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 ).     

Stabilization of solution to the Cauchy problem for a parabolic equation with lower order coefficients and an exponentially growing initial function

For the coefficients of lower order terms of a second-order parabolic equation, we obtain sharp sufficient conditions under which the solution of the Cauchy problem stabilizes to zero uniformly in x on each compact set K in ? ^N for any exponentially growing initial function.     

Conditions for the existence of a classical solution of a cauchy type problem for the diffusion equation with a Riemann-Liouville partial derivative

We study a Cauchy type problem for a differential equation containing a fractional Riemann-Liouville partial derivative of order α, 0 < α < 2. Conditions under which the solution of the problem tends to zero as |x| → ∞ are obtained. We prove an existence theorem for a classical solution of the Cauchy type problem and show that the solution has a singularity as t → 0 of order 1 ? α if 0 < α ≤ 1 and of order 2 ? α if 1 < α < 2.     

On ordered-covering mappings and implicit differential inequalities

We define the set of ordered covering of a mapping that acts in partially ordered spaces; we suggest a method for finding the set of ordered covering of vector functions of several variables and the Nemytskii operator acting in Lebesgue spaces. We prove assertions on operator inequalities in arbitrary partially ordered spaces. We obtain conditions that use a set of ordered covering of the corresponding mapping and ensure that the existence of an element u such that f(u) ≥ y implies the solvability of the equation f(x) = y and the estimate x ≤ u for its solution. We study the problem on the existence of the minimal and least solutions. These results are used for the analysis of an implicit differential equation. For the Cauchy problem, we prove a theorem on an inequality of the Chaplygin type.     

A two-stage difference method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped

A novel two-stage difference method is proposed for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. At the first stage, approximate values of the sum of the pure fourth derivatives of the desired solution are sought on a cubic grid. At the second stage, the system of difference equations approximating the Dirichlet problem is corrected by introducing the quantities determined at the first stage. The difference equations at the first and second stages are formulated using the simplest six-point averaging operator. Under the assumptions that the given boundary values are six times differentiable at the faces of the parallelepiped, those derivatives satisfy the Hölder condition, and the boundary values are continuous at the edges and their second derivatives satisfy a matching condition implied by the Laplace equation, it is proved that the difference solution to the Dirichlet problem converges uniformly as O(h ⁴lnh ^?1), where h is the mesh size.     

Convergence of a family of solutions to a Fujita-type equation in domains with cavities

The Dirichlet problem for a Fujita-type equation, i.e., a second-order quasilinear uniformly elliptic equation is considered in domains Ω_ε with spherical or cylindrical cavities of characteristic size ε. The form of the function in the condition on the cavities’ boundaries depends on ε. For ε tending to zero and the number of cavities increasing simultaneously, sufficient conditions are established for the convergence of the family of solutions {u _ε(x)} of this problem to the solution u(х) of a similar problem in the domain Ω with no cavities with the same boundary conditions imposed on the common part of the boundaries ?Ω and ?Ω_ε. Convergence rate estimates are given.     

A Stability Estimate for a Solution to the Wave Equation with the Cauchy Data on a Timelike Cylindrical Surface

We consider the linear wave equation in a domain of the x, t-space bounded from above and below by some smooth surfaces and from the sides by a cylindrical surface with generator parallel to the t-axis. We study the Cauchy problem for this equation with data on a piece of the timelike cylindrical surface and establish a stability estimate for a solution to the problem.     

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.     

Problem with analogs of the Frankl’ condition on a characteristic and the degeneration segment for an equation of mixed type with a singular coefficient

For the equation (sign y)|y| ^m u _xx +u _yy ?m(2y)^?1 u _y = 0, where m > 0, considered in some mixed domain, we prove existence and uniqueness theorems for the solution of the boundary value problem with an analog of the Frankl’ condition on a characteristic and on the degeneration segment of the equation.     

Criterion for the Solvability of the Weighted Cauchy Problem for an Abstract Euler–Poisson–Darboux Equation

In a Banach space E, we consider the abstract Euler–Poisson–Darboux equation u″(t) + kt^?1u′(t) = Au(t) on the half-line. (Here k ∈ ? is a parameter, and A is a closed linear operator with dense domain on E.) We obtain a necessary and sufficient condition for the solvability of the Cauchy problem u(0) = 0, lim _t→0+t ^k u′(t) = u₁, k < 0, for this equation. The condition is stated in terms of an estimate for the norms of the fractional power of the resolvent of A and its derivatives. We introduce the operator Bessel function with negative index and study its properties.     

Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a square is considered. A Neumann condition is specified on one side of the square, and a Dirichlet condition is set on the other three. It is assumed that the coefficient of the equation, its right-hand side, and the boundary values of the desired solution or its normal derivative on the sides of the square are smooth enough to ensure the required smoothness of the solution in a closed domain outside the neighborhoods of the corner points. No compatibility conditions are assumed to hold at the corner points. Under these assumptions, the desired solution in the entire closed domain is of limited smoothness: it belongs only to the Hölder class C ^μ, where μ ∈ (0, 1) is arbitrary. In the domain, a nonuniform rectangular mesh is introduced that is refined in the boundary domain and depends on a small parameter. The numerical solution to the problem is based on the classical five-point approximation of the equation and a four-point approximation of the Neumann boundary condition. A mesh refinement rule is described under which the approximate solution converges to the exact one uniformly with respect to the small parameter in the L _∞ ^h norm. The convergence rate is O(N ^?2ln² N), where N is the number of mesh nodes in each coordinate direction. The parameter-uniform convergence of difference schemes for mixed problems without compatibility conditions at corner points was not previously analyzed.     

Feynman and quasi-Feynman formulas for evolution equations

New methods for obtaining representations of solutions of the Cauchy problem for linear evolution equations, i.e., equations of the form u _t '(t, x) = Lu(t, x), where the operator L is linear and depends only on the spatial variable x and does not depend on time t, are proposed. A solution of the Cauchy problem, that is, the exponential of the operator tL, is found on the basis of constructions proposed by the author combined with Chernoff’s theorem on strongly continuous operator semigroups.     

Solution of Quasilinear Stochastic Problems in Abstract Colombeau Algebras

The main object of study is the stochastic Cauchy problem for a quasilinear equation with random disturbances in the form of a Hilbert-valued white noise process and with an operator generating an integrated semigroup in the space L²(R). We use the Colombeau theory of multiplication of distributions to introduce an abstract stochastic factor algebra and construct an approximate solution of the problem in this algebra.     

Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length

We consider the weak solution of the Laplace equation in a planar domain with a straight crack, prescribing a homogeneous Neumann condition on the crack and a nonhomogeneous Dirichlet condition on the rest of the boundary. For every k we express the k-th derivative of the energy with respect to the crack length in terms of a finite number of coefficients of the asymptotic expansion of the solution near the crack tip and of a finite number of other parameters, which only depend on the shape of the domain.     

Extinction for a fast diffusion equation with a nonlinear nonlocal source

In this article, the authors establish conditions for the extinction of solutions, in finite time, of the fast diffusion equation \({u_t=\Delta u^m+a\int_\Omega u^p(y,t)\,{d}y,\ 0 < m < 1,}\) in a bounded domain \({\Omega\subset R^N}\) with N > 2. More precisely speaking, it is shown that if p > m, any solution with small initial data vanishes in finite time, and if p < m, the maximal solution is positive in Ω for all t > 0. For the critical case p = m, whether the solutions vanish in finite time or not depends on the value of aμ, where \({\mu=\int_{\Omega}\varphi(x)\,{d}x}\) and \({\varphi}\) is the unique positive solution of the elliptic problem \({-\Delta\varphi(x)=1,\ x\in \Omega; \varphi(x)=0,\ x\in\partial\Omega}\) .     

Finite difference schemes for the singularly perturbed reaction-diffusion equation in the case of spherical symmetry

The boundary value problem for the singularly perturbed reaction-diffusion parabolic equation in a ball in the case of spherical symmetry is considered. The derivatives with respect to the radial variable appearing in the equation are written in divergent form. The third kind boundary condition, which admits the Dirichlet and Neumann conditions, is specified on the boundary of the domain. The Laplace operator in the differential equation involves a perturbation parameter ?², where ? takes arbitrary values in the half-open interval (0, 1]. When ? → 0, the solution of such a problem has a parabolic boundary layer in a neighborhood of the boundary. Using the integro-interpolational method and the condensing grid technique, conservative finite difference schemes on flux grids are constructed that converge ?-uniformly at a rate of O(N ^?2ln² N + N ₀ ^?1 ), where N + 1 and N ₀ + 1 are the numbers of the mesh points in the radial and time variables, respectively.     

On Necessary and Sufficient Conditions for the Existence of a Classical Solution of an Inhomogeneous Cauchy–Riemann System

The question of necessary and sufficient conditions for the existence of a classical solution of inhomogeneous Cauchy–Riemann systems in a domain bounded by a piecewise smooth contour is studied. It is proved that the condition of the uniform stronger continuity of that inhomogeneity is a necessary condition, but it is also a sufficient condition if the inhomogeneity belongs to the L₁ space.     

Algorithm for approximating solutions of Hammerstein integral equations with maximal monotone operators

Let X be a uniformly convex and uniformly smooth real Banach space with dual space X*. Let F: X → X* and K: X* → X be bounded monotone mappings such that the Hammerstein equation u + KFu = 0 has a solution. An explicit iteration sequence is constructed and proved to converge strongly to a solution of this equation.     

Well-posedness of a class of operator-differential equations

We consider a linear first-order ordinary operator-differential equation A(t)u′(t) + B(t)u(t) = f(t) in a Banach space, where the operator A(t) is not invertible in general. Sufficient conditions for the existence, uniqueness, and well-posedness of the Cauchy problem for this equation are obtained.