ISSN 1064–5624, Doklady Mathematics, 2006, Vol 74, No 3, pp 874–877 © Pleiades Publishing, Inc., 2006 Original Russian Text © Yu.V Egorov, Nguyen Minh Chuong, Dang Anh Tuan, 2006, published in Doklady Akademii Nauk, 2006, Vol 411, No 6, pp 732–735 MATHEMATICS Nonclassical Semilinear Boundary Value Problem for Parabolic Pseudodifferential Equations in Sobolev Spaces Yu V Egorova, Nguyen Minh Chuongb, and Dang Anh Tuanc Presented by Academician V.S Vladimirov March 23, 2006 Received March 24, 2006 DOI: 10.1134/S1064562406060226 We consider a nonlinear boundary value problem for parabolic pseudodifferential equations of an arbitrary order By applying the Laplace transform, the problem is reduced to a boundary value problem for an elliptic equation The existence of a solution is proved by using the Schauder theorem A nonclassical nonlinear boundary value problem for elliptic pseudodifferential equations in the Sobolev spaces Hl, p, < p < ∞ was considered in [5] In this paper, we use the Laplace transform to study a similar problem for parabolic pseudodifferential equations in Sobolev spaces The proof is substantially simplified by applying the Schauder theorem (instead of the Leray– Schauder one) Suppose that q ∈ ރand Re q > Let ∞ as the completion of P(ޒn) = {u ∈ C (ޒn × (–∞, +∞)): supp u ⊂ ޒn × (0, +∞)} in the norm u ⎛ 1/λ lp = ⎜ ( + ξ + µ + iτ ) ⎜ ⎝ ޒnξ,+τ ∫ × Ᏺn + [ e n u l, p, q, ޒ where ∑x ξ , i i ∫ U e u ( x, t ) dt, – qt n ∫ Ᏺ n + [ u ( x, t ) ] ( ξ, τ ) – ( n + )/2 ∫ ޒ e – i 〈 x, ξ〉 – itτ n E l, p ( λ, µ, ) ޒ ⎛ = ⎜ ⎝ ∫ ޒτ ⎞ U l, p, q1/λ, ޒn dτ⎟ ⎠ , Using an extension of functions from ޒ+ and Ω to ޒn, where Ω is a bounded domain in ޒn, we define the n u ( x, t ) d x dt spaces Pl, p(, à, + ì (0, +)), Pl, p(, à, ì (0, +)), n n+1 x, t Hl, p( ޒ+ ), Hl, p, q(Ω), El, p(λ, µ, ޒ+ ), and El, p(λ, µ, Ω) n We use the following spaces Let l ≥ 0; < p < +∞; and < λ, µ The space Pl, p(, à, n ì (0, +)) is defined n Remark (i) If U(x, q) ∈ El, p(λ, µ) and Req = µ, then U(x, q) ∈ H l, p, q1/λ for almost all q (ii) If u ∈ Pl, p(λ, µ), then ∂ -uk ∂t k ≤ l k a 1/ p p where q = µ + iτ = ( 2π ) El, p(λ, µ, ޒn) is the completion of ᏸP(ޒn) in the norm ᏸ [ u ( x, t ) ] ( x, q ) +∞ – 1/2 1/ p ⎛ ⎞ 1/ p p lp = ⎜ ( + ξ + q ) Ᏺ n u ( ξ, q ) dξ⎟ ; ⎜ ⎟ ⎝ ޒnξ ⎠ i=1 = ( 2π ) ⎞ u ( x, t ) ] ( ξ, τ ) dξ dτ ⎟ ⎠ p ∞ ޒx n – µt Hl, p, q(ޒn) is the completion of C (ޒn) in the norm Ᏺ n U ( ξ, q ) = ( 2π ) –n/2 ∫ e –i 〈 x, ξ〉 U ( x, q ) dx, 〈 x, ξ〉 = n P l, p ( , à, ( ì 0, + ) ) Universitộ Paul Sabatier, Toulouse, France b Institute of Mathematics, Hanoi, Vietnam c Hanoi National University, Hanoi, Vietnam 874 = for λ ≥ and t=0 NONCLASSICAL SEMILINEAR BOUNDARY VALUE PROBLEM Theorem The Laplace transform induces an isometric isomorphism of Pl, p(λ, µ) and El, p(λ, µ) vector of ∂Ω} If Γ0 is attracting, then Ᏹl, p(λ, µ, Ω) denotes the space { U ∈ E l, p ( λ, µ, Ω ) D ν ( hU ) ∈ E l, p ( λ, µ, Ω ), To prove this theorem, it is sufficient to note that ( hU ) Ᏺ n + [ e –µt u(x, t) ](ξ, τ) = Ᏺ n [ ᏸ [ u(x, t) ](x, q) ] ( ξ, τ ), µ = Req ≥ Assume that a field ν touches ∂Ω only at points of a manifold Γ0 of dimension n – on ∂Ω but does not touch Γ0 The set of such manifolds Γ0 is naturally divided into three classes: attracting, repelling, and neutral (the first, second, or third classes in the nomenclature of [6]) In what follows, we assume that all submanifolds Γ0 are either attracting or neutral Consider the following elliptic problem with a parameter q ∈ ރ: Ꮽ(x, D x, q 1/λ)U(x, q) = A(x, D x, q)U(x, q) = F(x, q), (1) x ∈ Ω, Ꮾ j ( x, D x, q 1/λ ) ( D ν U ( x, q ) ) = B j ( x, D x, q ) ( D ν U ( x, q ) ) = G j ( x, q ), (2) x ∈ ∂Ω, π where j = 1, 2, …, s; |argq| ≤ - , and Ꮽ(x, Dx, q1/λ), and Ꮾj(x, Dx, q1/λ) are defined as in [5, p 452] Specifically, in a local coordinate system, Ꮽ(x, Dx, q1/λ) is a pseudodifferential operator with the symbol 1/λ ) = ∑ β α q g αβ ( x, ξ )ξ : U Ᏹ l, p ( λ, µ, Ω ) = ( 2π ) ∫e ޒ σ Ꮽ ( x, ξ, q k ) Ᏺ n u ( ξ, q ) dξ x ∈ Γ0 , k = 0, 1, …, s – ⎧ set ⎨ (F, Gj) l 2s, p(, à, ) ì ⎩ (5) s ∏E l – m j – + 1/ p, p (λ, j=1 ⎫ µ, ∂Ω) such that hGj ∈ E l – m j – + 1/ p, p (λ, µ, ∂Ω) ⎬ ⎭ Theorem Let (Ꮽ(x, Dx, q1/λ), Ꮾj(x, Dx, q1/λ)|∂Ω) π be an elliptic operator for | arg q | ≤ - , l ≥ l1 = max{2s, mj + 2}, and < p < +∞ Then, for sufficiently large µ = Req, problem (1), (2) (or problem (1), (2), (3) when Γ0 is attracting) has a unique solution U(x, q) ∈ Ᏹl, p(λ, µ, Ω) if (F, Gj) ∈ Ᏹl, p(λ, µ, Ω, ∂Ω) When Γ0 is attracting, we have to add the condition U0k ∈ El – k – + 1/p, p(λ, µ, Γ0) Proof This result is proved using Theorem 5.1 in [5] Consider the following problem in Ω × (0, +∞): ∂ A ⎛ x, D x, -⎞ u ( x, t ) = f ( x, t ), ⎝ ∂t⎠ x ∈ Ω, t > 0, (6) x ∈ ∂Ω, (7) j = 1, 2, …, s k x ∈ Γ0 , t > 0, k = 0, 1, …, s – (8) Using the initial conditions ∂u -k∂t k (3) ∞ of Γ0, and ᏺ = ᏺd = {x ∈ Γd| ∃y ∈ Γ0, xy is a normal No E l, p ( λ, µ, Ω ) ᏺ E l, p ( λ, µ, ᏺ ) D n u ( x, t ) = u 0k ( x, t ), Below, a function h(x) ∈ C (Ω) is used such that d h(x) = in the - -neighborhood Γd/2 of Γ0 (d > is suf2 ficiently small), h(x) = outside the d-neighborhood Γd Vol 74 + D ν ( hU ) If Γ0 is attracting, then we add the conditions If Γ0 is attracting, then we add the conditions DOKLADY MATHEMATICS (4) If Γ0 is neutral, then the terms with (hU)|ᏺ are not necessary in (4) and (5) Denote by Ᏹl, p(λ, µ, Ω, ∂Ω) the n ξ D n U ( x, q ) = U 0k ( x, q ), E l, p ( λ, µ, Ω ) t > 0, Ꮽ ( x, D x, q )u ( x, λ ) 1/λ = U + ( hU ) 1/λ i 〈 x , ξ〉 ∈ E l, p ( λ, µ, ᏺ ) } ∂ B j ⎛ x, D x, -⎞ ( D ν u ( x, t ) ) = g j ( x, t ), ⎝ ∂t⎠ α + λβ ≤ 2s – n/2 ᏺ with the norm Nonclassical linear problem Let Ω be a bounded domain with a smooth boundary ∂Ω in ޒn σ Ꮽ ( x, ξ, q 875 2006 = 0, t=0 l k = 0, 1, …, -0 , λ (9) l = max { 2s, m j + } and formally applying