Singular solutions with exponential growth to Protter’s problems

Four-dimensional boundary value problems which were formulated by Proter for the nonhomogeneous wave equation are studied. They can be considered as multidimensional versions of the Darboux problems in ?². Protter’s problem is not well posed in the frame of classical solvability. On the other hand, it is known that the unique generalized solution may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. Some known results suggest that the solution may have at most exponential growth. We construct an infinitely smooth right-hand side function such that the corresponding generalized solution to Protter’s problem has an exponential singularity.     

The semilinear second-order hyperbolic equation with data strongly singular at one point

In this paper we study the initial value problem for the scalar semilinear strictly hyperbolic equation in multidimensional space with data strongly singular at one point. Under the assumption of the initial data being conormal with respect to one point and bounded or regular with a certain low degree, the existence of the solution to this problem is obtained; meanwhile, it is proved that the singularity of the solution will spread on the forward characteristic cone of the hyperbolic operator issuing from this point, and the solution is bounded and conormal with respect to this cone.     

Solution of a mixed problem with oblique derivatives in the boundary conditions for the inhomogeneous string vibration equation without continuing the data

We find closed-form recursion formulas for the unique classical solution of a mixed problem describing forced vibrations of a bounded string under two boundary modes with timedependent oblique derivatives. The formulas do not use any continuation of the problem data. We obtain conditions on the right-hand side of the equation necessary and sufficient for the well-posed global solvability of the problem.     

Boundary value problems with operator right-hand side

For a second-order boundary value problem with operator right-hand side and with functional boundary conditions, we prove solvability theorems with mixed and Dirichlet boundary conditions assuming the existence of a lower and an upper function. These theorems are analogs of theorems for the corresponding boundary value problems for an ordinary second-order differential equation with right-hand side satisfying the Carathéodory conditions.     

“Destruction” of the solution of a strongly nonlinear equation of pseudoparabolic type with double nonlinearity

We consider the first initial-boundary value problem for multidimensional strongly nonlinear equations with double nonlinearity of pseudoparabolic type in a bounded domain with sufficiently smooth boundary. We prove the local solvability of this problem in the weak generalized sense. Depending on the nonlinearity and initial conditions under consideration, we prove the solvability of the equation in any finite cylinder (x, t) ∈ Ω × [0, T] or the destruction of the solution in finite time.     

Bitsadze–Samarskii problem for a parabolic system on the plane

The solvability (in classical sense) of the Bitsadze–Samarskii nonlocal initial–boundary value problem for a one-dimensional (in x) second-order parabolic system in a semibounded domain with a nonsmooth lateral boundary is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the nonlocal boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

On the solvability of the dirichlet problem for elliptic nondivergent equations of the second order in a domain with conical point

We study the problem of solvability of the Dirichlet problem for second-order linear and quasilinear uniformly elliptic equations in a bounded domain whose boundary contains a conical point. We prove new theorems on the unique solvability of a linear problem under minimal smoothness conditions for the coefficients, right-hand sides, and the boundary of the domain. We find classes of solvability of the problem for quasilinear equations under natural conditions.     

Dirichlet problem for parabolic systems with Dini continuous coefficients on the plane

The Dirichlet problem for a one-dimensional (with respect to x) second-order parabolic system with Dini continuous coefficients is considered in an x-semibounded domain with a nonsmooth lateral boundary from the Dini–Hölder class. The classical solvability of the problem is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

On the construction of solutions of terminal problems for multidimensional affine systems in quasicanonical form

We consider the construction of solutions of terminal problems for multidimensional affine systems. We show that the terminal problem for a regular system in quasicanonical form can be reduced to a boundary value problem for a system of ordinary differential equations of lower order with right-hand side depending on a vector parameter. We prove a sufficient condition for the existence of a solution of the above-mentioned boundary value problem. A method for constructing a numerical solution is developed.     

Well-posedness of the Poincar�� problem for a multidimensional hyperbolic equation with the Chaplygin operator

In this paper, the unique solvability of the Poincaré problem for multidimensional hyperbolic equation with the Chaplygin operator in the domain with a deviation from the characteristic is proved. In the theory of partial differential equations of the hyperbolic type, boundary-value problems with the data on the whole boundary of the domain serve as an example of problems that are not well posed [3].     

Factorized equation of vibrations of a finite string with nonstationary second directional derivatives at the endpoints

We obtain closed-form recursion formulas for the classical solutions of a mixed problem for the general inhomogeneous factorized equation of vibrations of a bounded string with second directional derivatives in the boundary conditions, in which the coefficients multi-plying the first of the two directional derivatives are independent of time. We study the case of boundary conditions in which all first directional derivatives are not directed along the characteristics of the equation. We obtain necessary and sufficient conditions on the right-hand side and the initial and boundary data of the problem for its well-posed global solvability in the set of classical solutions.     

