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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