Initial-boundary value problem for an equation of mixed parabolic-hyperbolic type in a rectangular domain

The spectral analysis method is used to establish a uniqueness criterion and prove the existence of a solution of the first initial-boundary value problem for a special equation of mixed type.     

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in an L-shaped domain

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in an L-shaped domain is considered for when the solution has singularities at the corners of the domain. The densification of the Shishkin mesh near the inner corner where different boundary conditions meet is such that the solution obtained by the classical five-point difference scheme converges to the solution of the initial problem in the mesh norm L _?? ^h uniformly with respect to the small parameter with almost second order, i.e., as a smooth solution. Numerical analysis confirms the theoretical result.     

The initial boundary value problem for the nonlinear parabolic equation in unbounded domain

In this paper we consider the mixed problem for the equation u _tt  + A ₁ u + A ₂(u _t ) + g(u _t ) = f(x, t) in unbounded domain, where A ₁ is a linear elliptic operator of the fourth order and A ₂ is a nonlinear elliptic operator of the second order. Under natural assumptions on the equation coefficients and f we proof existence of a solution. This result contains, as a special case, some of known before theorems of existence. Essentially, in difference up to previous results we prove theorems of existence without the additional assumption on behavior of solution at infinity.      

On a problem with shift for mixed type equation with two degeneration lines

For mixed type equation with two perpendicular lines of degeneracy we consider the boundary-value problem with nonlocal condition, connecting with the help of generalized operators of fractional integro-differentiation the trace of the normal derivative of the unknown function on the transition line and its own trace on the control characteristics and the line of degeneracy. The author proves the unique solvability of the problem.     

On the boundary-value problem for a class of equations of mixed type in an unbounded domain

In this paper, we study the boundary-value problem for an equation of mixed type with singular coefficient. The uniqueness of the solution of the problem is proved using the extremum principle and the existence of a solution to the problem is established by the method of integral equations.     

Existence for a characteristic boundary value problem for an equation of mixed type in three-dimensional space

The equation of mixed type With k(x₃) = sign x₃|x₃|^m, m > 0, d?C¹(?), x = (x₁, x₂, x₃), is considered in the threedimensional region G which is bounded by the surfaces: a piecewise smooth surface Γ₀ lying in the half-space x₃ > 0 which intersects the plane x₃ = 0 in the unit circle, and for x₃ < 0 by the characteristic surfaces We prove existence of a generalized solution for the characteristic boundary value problem: Lu = fin G, u_Γ0∪Γ1 = 0. The result is obtained by using a variant of the energy-integral method.     

On a problem with three variants of shift conditions for mixed type equation

For mixed type equation we study a problem where shift conditions are given in inner characteristics, on degeneration line, and on the boundary of elliptic domain. With the help of the maximum principle we prove the uniqueness of a solution to the problem, and with the help of the method of integral equations we prove the existence of a solution to the problem.     

A boundary value problem for a class of mixed-type equations in an unbounded domain

We consider a boundary value problem for an equation of the mixed type with a singular coefficient in an unbounded domain. The uniqueness of the solution of the problem is proved with the use of the extremum principle. In the proof of the existence of a solution of the problem, we use the method of integral equations.     

Nonlocal boundary value problem for an equation of the mixed type with two degeneration lines

For the equation

$$xu_{xx} + yu_{yy} + \alpha u_x + \beta u_y = 0,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} < \alpha ,\beta < 1,$$

(1)

in the domain D bounded by a Jordan curve σ with endpoints A(1, 0) and B(0, 1) and the segment OB(x = 0, 0 ≤ y ≤ 1) for x > 0 and y > 0 and by the characteristics OC: x + y = 0 and √x + √?y = 1 of Eq. (1) for x > 0 and y < 0, we consider a nonlocal boundary value problem with data on the curve σ and the segment OB and with a boundary condition containing a generalized fraction integro-differentiation operator in the characteristic domain of Eq. (1) for x > 0 and y < 0.

We prove the existence of a regular solution of this problem for the case in which the “normal curve” x + y = 1 belongs to the elliptic part of the domain.     

A second-order boundary value problem with impulsive effects on an unbounded domain

We study a second-order nonlinear differential equation on an unbounded domain with solutions subject to impulsive conditions and the Sturm–Liouville type boundary conditions. The existence results are obtained via applications of Krasnosel’ski?’s fixed point theorem for the sum of a completely continuous operator and a contraction.     

Boundary value problem for a third-order equation of mixed type in a rectangular domain

For a third-order equation of the parabolic-hyperbolic type, we suggest a method for studying a boundary value problem by solving the inverse problem for a second-order equation of the mixed parabolic-hyperbolic type with unknown right-hand side depending implicitly on time. We prove a criterion for the uniqueness of the solution of the boundary value problem constructed in the form of the sum of a series in the eigenfunctions of the corresponding one-dimensional Sturm-Liouville problem. We prove the stability of the solution with respect to the boundary data in the norms of the spaces W ₂ ⁿ [0, 1] and $C\left( {\bar D} \right)$ .     

Dirichlet problem for a divergence form elliptic equation with unbounded coefficients in an unbounded domain

We prove existence and uniqueness of the solution of the Dirichlet problem for a class of elliptic equations in divergence form with discontinuous and unbounded coefficients in unbounded domains. Entrata in Redazione il 22 aprile 1999.     

Boundary value problem for a mixed type equation with an advanced-retarded argument

We consider the Tricomi problem for the Lavrent??ev-Bitsadze equation with a mixed deviation of the argument. The uniqueness theorem for the problem is proved under constraints on the deviation of the argument. The existence of a solution is related to the solvability of a difference equation. We obtain integral representations of solutions in closed form.     

Mixed problem for a semilinear ultraparabolic equation in an unbounded domain

We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation {fx1870-01} in an unbounded domain with respect to the variables x. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1661–1673, December, 2007.     

Unilateral estimates for a solution to the first boundary value problem for a strongly dissipative second-order parabolic equation over an unbounded domain

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 35, No. 1, pp. 105–117, January–February, 1994.