Dirichlet-Cauchy problem for second-order parabolic equations in domains with edges

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2011, Vol. 56, No. 3, pp. 13-24 DIRICHLET-CAUCHY PROBLEM FOR SECOND-ORDER PARABOLIC EQUATIONS IN DOMAINS WITH EDGES Do Van Loi Hong Duc University E-mail: 37loilinh@gmail.com Abstract. In this paper, we study the first initial boundary value prob- lem for second-order parabolic equations in cylinders with piecewise smooth base. Some results on the unique existence and on the smoothness with respect to time of the solution are given. Keywords: Domains with edges, generalized solution, regularity. 1. Introduction Initial boundary value problems for non-stationary systems and equations in cylinders with non-smooth bases have been investigated in [2-8], where some impor- tant results on the unique existence, smoothness and asymptotic of the solution for the problems in Sobolev spaces were given. The initial boundary value problems for parabolic equations in a cylinder with base containing conical points [5] or in a dihe- dral angle [8] have been studied in Sobolev spaces with weights. In the present paper, we consider the first initial boundary value problem for second-order parabolic equa- tions in domains with edges. We study the existence, uniqueness and smoothness with respect to time of the generalized solution for this particular problem. The paper is organized as follows. In the second section we define Dirichlet- Cauchy problem for second-order parabolic equations in domains with edges. In the 3rd section we study the solvability of the problem. The last section is intended to regularity with respect to time of the Generani Zaisedzed solution. 2. Formulation of the problem Let Ω be a bounded domain in Rn, n ≥ 2. Its boundary ∂Ω is a piecewise smooth surface consisting of finitely many (n− 1)-dimensional smooth surfaces Γi. We assume that the surface Γi intersects only the surfaces Γi−1, Γi+1 along smooth (n − 2)-Dimensional manifolds li−1, li+1. Without loss of generality we shall deal 13 Do Van Loi explicitly with the case when ∂Ω consists two surfaces Γ1, Γ2 which intersects along a manifold l0. Assume that in a neighbourhood of each point of l0 the set Ω is diffeo- morphic to a dihedral angle. For any point P ∈ l0 two half-spaces T1(P ), T2(P ) tangent to Ω and a two-dimensional plane π(P ) normal to l0 are defined. We denote by ν(P ) the angle in the plane π(P ) (on the side of Ω) bounded by the rays T1(P )∩ π(P ), T2(P ) ∩ π(P ) and by β(P ) the aperture of this angle. Denote QT = Ω × (0, T ), ST = ∂Ω × (0, T ) for each T : 0 < T ≤ ∞. For each multi-index p = (p1, . . . , pn) ∈ N n, |p| = p1 + · · · + pn, the symbol Dpu = ∂|p|u/∂xp11 ...