On the convergence of the formal Fourier solution of the wave equation with a summable potential

The convergence of the formal Fourier solution to a mixed problem for the wave equation with a summable potential is analyzed under weaker assumptions imposed on the initial position u(x, 0) = φ(x) than those required for a classical solution up to the case φ(x) ∈ L_p[0,1] for p > 1. It is shown that the formal solution series always converges and represents a weak solution of the mixed problem.     

Behavior of the formal Fourier solution of the wave equation with a summable potential

The convergence of the formal Fourier solution to a mixed problem for the wave equation with a summable potential is analyzed under weaker assumptions imposed on the initial position u(x, 0) = φ(x) than those required for a classical solution up to the case φ(x)∈ L_p[0,1] for p > 1. It is shown that the formal solution series always converges and represents a weak solution of the mixed problem.     

Mixed problem for the wave equation with a summable potential and nonzero initial velocity

The resolvent approach in the Fourier method, combined with Krylov’s ideas concerning convergence acceleration for Fourier series, is used to obtain a classical solution of a mixed problem for the wave equation with a summable potential, fixed ends, a zero initial position, and an initial velocity ψ(x), where ψ(x) is absolutely continuous, ψ'(x) ∈ L ₂[0,1], and ψ(0) = ψ(1) = 0. In the case ψ(x) ∈ L[0,1], it is shown that the series of the formal solution converges uniformly and is a weak solution of the mixed problem.     

On a singular minimizing problem

We investigate the non-homogeneous modular Dirichlet problem Δ _p (·)u(x) = f (x) (where Δ _p (·)u(x) = div(|?u|^p(x-2)?u(x)) from the functional analytic point of view and we prove the stability of the solutions \({\left( {{u_{{p_i}}}} \right)_i}\) of the equation \({\Delta _{{p_i}\left( \cdot \right)}}{u_{{p_i}\left( \cdot \right)}} = f\) as p _i (·) → q(·) via Gamma-convergence of sequence of appropriate functionals.     

Asymptotics of the Velocity Potential of an Ideal Fluid Flowing around a Thin Body

We consider the Neumann problem outside a small neighborhood of a planar disk in the three-dimensional space. The surface of this neighborhood is assumed to be smooth, and its thickness is characterized by a small parameter ε. A uniform asymptotic expansion of the solution of this problem with respect to ε is constructed by the matching method. Since the problem turned out to be bisingular, an additional inner asymptotic expansion in the so-called stretched variables is constructed near the edge of the disk. A physical interpretation of the solution of this boundary value problem is the velocity potential of a laminar flow of an ideal fluid around a thin body, which is the neighborhood of the disk. It is assumed that this flow has unit velocity at a large distance from the disk, which is equivalent to the following condition for the potential: u(x₁, x₂, x₃, ε) = x₃+O(r^?2) as r → ∞, where r is the distance to the origin. The boundary condition of this problem is the impermeability of the surface of the body: ?u/?n = 0 at the boundary. After subtracting x₃ from the solution u(x₁, x₂, x₃, ε), we get a boundary value problem for the potential ?(x₁, x₂, x₃, ε) of the perturbed motion. Since the integral of the function ??/?n over the surface of the body is zero, we have ?(x₁, x₂, x₃, ε) = O(r^?2) as r → ∞. Hence, all the coefficients of the outer asymptotic expansion with respect to ε have the same behavior at infinity. However, these coefficients have growing singularities at the approach to the edge of the disk, which implies the bisingularity of the problem.     

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.     

The one dimensional parabolic <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)-Laplace equation

The Dirichlet problem for the degenerate and singular parabolic p(x)-Laplace equation with one spatial variable is considered. We prove the existence of the unique weak solution such that the derivatives u _t and u _x of a solution u belong to \({L_{\infty}}\). Moreover for the singular case we prove the existence of the strong solution i.e. such that u _t , u _x and u _xx belong to \({L_{\infty}}\).     

Fujita type conditions to heat equation with variable source

This paper studies heat equation with variable exponent u _t = Δu + u^p(x) + u ^q in ? ^N × (0, T), where p(x) is a nonnegative continuous, bounded function, 0 < p_? = inf p(x) ≤ p(x) ≤ sup p(x) = p₊. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max {p₊, q} ≤ 1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 < q ≤ 1 with p₊ > 1, or 1 < q < 1 + \(\frac{2}{N}\). In addition, if q > 1 + \(\frac{2}{N}\), then (i) all solutions blow up in finite time with 0 < p_? ≤ p₊ ≤ 1 + \(\frac{2}{N}\); (ii) there are both global and nonglobal solutions for p_? > 1 + \(\frac{2}{N}\); and (iii) there are functions p(x) such that all solutions blow up in finite time, and also functions p(x) such that the problem possesses global solutions when p_? < 1 + \(\frac{2}{N}\) < p₊.     