A numerical method for solving one nonlocal boundary value problem for a third-order hyperbolic equation

A nonlocal boundary value problem for a third-order hyperbolic equation with variable coefficients is considered in the one- and multidimensional cases. A priori estimates for the nonlocal problem are obtained in the differential and difference formulations. The estimates imply the stability of the solution with respect to the initial data and the right-hand side on a layer and the convergence of the difference solution to the solution of the differential problem.     

Boundary Value Problems for Degenerate Hyperbolic Systems of First Order Equations

In this paper we discuss some boundary value problems for degenerate hyperbolic complex equations of first order in a simply connected domain, in which the boundary value problems include the Riemann-Hilbert problem and the Cauchy problem. We first give the representation of solutions of the boundary value problems for the equations, and then prove the uniqueness and existence of solutions for the problems. In [A.V. Bitsadze (1988). Some Classes of Partial Differential Equations . Gordon and Breach, New York; A.V. Bitsadze and A.N. Nakhushev (1972). Theory of degenerating hyperbolic equations. Dokl. Akad. Nauk, SSSR , 204 , 1289-1291 (Russian); M.H. Protter (1954). The Cauchy problem for a hyperbolic second order equation. Can. J. Math ., 6 , 542-553], the authors discussed some boundary value problems for hyperbolic equations of second order.     

Inverse problems for elliptic equations in the space: I

We consider inverse problems of finding the right-hand side of a linear secondorder elliptic equation of general form. We study the first boundary value problem. Two ways of representation of the additional information (over-determination) are considered: specification of the trace of the solution of the boundary value problem inside the domain on some manifold of smaller dimension and specification of values of the normal derivative on a part of the boundary. The Fredholm alternative is proved for the considered problems. Stability estimates are given for the case of unique solvability. The analysis is performed in classes of continuous functions whose derivatives satisfy the Hölder condition.     

To the theory of boundary-value problems for elliptic equations with superposition operators in the boundary condition. I

We study the existence, uniqueness, and constant sign property of classical solutions to a nonlocal boundary-value problem for a second-order elliptic equation in a bounded domain of the Euclidean space. Using the system of maps that define superposition operators, we construct some subset of the domain boundary and establish the connection between the solvability of the problem under consideration and the solvability of the boundary value equation on the constructed subset.     

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

     

A parametrization method for solving nonlinear two-point boundary value problems

A sharper version of the local Hadamard theorem on the solvability of nonlinear equations is proved. Additional parameters are introduced, and a two-parameter family of algorithms for solving nonlinear two-point boundary value problems is proposed. Conditions for the convergence of these algorithms are given in terms of the initial data. Using the right-hand side of the system of differential equations and the boundary conditions, equations are constructed from which initial approximations to the unknown parameters can be found. A criterion is established for the existence of an isolated solution to a nonlinear two-point boundary value problem. This solution is shown to be a continuous function of the data specifying the problem.     

Variational Ultraparabolic Inequalities

In a bounded domain of the space ℝ ^{n +2}, we consider variational ultraparabolic inequalities with initial condition. We establish conditions for the existence and uniqueness of a solution of this problem. As a special case, we establish the solvability of mixed problems for some classes of nonlinear ultraparabolic equations with nonclassical and classical boundary conditions.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1616–1628, December, 2004.     

Criterion for the Uniqueness of a Solution of the Darboux–Protter Problem for Degenerate Multidimensional Hyperbolic Equations

We obtain a criterion for the uniqueness of a regular solution of the Darboux–Protter problem for degenerate multidimensional hyperbolic equations.     

Generalized Harmonic Functions and the Dewetting of Thin Films

This paper describes the solvability of Dirichlet problems for Laplace's equation when the boundary data is not smooth enough for the existence of a weak solution in H¹Ω. Scales of spaces of harmonic functions and of boundary traces are defined and the solutions are characterized as limits of classical harmonic functions in special norms. The generalized harmonic functions, and their norms, are defined using series expansions involving harmonic Steklov eigenfunctions on the domain. It is shown that the usual trace operator has a continuous extension to an isometric isomorphism of specific spaces. This provides a characterization of the generalized solutions of harmonic Dirichlet problems. Numerical simulations of a model problem are described. This problem is related to the dewetting of thin films and the associated phenomenology is described.