We consider a mixed problem for a nonlinear ultraparabolic equation that is a nonlinear generalization of the diffusion equation with inertia and the special cases of which are the Fokker-Planck equation and the Kolmogorov equation. Conditions for the existence and uniqueness of a solution of this problem are established.Ultraparabolic equations arise in the investigation of Markov diffusion processes and scattering of electrons, in financial mathematics, etc. (see, e.g., [1][2][3] and references in [4,5]).The Cauchy problems for linear ultraparabolic equations that are generalizations of the diffusion equation with inertia and for equations that describe processes in financial mathematics were considered in [4][5][6][7][8]. For their investigation, the theory of Lie groups and properties of volume potentials were used.Mixed problems for linear and nonlinear ultraparabolic equations in bounded domains were investigated in [9][10][11][12][13][14]. Under certain conditions imposed on the coefficients, conditions for the existence and uniqueness of solutions of these problems were obtained.In the present paper, we consider a problem in a bounded domain for a nonlinear ultraparabolic equation, which, in particular, contains an unknown function of order q ∈ (1, ∞) and its derivatives with respect to the group of space variables raised to power p ∈ (1, 2]. In contrast to [11][12][13], the hyperbolic part of this equation contains the first derivatives with respect to a group of l + 1, l ≥ 1, independent variables. Also note that it was assumed that p ∈ (2; ∞) in [13] and q ∈ (1, 2) and p = 2 in [12]. The equation considered is a nonlinear generalization of the diffusion equation with inertia, and its special cases are the Fokker-Planck equation and the Kolmogorov equation.Let Ω ⊂ R n and D ⊂ R l be bounded domains with boundaries ∂Ω ∈ C 1 and ∂D ∈ C 1 , respectively, and let n, l ∈ N and T ∈ (0, ∞). We introduce the following notation: τ is an arbitrary fixed time from the segment (0, T ], G = Ω × D,is the outward normal to the surface S T , and ∂G is the boundary of the domain G.We consider functions that satisfy the following conditions:(A) a i ∈ L ∞ 0, T ; C(G) , a i (x, y, t) ≥ a 0 for almost all (x, y, t) ∈ Q T and all i ∈ {1, . . . , n}, and a 0 is a positive constant;(P) the numbers p and q are such that q ∈ (1, ∞) and p ∈ (1, 2];(C) c ∈ L ∞ (Q T ), c(x, y, t) ≥ c 0 for almost all (x, y, t) ∈ Q T , and c 0 is a constant;

The problem of determining of the right-hand side function and the time depended minor coefficient in semilinear ultraparabolic equation from the initial, boundary and overdetermination conditions, is considered in this paper. The sufficient conditions of the existence and the uniqueness of solution on some interval [0, T ], where T depends on the coefficients of the equation, for the problem are obtained.

This article is devoted to the solvability of the inverse problem for linear ultraparabolic equation. The problem contains the unknown function in the boundary condition. The existence and the uniqueness of the solution for the mixed problem for linear ultraparabolic equation with the non homogeneous boundary conditions on the space variables are also obtained.

This paper describes the solvability of mixed problem for nonlinear ultraparabolic equation with degeneration. The result is obtained in the cases of the weak and the strong degenerations. The conditions of solvability depend on the relations between the coefficients of the equation and the parameter of degeneration.

Abstract:In the paper the conditions of the existence and uniqueness of the solution for the inverse problem for higher order ultraparabolic equation are obtained. The equation contains two unknown functions of spatial and time variables in its right-hand side. The overdetermination conditions of the integral type are used.