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 blow-up solutions of <Emphasis Type="Italic">p</Emphasis>-Laplacian elliptic equations with lower order terms

We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type ?Δ _p u = a(x)u ^m ?b(x)f(u) with p >  1 and 0 <  m <  p?1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p =  2 to the quasilinear case with p >  1.     

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].

     

Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

For the system of root functions of an operator defined by the differential operation ?u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L₂(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L₂ and W₂^?1, respectively, and may have singularities at the endpoints of G such that q(x) = q_R(x) +q′S(x) and the functions q_S(x), p(x), q ₂ ^S (x)w(x), p²(x)w(x), and q_R(x)w(x) are integrable on the whole interval G, where w(x) = x(1 ? x).     

Convergence of ground state solutions for nonlinear Schrdinger equations on graphs

We consider the nonlinear Schr¨odinger equation-?u +(λa(x) + 1)u = |u|~(p-1) u on a locally finite graph G =(V, E). We prove via the Nehari method that if a(x) satisfies certain assumptions, for any λ 1, the equation admits a ground state solution uλ. Moreover, as λ→∞, the solution uλconverges to a solution of the Dirichlet problem-?u + u = |u|~(p-1) u which is defined on the potential well ?. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.     

Nonlocal Dezin’s problem for Lavrent’ev–Bitsadze equation

Using the method of spectral analysis, for the mixed type equation u_xx + (sgny)u_yy = 0 in a rectangular domain we establish a criterion of uniqueness of its solution satisfying periodicity conditions by the variable x, a nonlocal condition, and a boundary condition. The solution is constructed as the sum of a series in eigenfunctions for the corresponding one-dimensional spectral problem. At the investigation of convergence of the series, the problem of small denominators occurs. Under certain restrictions on the parameters of the problem and the functions, included in the boundary conditions, we prove uniform convergence of the constructed series and stability of the solution under perturbations of these functions.     

On power series representing solutions of the one-dimensional time-independent Schrödinger equation

For the equation χ″(x) = u(x)χ(x) with infinitely smooth u(x), the general solution χ(x) is found in the form of a power series. The coefficients of the series are expressed via all derivatives u ^(m)(y) of the function u(x) at a fixed point y. Examples of solutions for particular functions u(x) are considered.     

General compactly supported solution of an integral equation of the convolution type

We find the general form of solutions of the integral equation ∫k(t ? s)u₁(s) ds = u₂(t) of the convolution type for the pair of unknown functions u₁ and u₂ in the class of compactly supported continuously differentiable functions under the condition that the kernel k(t) has the Fourier transform \(\widetilde {{P_2}}\), where \(\widetilde {{P_1}}\) and \(\widetilde {{P_2}}\) are polynomials in the exponential e^iτx, τ > 0, with coefficients polynomial in x. If the functions \({P_l}\left( x \right) = \widetilde {{P_l}}\left( {{e^{i\tau x}}} \right)\), l = 1, 2, have no common zeros, then the general solution in Fourier transforms has the form U_l(x) = P_l(x)R(x), l = 1, 2, where R(x) is the Fourier transform of an arbitrary compactly supported continuously differentiable function r(t).     

A tracing of the fractional temperature field

This paper is devoted to a study of L~q-tracing of the fractional temperature field u(t, x)—the weak solution of the fractional heat equation(?_t +(-?_x)~α)u(t, x) = g(t, x) in L~p(R_+~(1+n)) subject to the initial temperature u(0, x) = f(x) in L~p(R~n).     

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.

     

Equiconvergence of the trigonometric Fourier series and the expansion in eigenfunctions of the Sturm-Liouville operator with a distribution potential

We consider the Sturm-Liouville operator L = ?d ²/dx ² + q(x) with the Dirichlet boundary conditions in the space L ₂[0, π] under the assumption that the potential q(x) belongs to W ₂ ^?1 [0, π]. We study the problem of uniform equiconvergence on the interval [0, π] of the expansion of a function f(x) in the system of eigenfunctions and associated functions of the operator L and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function f(x) of class L ₁. We also consider the case of potentials belonging to the scale of Sobolev spaces W ₂ ^?θ [0, π] with ½ < θ ≤ 1. We show that if the antiderivative u(x) of the potential belongs to some space W ₂ ^θ [0, π] with 0 < θ < 1/2, then, for any function in the space L ₂[0, π], the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing u(x). We also give an explicit estimate for the rate of equiconvergence.     

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.