482 ISSN 1064–5624, Doklady Mathematics, 2009, Vol. 80, No. 1, pp. 482–486. © Pleiades Publishing, Ltd., 2009. Published in Russian in Doklady Akademii Nauk, 2009, Vol. 427, No. 2, pp. 155–159. In this article a semilinear boundary value problem is studied for a degenerate parabolic pseudodifferential equation. The main result generalizes the famous theo- rem of Agranovich and Vishik (see [1]). We prove the existence of a solution using the Rothe theorem on a fixed point (see [2]). 1. Let p ∈ (1, ∞), l ∈  + , q = μ + iτ ∈ , μ ≥ 0, d ≥ 1, d ∈ , δ = (δ 1 , δ 2 , …, δ n – 1 , 0), δ i ≥ 0, i = 1, 2, …, n – 1. For each ξ' = (ξ 1 , ξ 2 , …, ξ n – 1 ) ∈  n – 1 , x n ∈  + , set Define the Laplace transform ξ' q , ( ) δ d , ξ i p /1 δ i + ( ) x n p δ i i 1 = n 1 – ∑ q p / d + ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ 1/ p , = ξ' x n q , , ( ) δ d , ξ i p x n p δ i i 1 = n 1 – ∑ q p / d + ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ 1/ p , =  n 1 – u ξ' x n q , , ( ) = 1 2 π ( ) n 1 – ( ) /2 ------------------------- e i x' ξ' , 〈 〉 – ux' x n q , , ( ) x', d  n 1 – ∫  n 1 – v ξ' q , ( ) = 1 2 π ( ) n 1 – ( ) /2 ------------------------- e i x' ξ' , 〈 〉 – v x' q , ( ) x'. d  n 1 – ∫ The space H (l, p, δ) ( n – 1 ) is defined as the comple- tion of the space ( n – 1 ) with the norm The space H (l, p, δ) ( ) (l ∈  + ) is defined as the completion of the space P( ) = {u ∈ ( n ): suppu ∈ } using the norm The space P l, p (d, μ, δ, × (0, +∞)) is defined as the completion of the space P( n ) = {u ∈ ( n × (–∞, +∞)): suppu ⊂ × (0, +∞)} with the norm uxt , ( ) 1 2 π ( ) 1/2 ---------------- e qt – uxt , ( ) t . d 0 +∞ ∫ = C 0 ∞ v lpd δ  n 1 – , , , , = 1 ξ' q , ( ) δ d , + ( ) lp  n 1 – v ξ' q , ( ) p ξ' d  n 1 – ∫ ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ 1/ p .  + n  + n C 0 ∞  + n u lpd δ  + n , , , , = 1 ξ' q , ( ) δ d , ξ' x n q , , ( ) δ d , + + ( ) l j – ( ) p 0 +∞ ∫  n 1 – ∫ j 0 = l ∑ ⎝ ⎜ ⎛ ∫ × D n j  n 1 – u ξ' x n q , , ( ) p x n ξ' d d ⎠ ⎟ ⎞ 1 p -- .  + n C 0 ∞  + n u P lp , d μδ  + n , , , 0+∞ , ( ) × ( ) = 1 ξ' q , ( ) δ d , ξ' x n q , , ( ) δ d , + + ( ) pl j – ( ) 0 +∞ ∫  ξ' τ , n ∫ j 0 = l ∑ ⎝ ⎜ ⎜ ⎛ ∫ × D n j  n ue μt – uxt , ( ) ( )ξ' x n τ , , ( ) p x n ξ' τ d d d ⎠ ⎟ ⎞ 1 p -- . MATHEMATICS On a Semilinear Boundary Value Problem for Degenerate Parabolic Pseudodifferential Equations 1 Yu. V. Egorov a , Nguyen Minh Chuong b , and Dang Anh Tuan c Presented by Academician V.S. Vladimirov February 18, 2009 Received March 6, 2009 DOI: 10.1134/S1064562409040085 a Université Paul Sabatier, Toulouse, France b Institute of Mathematics, Hanoi, Vietnam c Hanoi National University, Hanoi, Vietnam 1 The article was translated by the authors.