operator is introduced, its some properties are obtained. A generation theorem for exponentially bounded fractional solution operators is given. It is proved that the homogeneous fractional Cauchy problem (FACP ₀) is well-posed if and only if its coefficient operator A generates an α-order fractional solution operator. Sufficient conditions are given to guarantee the existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem (FACP _f ).     

Asymptotic behavior at infinity for solutions of Emden-Fowler type equations

The semilinear equation Δu = |u|^σ?1 u is considered in the exterior of a ball in ? ⁿ , _n ≥ 3. It is shown that if the exponent σ is greater than a “critical” value (= n/n?2), then for x → ∞ the leading term of the asymptotics of any solution is a linear combination of derivatives of the fundamental solution. It is shown that there exist solutions with the indicated leading term of an asymptotics of such a type.     

Asymptotic expansion of the solution of the initial value problem for a singularly perturbed ordinary differential equation

We consider the Cauchy problem for the nonlinear differential equation

$$\varepsilon \frac{{du}}{{dx}} = f(x,u),u(0,\varepsilon ) = R_0 ,$$

where ? > 0 is a small parameter, f(x, u) ∈ C ^∞ ([0, d] × ?), R ₀ > 0, and the following conditions are satisfied: f(x, u) = x ? u ^p + O(x ² + |xu| + |u|^p+1) as x, u → 0, where p ∈ ? \ {1} f(x, 0) > 0 for x > 0; f _u ²(x, u) < 0 for (x, u) ∈ [0, d] × (0, + ∞); Σ ₀ ^+∞ f _u ²(x, u) du = ?∞. We construct three asymptotic expansions (external, internal, and intermediate) and prove that the matched asymptotic expansion approximates the solution uniformly on the entire interval [0, d].

     

Solution of the Ulam Stability Problem for an Euler Type Quadratic Functional Equation

In 1968 S.M. Ulam proposed the problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?’’. In 1978 according to P.M. Gruber this kind of problems is of particular interest in probability theory and in the case of functional equations of different types. In 1997 W. Schuster established a new vector quadratic identity on the basis of the well-known Euler type theorem on quadrilaterals: If ABCD is a quadrilateral and M, N are the mid-points of the diagonals AC, BD as well as A′, B′, C′, D′ are the mid-points of the sides AB, BC, CD, DA, then |AB|² + |BC|² + |CD|² + |DA|² = 2|A′C′|² + 2|B′D′|² + 4|MN|². Employing in this paper the above geometric identity we introduce the new Euler type quadratic functional equation

$\begin{array}{l}2{[}Q(x_{0} - x_{1}+Q(x_{1}-x_{2})+Q(x_{2}- x_{3})+Q(x_{3}-x_{0}){]}\\\qquad = Q(x_{0}-x_{1}-x_{2}+x_{3})+Q(x_{0} + x_{1}-x_{2}-x_{3})+2Q(x_{0}-x_{1}+ x_{2}-x_{3})\end{array}$

for all vectors (x₀, x₁, x₂, x₃) X⁴, with X and Y linear spaces. For every x ∈ R set Q(x) = x². Then the mapping Q : X → Y is quadratic. Note also that if Q : R → R is quadratic, then we have Q(x) = Q(1)x². Besides note that the geometric interpretation of the special example

$\begin{array}{l}2{[}(x_{0} - x_{1})^{2}+ (x_{1}-x_{2})^{2}+ (x_{2}-x_{3})^{2}+(x_{3}-x_{0})^{2}{]}\\\qquad = (x_{0}-x_{1}-x_{2} + x_{3})^{2}+(x_{0} + x_{1}-x_{2}-x_{3})^{2} + 2(x_{0}-x_{1}+ x_{2}-x_{3})^{2}\end{array}$

leads to the above-mentioned Euler type theorem on quadrilaterals ABCD with position vectors x₀, x₁, x₂, x₃ of vertices A, B, C, D, respectively. Then we solve the Ulam stability problem for the afore-mentioned equation, with non-linear Euler type quadratic mappings Q : X → Y.

     

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ?D. Perturbing the Cauchy problem for the Cauchy–Riemann system ??u = f in D with boundary data on a closed subset S ? ?D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter ε ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ?D\S, each of them has a unique solution in some appropriate Hilbert space H ⁺(D) densely embedded in the Lebesgue space L ₂(?D) and the Sobolev–Slobodetski? space H ^1/2?δ(D) for every δ > 0. The corresponding family of the solutions {u _ε} converges to a solution to the Cauchy problem in H ⁺(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H ⁺(D) is equivalent to boundedness of the family {u _ε} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.     

