Solvability of the Dirichlet problem for a mixed-type equation of the second kind

We obtain sufficient conditions for the solvability of the Dirichlet problem for a mixed-type equation of the second kind in a rectangular domain. The solution is represented by a convergent series constructed from the problem data. Some cases of nonuniqueness of the solution are described.     

The Dirichlet problem for a mixed-type equation with characteristic degeneration in a rectangular domain

We study the first boundary-value problem for the following mixed-type equation of the second kind:

Boundary-value problem for a differential-difference mixed-type equation with intersecting degeneration lines

For a mixed-type equation with the Lavrent??ev?CBitsadze operator, nonsmooth degeneration line, and delay in the derivative, an analog of the Tricomi problem is considered in a non-symmetric domain.     

On solvability of Dirichlet problem for second order elliptic equation

The paper gives some solvability conditions of the Dirichlet problem for the second order elliptic equation $$ - div(A(x)\nabla u) + (\bar b(x),\nabla u) - div(\bar c(x)u) + d(x)u = f(x) - divF(x),x \in Q,u|_{\partial Q} = u_0 \in L_2 (\partial Q) $$ in bounded domain Q ? R _n (n ≥ 2) with smooth boundary ?Q ∈ C ₁. In particular, it is proved that if the homogeneous problem has only the trivial solution, then for any u ₀ ∈L ₂(?Q) and f, F from the corresponding functional spaces the solution of the non-homogeneous problem exists, from Gushchin’s space $ C_{n - 1} (\bar Q) $ and the following inequality is true: $$ \begin{gathered} \left\| u \right\|_{C_{n - 1} (\bar Q)}^2 + \mathop \smallint \limits_Q r\left| {\nabla u} \right|^2 dx \leqslant \hfill \\ \leqslant C\left( {\left\| {u_0 } \right\|_{L_2 (\partial Q)}^2 + \mathop \smallint \limits_Q r^3 (1 + |\ln r|)^{3/2} f^2 dx + \mathop \smallint \limits_Q r(1 + |\ln r|)^{3/2} |F|^2 dx} \right) \hfill \\ \end{gathered} $$ where r(x) is the distance from a point x ∈ Q to the boundary ?Q and the constant C does not depend on u ₀, f and F.     

Inverse problem for a parabolic equation with strong power degeneration

We consider the inverse problem of determining the time-dependent coefficient of the leading derivative in a full parabolic equation under the assumption that this coefficient is equal to zero at the initial moment of time. We establish conditions for the existence and uniqueness of a classical solution of the problem under consideration. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1487–1500, November, 2006.     

On the solvability of a nonlocal problem for a hyperbolic equation of the second kind

In the characteristic triangle for a hyperbolic equation of the second kind we study a nonlocal problem, where the boundary value condition contains a linear combination of Riemann–Liouville fractional integro-differentiation operators. We establish variation intervals of orders of fractional integro-differentiation operators, taking into account parameters of the considered equation with which the mentioned problem has either a unique solution or more than one solution.     

The Gellerstedt problem for a parabolic-hyperbolic equation of the second kind

In this paper we consider the Gellerstedt problem for a parabolic-hyperbolic equation of the second kind. We prove the unique solvability of this problem by means of a new representation for a solution to the modified Cauchy problem in a generalized class R.     

The Bitsadze-Samarskii problem with the Frankl condition on the degeneration line for a mixed-type equation with singular coefficient

We consider the Bitsadze-Samarskii problem with the Frankl condition for the Gellerstedt equation with a singular coefficient and prove its well-posedness.     

Initial-boundary value problem for a mixed-type equation with nonsmooth degeneration line and with retarded argument in the derivative

For a mixed-type equation with the Lavrent’ev-Bitsadze operator, with a nonsmooth degeneration line, and with a delay in the derivative, we consider an analog of the Tricomi problem in a nonsymmetric domain.     

A nonlocal inverse problem for a mixed-type equation

We establish a criterion for the unique solvability of the inverse problem for the Lavrent’ev-Bitsadze equation with unknown right-hand side. We construct a solution as the sum of a series over a system of bi-orthogonal root functions of the corresponding adjoint problems on eigenvalues.     

On the Dirichlet problem for a nonlinear elastic beam equation

Using a direct variational approach with no global growth conditions on the nonlinear term, we consider the existence of solutions and their dependence on a functional parameter for the fourth order Dirichlet problem connected with the elastic beam equation. We investigate also the existence of an optimal process for such an optimal control problem in which the dynamics is described by the beam equation.     

Boundary value problem for a mixed-type equation with a difference-differential operator

The Tricomi problem for a mixed-type equation with retarded argument in an unbounded domain is considered. The unique solvability of the problem is proved without restrictions on the delay magnitude. The existence of a solution follows from the solvability of a difference equation. Closed-form integral representations for the solutions are derived.     

Dirichlet problem for an equation of mixed type with two degeneration lines in a rectangular domain

For an equation of elliptic-hyperbolic type with two perpendicular lines of power-law degeneration, we prove a criterion for the uniqueness of the solution of the first boundary value problem by the spectral expansion method. The solution is constructed in the form of the sum of a Fourier-Bessel series.     

An analog of the Frankl problem for a second-kind mixed-type equation

The present work is devoted to the investigation of a Frankl-type problem for the second-kind mixed-type equation. The formulated problem is equivalently reduced to the a nonlocal elliptic problem. The unique solvability is proved with the use of the integral energy method and the theory of singular integral equations.