∂x pn n denotes the generalized derivative of order α with respect to x = (x1, ..., xn); utk∂ ku/∂tk is the generalized derivative of order k with respect to t. Throughout the paper we need the following functional spaces (see [1]): H l(Ω) is the space consisting of all functions u(x) defined on Ω such that ‖u‖Hm(Ω) =  ∑ |p|≤m ∫ Ω |Dpu|2dx   1 2 < +∞. H˚ l(Ω) is the completion of o C∞(Ω) in the norm of the space H l(Ω). Let γ be a positive number. We denote the following spaces: H l,k(QT , γ) is the space consisting of all functions u(x, t) defined on QT such that ‖u‖2Hm,k(QT ,γ) = ∫ QT  ∑ |p|≤m |Dpu|2 + k∑ j=1 |utj | 2   e−γtdxdt < +∞, ‖u‖2Hm,0(QT ,γ) = ∫ QT  ∑ |p|≤m |Dpu|2   e−γtdxdt < +∞. H˚ l,k(QT , γ) is the closure in H l,k(QT , γ) of the set of all infinitely differentiable functions on QT which vanish near ST . L2(QT , γ) is the space of functions u(x, t) defined on QT with the normal ‖u‖L2(QT ,γ) = ( ∫ QT |u|2e−γtdxdt ) 1 2 . Now we introduce the differential operator 14 Dirichlet-Cauchy problem for second-order parabolic equations in domains with edges L(x, t,D) = − n∑ i,j=1 ∂ ∂xi ( aij(x, t) ∂u ∂xj ) + n∑ i=1 bi(x, t) ∂u ∂xi + c(x, t)u, where aij(x, t), bi(x, t) and c(x, t) are bounded real-valued functions on QT and aij(., t) belongs to C 1(Ω), t ∈ (0, T ). Moreover, suppose that aij, i, j = 1, ..., n, are continuous in x ∈ Ω uniformly with respect to t ∈ (0, T ) and n∑ i,j=1 aij(x, t)ξiξj ≥ µ0|ξ| 2 (2.1) for all ξ ∈ Rn\{0} and (x, t) ∈ QT , where µ0 = const > 0. We consider the following problem in the cylinder QT : ut + L(x, t,D)u = f (2.2) with the initial conditions u|t=0 = 0, (2.3) and boundary conditions u ∣∣∣ ST = 0. (2.4) Denote B(u, v; t) = n∑ i,j=1 ∫ Ω aij(., t) ∂u ∂xj ∂v ∂xi dx+ n∑ i=1 ∫ Ω bi(., t) ∂u ∂xi vdx+ ∫ Ω c(., t)uvdx. Applying condition (2.1) and similar arguments as the proof of Guarding in- equality it follows that B(u, u; t) ≥ µ0‖u‖ 2 H1(Ω) − λ0‖u‖ 2 L2(Ω) (2.5) for all u(x, t) ∈ o H1,1(QT , γ), where µ0 = const > 0, λ0 = const ≥ 0. Without loss of generality we shall deal explicitly with the case when λ0 = 0, since by a substitution v = eλ0tu problem (2.2) - (2.4) can be transformed to a problem with constant λ0 = 0. The function u(x, t) is called a generalized solution in the space H1,1(γ,QT ) of problem (2.2) - (2.4) if and only if u(x, t) ∈ o H1,1(QT , γ), u(x, 0) = 0 and the equality 〈ut, v〉+B(u, v; t) = 〈f, v〉 (2.6) holds for all v ∈ o H1(Ω). 15 Do Van Loi 3. The well-posedness Theorem 3.1. Let f ∈ L2(QT , γ0), γ0 > 0, and suppose that the coefficients of the operator L satisfy sup{|aij |, |bi|, |c| : i, j = 1, . . . , n; (x, t) ∈ QT} ≤ µ. Then for each γ > γ0, problem (2.2) - (2.4) has unique generalized solution u in the space H˚1,1(QT , γ) and the following estimate holds ‖u‖2H1,1(QT ,γ) ≤ C‖f‖ 2 L2(QT ,γ0), (3.1) where C is a constant independent of u and f . This solution depends continuously on f . Proof. Firstly, we will prove the existence by Galerkin’s approximating method. Let {ωk(x)} ∞ k=1 be an orthogonal basis of H˚ 