Uniqueness of the solution of a nonlinear problem

Ukrainian Mathematical Journal -     

Uniqueness of a solution to the Cauchy problem for parabolic systems

The Cauchy problem for a second-order Petrovskii parabolic system with bounded and continuous coefficients that are Dini-continuous with respect to space variables is proved to have a unique classical solution in the Tikhonov class.     

Uniqueness of the bounded solution to a strongly degenerate parabolic problem

This paper concerns the uniqueness of the bounded solution to a strongly degenerate parabolic problem. The equation considered may have two kinds of strong degeneracies and there is no restriction on the relation between the two degeneracies. By using Holmgren’s approach, we prove that the bounded solution of the associated initial–boundary value problem is unique under some essentially necessary condition on the growth of the convection.     

Uniqueness of solution to a free boundary problem from combustion

We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function defined in a domain and such that

0\}. \end{displaymath}">

We also assume that the interior boundary of the positivity set, \nobreak 0\}$">, so-called free boundary, is a regular hypersurface on which the following conditions are satisfied:

Here denotes outward unit spatial normal to the free boundary. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of . This problem arises in combustion theory as a limit situation in the propagation of premixed flames (high activation energy limit).

The problem admits classical solutions only for good data and for small times. Several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. We investigate conditions under which the three concepts agree and produce a unique solution.

     

Uniqueness of the solution to an inverse thermoelasticity problem

The inverse problem of coupled thermoelasticity is considered in the static, quasi-static, and dynamic cases. The goal is to recover the thermal stress state inside a body from the displacements and temperature given on a portion of its boundary. The inverse thermoelasticity problem finds applications in structural stability analysis in operational modes, when measurements can generally be conducted only on a surface portion. For a simply connected body consisting of a mechanically and thermally isotropic linear elastic material, uniqueness theorems are proved in all the cases under study.     

Uniqueness of a solution to the problem of evolution of a point vortex

Translated from Sibirskii Matematicheskii, Vol. 35, No. 3, pp. 696–701, May–June, 1994.     

Uniqueness of the least-energy solution for a semilinear Neumann problem

We prove that the least-energy solution of the problem

where is a ball, and if , if , is unique (up to rotation) if is small enough.

     

Uniqueness of a solution to the Cauchy problem for the Hopf equation in the two-dimensional case

We consider the Cauchy problem for the Hopf equation corresponding to the two-dimensional Navier–Stokes equations. The global uniqueness of a solution is proved under the condition that the mean energy of the initial measure is finite, without any additional conditions, i.e., under the same assumptions as in the existence theorem. Thereby we receive a sharp improvement of Foias’s result obtained in the 1970’s about the uniqueness of a spatial statistical solution. Bibliography: 12 titles.     

Uniqueness of the solution of a partial differential equation problem with a non-constant coefficient

We consider the problem , , , where is bounded below by a positive constant. The solution on the boundary is a known function and . This is an ill-posed problem in the sense that a small disturbance on the boundary specification can produce a big change in its solution, if it exists. In a previous work, we used a Wavelet Galerkin Method with the Meyer Multiresolution Analysis to generate a sequence of well-posed approximating problems to it. In the present work, by assuming that is Lipschitz, we are able to prove that the existence of a solution , for this problem, implies its uniqueness.

     

Uniqueness of the solution of Christoffel's problem for nonclosed surfaces

Conditions satisfied by a region of a sphere are found under which the differential equation arising in the solution of Christoffel's problem has not more than one solution in the region taking given boundary values.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 251–257, August, 1970.