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Рассматривается краевая задача для уравнения третьего порядка, не разрешенного относительно старшей производной. Уравнения такого типа, часто называемые уравнениями соболевского типа, встречаются во многих прикладных задачах. С помощью нестационарного метода Галёркина и метода регуляризации доказана теорема существования и единственности регулярного решения краевой задачи. Также получена оценка погрешности метода Галёркина через параметр регуляризации и собственные значения спектральной задачи для оператора Лапласа. We consider a boundary value problem for the third-order equation not solvable with respect to the highest-order derivative. Equations of this type, often called Sobolev type equations, occur in many applied problems. The nonstationary Galerkin method and regularization method are applied to prove the existence and uniqueness theorem for a regular solution of the boundary value problem. Also we obtain an error estimate via regularization parameter and in terms of eigenvalues of the spectral problem for the Laplace operator.

UDC 517.95We prove the existence and uniqueness of a solution in the space W 2,1 2 (Q) to the first boundary value problem for a semilinear parabolic equation in a cylindrical domain Q ⊆ R n+1 . We obtain an error estimate in the W 1,0 2 (Q)-norm in terms of eigenvalues of the selfadjoint spectral problem for a second order elliptic equation. Bibliography: 5 titles.Boundary value problems for parabolic equations with alternating time direction were studied, for example, in [1]-[4] (cf. also the references therein). As is known, the Galerkin method is a universal method for studying nonstationary equations. In particular, error estimates for nonstationary equations solved by the Galerkin method were obtained in [5].In this paper, we establish the existence and uniqueness of a solution to the boundary value problem for a semilinear parabolic equation with alternating time direction. For solving this problem we use the stationary Galerkin method. For basis functions we take solutions to the adjoint spectral problem for a second order elliptic equation. For approximate solutions we obtain an error estimate for the semilinear parabolic equation under consideration.Let Ω be a bounded domain in R n with smooth boundary S, Ω t = Ω × {t}, 0 t T , S T = S × (0, T ). In the cylindrical domain Q = Ω × (0, T ), we consider the semilinear parabolic type equationLu ≡ k(x, t)u t − Δu + c(x, t)u + |u| ρ u = f (x, t).

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