the Laplace transform to (6) and (7), we obtain (1) and (2) The spaces ᏼl, p(λ, µ, Ω × (0, +∞), ∂Ω × (0, +∞)) and ᏼl, p(λ, µ, Ω × (0, +∞)) are defined in a similar manner to Ᏹl, p(λ, µ, Ω, ∂Ω) and Ᏹl, p(λ, µ, Ω) 876 EGOROV et al Applying Theorem 1, we obtain the following result Theorem Let l ≥ l1 and < p < +∞ Then the Laplace transform induces an isometric isomorphism between the following pairs of spaces: l, p ( , à, ì ( 0, +∞ ) ) Ë Ᏹ l, p ( λ, µ, Ω ), - Ꮾ j ⎛⎝ x, D x, q λ⎞⎠ ( D ν U ( x, q ) ) = B j ( x , D x, q ) ( D ν U ( x , q ) ) –1 D n U ( x, q ) = U 0k ( x, q ), k ᏼ l, p ( , à, ì ( 0, + ), ∂Ω × ( 0, +∞ ) ) and 2s – j à, ì ( 0, + ) ) Ë Ᏹ l, p ( λ, µ, Ω, ∂Ω ) × ∏ E l – k – + 1/ p, p ( λ, µ, Γ ) k=0 Using Theorems 2, 3, and Remark 1, we obtain the following assertion Ꮾj(x, be π an elliptic operator for | arg q | ≤ - , l ≥ l1, and < p < +∞ q1/λ), Dx, q1/λ)|∂Ω) Then, for sufficiently large µ = Req, problem (6), (7) (or problem (6), (7), (8) when Γ0 is attracting) has a unique solution u l, p(, à, ì (0, +∞)) for any (f, gj) from ᏼl, p(λ, µ, Ω × (0, +∞), ∂Ω × (0, +∞)) (if Γ0 is attracting, then u0k ∈ Pl – k – + 1/p, p(, à, ì (0, +)) Nonclassical semilinear problem Consider the problem ∂ A ⎛ x, D x, -⎞ u ( x, t ) = f ( x, t, u ( x, t ) ), ⎝ ∂t⎠ x ∈ Ω, are continuous with respect to u = (u1, u2, …, uN) and j u = (u1, u2, …, u N j ) for almost all (x, t) and are meaj surable with respect to (x, t) for every u and u (ii) The mapping u(x, t) ( ۋf (x, t, u (x, t)), gi(x, t, j u (x, t))) (from ᏼl, p(λ, µ, ì (0, +)) to l, p(, à, ì (0, +∞), ∂Ω × (0, +∞))) maps each bounded set into a relatively compact one (iii) > lim inf lim sup ||(A, Bj)–1||||( f , gj))||M, µ, M → +∞ (11) j m –1 (u(x, t), …, D x j u(x, t)) Assume that conditions (8) and (9) are satisfied Applying the Laplace transform for Req > 0, we can reduce problem (10), (11) (or (10), (11), (8) for an attracting Γ0) to the problem - = ᏸ [ f ( x, t, ᏸ U ( x, t ) ) ] ( x, q ), µ → +∞ where M, µ ⎧1 = sup ⎨ - ( f ( x, t, u ( x, t ) ), ⎩M ᏼ l, p ( λ, µ, Ω × ( 0, +∞ ), ∂Ω × ( 0, +∞ ) ) for all u such that ||u || ᏼl, p ( , à, ì ( 0, + ) ) ≤ M} u(x, t)), and u (x, t) = Ꮽ ⎛⎝ x, D x, q λ⎞⎠ U ( x, q ) = A ( x, D x, q )U ( x, q ) j to )ޒhave Carathéodory property; i.e., they j where A and Bj are the same operators as in (6) and (7), –1 Nj (10) j = 1, 2, …, s, 2s – ޒ+ × ޒ g j ( x, t , u ( x, t ) ) ) j ∂ B j ⎛ x, D x, -⎞ ( D ν u ( x, t ) ) = g j ( x, t, u ( x, t ) ), ⎝ ⎠ ∂t u (x, t) = (u(x, t), …, D x j ޒ+ × ޒN to )ޒand (x, t, u ) ۋgj(x, t, u ) (from ∂Ω × ( f , g j) t > 0, t > 0, U(x, t)) (i) The mappings (x, t, u ) ۋf (x, t, u ) (from Ω × s–1 x ∈ ∂Ω, U(x, t)) Assume that the following conditions hold: k=0 Theorem Let (Ꮽ(x, Dx, m –1 and ᏸ–1 U (x, t) = (ᏸ–1U(x, t), …, ᏸ–1 D x j s–1 l – k – + 1/ p, p ( λ, (14) where ᏸ–1 U (x, t) = (ᏸ–1U(x, t), …, ᏸ–1 D x ᏼ l, p ( λ, µ, Ω × ( 0, +∞ ), ∂Ω × ( 0, +∞ ) ) ∏P x ∈ Γ0 , k = 0, 1, …, s – 1, Ᏹ l, p ( , à, , ), ì j = [ gj( x, t, ᏸ U ( x, t))]( x, q), x ∈ ∂Ω, j = 1, 2, …, s, (13) (12) To prove Theorem 5, we need the following assertion Lemma The mapping U ( x, q ) ( ۋᏸ [ f ( x, t, ᏸ U ( x, t ) ) ] ( x, q ), –1 j ᏸ [ g j ( x, t , ᏸ U ( x, t ) ) ] ( x, q ) ) –1 is a compact mapping from Ᏹl, p(λ, µ, Ω) to Ᏹl, p(λ, µ, Ω, ∂Ω) Theorem Assume that the operator (Ꮽ(x, Dx, π q1/λ), Ꮾj(x, Dx, q1/λ)|∂Ω) is elliptic for | arg q | ≤ - Let assumptions (i), (ii), and (iii) be satisfied Let l ≥ l1, < p < +∞, and U0k ∈ El – k – + 1/p, p(λ, µ, Γ0) DOKLADY MATHEMATICS Vol 74 No 2006 NONCLASSICAL SEMILINEAR BOUNDARY VALUE PROBLEM Then, for sufficiently large µ = Req, problem (12), (13) (or problem (12)–(14) when Γ0 is attracting) has a solution U(x, q) ∈ Ᏹl, p(λ, µ, Ω) Proof sketch We prove Theorem for an attracting manifold Γ0 For every W ∈ Ᏹl, p(λ, µ, Ω), we obtain ( ᏸ [ f ( x, t, ᏸ W ( x, t ) ) ] ( x, q ), 877 The following result is proved by applying Remark and Theorems and Theorem Under the assumptions of Theorem 5, if Γ0 is neutral (or attracting), then, for sufficiently large µ, problem (10), (11) (or problem (10), (11), (8) with additional data u0k ∈ Pl k + 1/p, p(, à, ì (0, +∞))) has a solution u ∈ ᏼl, p(λ, µ, Ω × (0, +∞)) –1 REFERENCES j ᏸ [ g j ( x, t, ᏸ W ( x, t ) ) ] ( x, q ) ) ∈ Ᏹ l, p ( λ, µ, Ω, ∂Ω ) –1 Therefore, for sufficiently large µ = Req, problem (1)– (3) with the right-hand sides (ᏸf, ᏸgj, U0k) has a unique solution U ∈ Ᏹl, p(λ, µ, Ω) By Lemma 1, the mapping W ۋU is compact in Ᏹl, p(λ, µ, Ω) Using (iii), we obtain R > such that this mapping is compact in the closed ball BR = {U ∈ Ᏹl, p(λ, µ, Ω)| ||U || Ᏹl, p ( λ, µ, Ω ) ≤ R} for sufficiently large µ Therefore, by the Schauder theorem [3, p 60], this mapping has a fixed point in BR, which solves problem (12), (13), (4) DOKLADY MATHEMATICS Vol 74 No 2006 Nguyen Minh Chuong, Dokl Akad Nauk SSSR 258, 1308–1312 (1981) Nguyen Minh Chuong, Dokl Akad Nauk SSSR 268, 1055–1058 (1983) K Deimling, Nonlinear Functional Analysis (SpringerVerlag, Berlin, 1985) Yu V Egorov and Nguyen Minh Chuong, Usp Mat Nauk 53, 249–250 (1998) Yu V Egorov, Nguyen Minh Chuong, and Dang Anh Tuan, S.R Acad Sci Ser 337, 451–456 (2003) Yu V Egorov and V A Kondrat’ev, Mat Sb 78 (120), 148–176 (1969) ... Γ0) DOKLADY MATHEMATICS Vol 74 No 2006 NONCLASSICAL SEMILINEAR BOUNDARY VALUE PROBLEM Then, for sufficiently large µ = Req, problem (12), (13) (or problem (12)–(14) when Γ0 is attracting) has a... manifolds Γ0 is naturally divided into three classes: attracting, repelling, and neutral (the first, second, or third classes in the nomenclature of [6]) In what follows, we assume that all submanifolds.. .NONCLASSICAL SEMILINEAR BOUNDARY VALUE PROBLEM Theorem The Laplace transform induces an isometric isomorphism of Pl, p(λ, µ) and El, p(λ, µ) vector of ∂Ω} If Γ0 is attracting, then