A numerical algorithm for solving a nonstationary problem of optimal control

Let {φ _n ^(α,β) (z)} _n=0 ^∞ be a system of Jacobi polynomials orthonormal on the circle |z| = 1 with respect to the weight (1 ? cos τ)^α+1/2(1 + cos τ)^β+1/2 (α, β > ?1), and let \(\psi _n^{\left( {\alpha ,\beta } \right)*} \left( z \right): = z^n \overline {\psi _n^{\left( {\alpha ,\beta } \right)} \left( {{1 \mathord{\left/ {\vphantom {1 {\bar z}}} \right. \kern-\nulldelimiterspace} {\bar z}}} \right)}\)). We establish relations between the polynomial φ _n ^(α,?1/2) (z) and the nth (C, α ? 1/2)-mean of the Maclaurin series for the function (1 ? z)^?α?3/2 and also between the polynomial φ _n ^(α,?1/2)* (z) and the nth (C, α + 1/2)-mean of the Maclaurin series for the function (1 ? z)^?α?1/2. We use these relations to derive an asymptotic formula for φ _n ^(α,?1/2) (z); the formula is uniform inside the disk |z| < 1. It follows that φ _n ^(α,?1/2) (z) ≠ 0 in the disk |z| ≤ ρ for fixed φ ∈ (0, 1) and α > ?1 if n is sufficiently large.     

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.     

Uniqueness of a positive radially symmetric Solution of the Dirichlet problem for certain nonlinear differential equation of the second order in a ball

We consider the Dirichlet problem

$u_\Gamma = 0$

for the nonlinear differential equation

$\Delta u + \left| x \right|^m \left| u \right|^p = 0, x \in S,$

with constant m ≥ 0 and p > 1 in the unit ball S = {x ∈ R ⁿ : |x| < 1}(n ≥ 3) with the boundary Γ. We prove that with p ≤ m+n/n?2 this problem has a unique positive radially symmetric solution.

     

A priori estimates and continuous dependence of solutions of mixed problems for parabolic equations as nonlocal boundary conditions pass into local ones

In the rectangle G = (0, 1) × (0, T), we consider the family of problems

$$\begin{gathered} \frac{1}{{a(x,t)}}\frac{{\partial u_\alpha }}{{\partial t}} - \frac{{\partial ^2 u_\alpha }}{{\partial x^2 }} = f(x,t), u_\alpha (x,0) = \phi _\alpha (x), u_\alpha (0,t) = 0, 0 \leqslant \alpha \leqslant 1, \hfill \\ u_0 (1,t) = h(t), \frac{{\partial u_1 (1,t)}}{{\partial x}} = h(t), \frac{{u_\alpha (1,t) - u_\alpha (\alpha ,t)}}{{1 - \alpha }} = h(t), 0 < \alpha < 1, \hfill \\ a_1 \geqslant a(x,t) \geqslant a_0 > 0, h \in W_2^1 (0,T), \phi _\alpha \in W_2^1 (0,T), \phi _\alpha (0) = 0, 0 \leqslant \alpha \leqslant 1, \hfill \\ \phi _0 (1) = h(0), \phi '_1 (1) = h(0), \frac{{\phi _\alpha (1) - \phi _\alpha (0)}}{{1 - \alpha }} = h(0), 0 < \alpha < 1, f \in L_2 (G) \hfill \\ \end{gathered} $$

. It is well known that, for α = 0 and α = 1, the corresponding problems with local conditions are solvable, and the solutions are unique and belong to W ₂ ^2,1 (G).

We prove the existence and uniqueness of solutions of the family of problems with nonlocal conditions for each α ∈ (0, 1). For the differences u _α ? u ₀ and u _α ? u ₁ (0 < α < 1), we establish a priori estimates and use them to prove that if ? _α → ? ₀ as α → 0, then u _α → u ₀ and if ? _α → ? ₁ as α → 1, then u _α → u ₁.     

Some algorithms for the dynamic reconstruction of inputs

For the weight \(v_k \left( x \right) = \prod _{\alpha \in \mathbb{R}_ + } \left| {\left( {\alpha ,x} \right)} \right|^{2k\left( \alpha \right)}\) defined by a positive subsystem R ₊ of a finite root system R ? ? ^d and by a function k(α): R → ?₊ invariant under the reflection group generated by R, a sharp Jackson inequality in L ₂(? ^d ) is proved.     

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.     

A transmission problem on $${\mathbb{R}^{2}}$$ with critical exponential growth

In this work we prove the existence of a nontrivial solution for a transmission problem on \({\mathbb{R}^{2}}\) with critical exponential growth, that is, the nonlinearity behaves like exp(α₀ s ²) as |s| → ∞, for some α₀ > 0.     

Catenoidal Layers for the Allen-Cahn Equation in Bounded Domains

This paper presents a new family of solutions to the singularly perturbed Allen-Cahn equation α~2Δu + u(1- u~2) = 0 in a smooth bounded domain Ω R~3, with Neumann boundary condition and α 0 a small parameter. These solutions have the property that as α→ 0, their level sets collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature intersecting ?Ω orthogonally and that is non-degenerate respect to ?Ω. The authors provide explicit examples of surfaces to which the result applies.     

