Uniqueness Criterion for the Solution of Boundary-Value Problems for the Abstract Euler–Poisson–Darboux Equation on a Finite Interval

Mathematical Notes - For the abstract Euler–Poisson–Darboux equation, boundary-value problems with Dirichlet and Neumann conditions are considered. The criterion for the uniqueness of...     

Criterion for the solvability of the cauchy problem for an abstract Euler–Poisson–Darboux equation

We consider the Cauchy problem for an abstract Euler–Poisson–Darboux equation in a Banach space and prove a necessary and a sufficient condition for the solvability of this problem. The conditions are stated in terms of an estimate for the norm of a fractional power of the resolvent and its derivatives. The properties of solutions are established, and examples are given.     

A singular Cauchy problem for the Euler–Poisson–Darboux equation

We consider a singular Cauchy problem for the Euler–Poisson–Darboux equation of Fuchsian type in the time variable with ramified Cauchy data. In this paper we establish an expansion of the solutions in a series of hypergeometric functions and then investigate the nature of the singularities of the solutions.     

Abstract Euler–Poisson–Darboux equation with nonlocal condition

We find conditions for the unique solvability of nonlocal problems for abstract differential equation of the Euler–Poisson–Darboux equation. Nonlocal conditions contain either Erdelyi–Kober operator or Riemann–Liouville fractional integration operator.     

Cauchy problems with modified conditions for the Euler–Poisson–Darboux equations in the hyperbolic space

In this note, we give the solutions of the Cauchy problems for the Euler–Poisson–Darboux equations (EPD) with modified conditions in the hyperbolic space with application to the wave equation.     

On the Cauchy Problem for a Generalized Emden–Fowler-Type Equation

Mathematical Notes -     

On the Relation between the Properties of a Degenerate Linear-Quadratic Control Problem and the Euler–Poisson Equation

Computational Mathematics and Mathematical Physics - A quadratic functional with linear constraints in the form of differential equations with identically degenerate matrices multiplying the...     

On Nonuniqueness of Probability Solutions to the Cauchy Problem for the Fokker–Planck–Kolmogorov Equation

Doklady Mathematics - In this paper we give a positive answer to the question about the possibility of existence of several probability solutions to the Fokker–Planck–Kolmogorov...     

The Cauchy Problem for the Fractional Diffusion Equation in a Weighted Hölder Space

Under study is the Cauchy problem for the fractional diffusion equation with a Caputo derivative. The existence and uniqueness theorems for a smooth solution are proven in a weighted H¨older space.     

A Criterion for the Uniqueness of the Solution of the Tricomi Problem for the Multidimensional Lavrent’ev–Bitsadze Equation

Differential Equations - An example of a mixed domain in which the solution of the Tricomi problem is trivial is given. A criterion for the uniqueness of the classical solution of this problem is...     

Global Entropy Solutions of the Cauchy Problem for Nonhomogeneous Relativistic Euler System

We analyze the 2×2 nonhomogeneous relativistic Euler equations for perfect fluids in special relativity. We impose appropriate conditions on the lower order source terms and establish the existence of global entropy solutions of the Cauchy problem under these conditions.     

Collapse Rate of Solutions of the Cauchy Problem for the Nonlinear Schrödinger Equation

We prove that solutions of the Cauchy problem for the nonlinear Schrödinger equation with certain initial data collapse in a finite time, whose exact value we estimate from above. We obtain an estimate from below for the solution collapse rate in certain norms.

     

Solvability of the clamped Euler–Bernoulli beam equation

In this study, solvability of the initial boundary value problem for general form Euler–Bernoulli beam equation which includes also moving point-loads is investigated. The complete proof of an existence and uniqueness properties of the weak solution of the considered equation with Dirichlet type boundary conditions is derived. The method used here is based on Galerkin approximation which is the main tool for the weak solution theory of linear evolution equations as well as in derivation of a priori estimate for the approximate solutions. All steps of the proposed technique are explained in detail.     

Abstract Cauchy problem for the Bessel–Struve equation

We consider the Cauchy problem for the Bessel–Struve equation in a Banach space. A sufficient condition for the solvability of this problem is proved, and the solution operator is written in explicit form via the Bessel and Struve operator functions. A number of properties is established for the solutions.     

Extremum Problem for the Wiener–Hopf Equation

The extremum problem for the Wiener–Hopf equation obtained by replacing the condition u(x) = 0, x < 0, by the condition of the minimum of the quadratic functional of the function u(x)exp(–x), – < x < , is solved in closed form.     

On the Collapse of Solutions of the Cauchy Problem for the Cubic Schrödinger Evolution Equation

It is proved that, for some initial data, the solutions of the Cauchy problem for the cubic Schrödinger evolution equation blow up in finite time whose exact value is estimated from above. In addition, lower bounds for the blow-up rate of the solution in certain norms are obtained.

     

On Conditions for the Applicability of the Lax–Phillips Scattering Scheme to the Investigation of an Abstract Wave Equation

We find necessary and sufficient conditions under which orthogonal incoming and outgoing subspaces exist for a group of solutions of an abstract wave equation and possess an additional property of equivalence with respect to the operator of time reversion.     

Blow-up of Solutions of the Cauchy Problem for a Nonlinear Schrödinger Evolution Equation

The solution of the Cauchy problem for a nonlinear Schrödinger evolution equation with certain initial data is proved to blow up in a finite time, which is estimated from above. Additionally, lower bounds for the blow-up rate are obtained in some norms.