Neighborhood Conditions for Fractional ID-<Emphasis Type="Italic">k</Emphasis>-factor-critical Graphs

Let G be a graph and k ≥ 2 a positive integer. Let h: E(G) → [0, 1] be a function. If \(\sum\limits_{e \mathrel\backepsilon x} {h(e) = k} \) holds for each x ∈ V (G), then we call G[F_h] a fractional k-factor of G with indicator function h where F_h = {e ∈ E(G): h(e) > 0}. A graph G is fractional independent-set-deletable k-factor-critical (in short, fractional ID-k-factor-critical), if G ? I has a fractional k-factor for every independent set I of G. In this paper, we prove that if n ≥ 9k ? 14 and for any subset X ? V (G) we have

$${N_G}(X) = V(G)if|X| \geqslant \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ;or|{N_G}(X)| \geqslant \frac{{3k - 1}}{k}|X|if|X| < \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ,$$

then G is fractional ID-k-factor-critical.

     

Probabilities of high extremes for a Gaussian stationary process in a random environment

Let ξ(t) be a zero-mean stationary Gaussian process with the covariance function r(t) of Pickands type, i.e., r(t) = 1 ? |t| ^α + o(|t| ^α ), t → 0, 0 < α ≤ 2, and η(t), ζ(t) be periodic random processes. The exact asymptotic behavior of the probabilities P(max _t∈[0,T] η(t)ξ(t) > u), P(max _t∈[0,T] (ξ(t) + η(t)) > u) and P(max _t∈[0,T] (η(t)ξ(t) + ζ(t)) > u) is obtained for u → ∞ for any T > 0 and independent ξ(t), η(t), ζ(t).     

Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation u_t-div(|?u|~(p-2)?u) =-|u|~(β-1) u + α|u|~(q-2 )u,where p 1, β 0, q≥1 and α 0. By using Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.     

Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone

In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u _t  = Δu ^m  + V(x)h(t)u ^p in a cone D = (0, ∞) × Ω, where \({V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}\). 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 \({m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if \({p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has global solutions for some u ₀ ≥ 0.     

Averaging of boundary-value problem in domain perforated along (<Emphasis Type="Italic">n</Emphasis> − 1)-dimensional manifold with nonlinear third type boundary conditions on the boundary of cavities

The research considers the asymptotic behavior of solutions u _? of the Poisson equation in a domain ?-periodically perforated along manifold γ = ω ∩ {x ₁ = 0} ≠ Ø with a nonlinear third type boundary condition ? _v u _? + ?^?ασ(x, u _?) = 0 on the boundary of the cavities. It is supposed that the perforations are balls of radius C ₀?^α, C ₀ > 0, α = n ? 1 / n ? 2, n ≥ 3, periodically distributed along the manifold γ with period ? > 0. It has been shown that as ? → 0 the microscopic solutions can be approximated by the solution of an effective problem which contains in a transmission conditions a new nonlinear term representing the macroscopic contribution of the processes on the boundary of the microscopic cavities. This effect was first noticed in [1] where the similar problem was investigated for n = 3 and for the case where Ω is a domain periodically perforated over the whole volume. This paper provides a new method for the proof of the convergence of the solutions {u _?} to the solution of the effective problem is given. Furthermore, an improved approximation for the gradient of the microscopic solutions is constructed, and more accurate results are obtained with respect to the energy norm proved via a corrector term. Note that this approach can be generalized to achieve results for perforations of more complex geometry.     

On the positive definiteness of some functions related to the Schoenberg problem

For a broad class of functions f: [0,+∞) → ?, we prove that the function f(ρ ^λ(x)) is positive definite on a nontrivial real linear space E if and only if 0 ≤ λ ≤ α(E, ρ). Here ρ is a nonnegative homogeneous function on E such that ρ(x) ? 0 and α(E, ρ) is the Schoenberg constant.     

Existence of Nonnegative Solutions for a Class of Systems Involving Fractional (p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional (p, q)-Laplacian operators:

$$\left\{ {\begin{array}{*{20}{c}} {\left( { - \Delta } \right)_p^su = \lambda a\left( x \right){{\left| u \right|}^{p - 2}}u + \lambda b\left( x \right){{\left| u \right|}^{\alpha - 2}}{{\left| v \right|}^\beta }u + \frac{{\mu \left( x \right)}}{{\alpha \delta }}{{\left| u \right|}^{\gamma - 2}}{{\left| v \right|}^\delta }uin\Omega ,} \\ {\left( { - \Delta } \right)_q^sv = \lambda c\left( x \right){{\left| v \right|}^{q - 2}}v + \lambda b\left( x \right){{\left| u \right|}^\alpha }{{\left| v \right|}^{\beta - 2}}v + \frac{{\mu \left( x \right)}}{{\beta \gamma }}{{\left| u \right|}^\gamma }{{\left| v \right|}^{\delta - 2}}vin\Omega ,} \\ {u = v = 0on{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$

where λ > 0 is a real parameter, Ω is a bounded domain in R ^N , with boundary ?Ω Lipschitz continuous, s ∈ (0, 1), 1 < p ≤ q < ∞, sq < N, while (?Δ) _p ^s u is the fractional p-Laplacian operator of u and, similarly, (?Δ) _q ^s v is the fractional q-Laplacian operator of v. Since possibly p ≠ q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalue λ₁ for a related system, they prove that there exists a positive solution for the problem when λ < λ₁ by the modified definitions. Moreover, the authors obtain the bifurcation property when λ → λ₁^-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ ≥ λ₁.