1(Ω) which is orthonormal in L2(Ω). Puting uN(x, t) = N∑ k=1 CNk (t)ωk(x), where CNk (t), t ∈ [0, T ), k = 1, ..., N, are the solution of the system of the following ordinary differential equations: (uNt , ωk) +B(u N , ωk; t) = (f, ωk), t ∈ [0, T ), k = 1, ..., N, (3.2) with the initial conditions CNk (0) = 0, k = 1, ..., N, (3.3) here (., .) is the inner products in the space L2(Ω). Let us multiply (3.2) by CNk (t), then take the sum with respect to k from 1 to N , we arrive at (uNt , u N) +B(uN , uN ; t) = (f, uN), t ∈ [0, T ). Now adding this equality to its complex conjugate, we get d dt ( ‖uN‖2L2(Ω) ) + 2ReB(uN , uN ; t) = 2Re(f, uN). (3.4) From (2.5) with λ0 = 0, we obtain ReB(uN , uN ; t) ≥ µ0‖u N‖2H1(Ω). 16 Dirichlet-Cauchy problem for second-order parabolic equations in domains with edges On the other hand, by the Cauchy inequality, for an arbitrary positive number ε, we have 2|(f, uN)| ≤ 2‖f‖L2(Ω)‖u N‖L2(Ω) ≤ C‖f‖ 2 L2(Ω) + ε‖uN‖2L2(Ω), where C = C(ε) is a constant independent of uN , f and t. Combining the estimations above, we get from (3.4) that d dt ( ‖uN(., t)‖2L2(Ω) ) + 2µ‖uN(., t)‖2H1(Ω) ≤ C‖f(., t)‖ 2 L2(Ω) + ε‖u N(., t)‖2L2(Ω) (3.5) for a.e. t ∈ [0, T ). Now write η(t) := ‖uN(., t)‖2L2(Ω); ξ(t) := ‖f(., t)‖ 2 L2(Ω) , t ∈ [0, T ). Then (3.5) implies η′(t) ≤ ε.η(t) + ξ(t), for a.e. t ∈ [0, T ). Thus the differential form of Gronwall-Belmann’s inequality yields the estimate η(t) ≤ Ceεt t∫ 0 ξ(s)ds), t ∈ [0, T ). (3.6) We obtain from (3.6) the estimate ‖uN(., t)‖2L2(Ω) ≤ Ce (ε+γ0)t t∫ 0 e−γ0s‖f‖2L2(Ω)ds ≤ Ce (γ0+ε)t‖f‖2L2(QT ,γ0). Now multiplying both sides of the inequality above by e−γt, γ > γ0 + ε, then inte- grating them with respect to t from 0 to T , we obtain ‖uN‖2L2(QT ,γ) ≤ C‖f‖ 2 L2(QT ,γ0) . (3.7) Returning once more to inequality (3.5), we multiply both sides of the inequality above by e−γt, then integrating them with respect to t from 0 to τ, τ ∈ (0, T ), we obtain τ∫ 0 e−γt ( d dt ‖uN‖2L2(Ω) ) dt+ 2µ τ∫ 0 e−γt‖uN‖2H1(Ω)dt ≤ C ( ‖f‖2L2(QT ,γ0) + ‖u N‖2L2(QT ,γ) ) . 17 Do Van Loi Noting that τ∫ 0 e−γt ( d dt ‖uN‖2L2(Ω) ) dt = τ∫ 0 d dt ( e−γt‖uN‖2L2(Ω) ) dt+ γ τ∫ 0 e−γt‖uN‖2L2(Ω)dt = e−γτ‖uN(x, τ)‖2L2(Ω) + γ τ∫ 0 e−γt‖uN‖2L2(Ω)dt ≥ 0. We employ the inequalities above to find 2µ τ∫ 0 e−γt‖uN‖2H1(Ω)dt ≤ C‖f‖ 2 L2(QT ,γ0) , ∀τ ∈ (0, T ). (3.8) Since the right-hand side of (3.8) is independent of τ , we get ‖uN‖2H1,0(QT ,γ) ≤ C‖f‖ 2 L2(QT ,γ0), (3.9) where C is a constant independent of u, f and N . Fix any v ∈ H˚1(Ω), with ‖v‖2H1(Ω) ≤ 1 and write v = v 1+v2 where v1 ∈ span{ωk} N k=1 and (v2, ωk) = 0, k = 1, ..., N, (v 2 ∈ span{ωk} N k=1 ⊥ ). We have ‖v1‖H1(Ω) ≤ ‖v‖H1(Ω) ≤ 1. From (3.2), we get (uNt , v 1) +B(uN , v1; t) = (f, v1), for a.e. t ∈ [0, T ). From uN(x, t) = N∑ k=1 CNk (t)ωk, we can see that (uNt , v) = (u N t , v 1) = (f, v1)− B(uN , v1; t), consequently, |(uNt , v)| ≤ C ( ‖f‖2L2(Ω) + ‖u N‖2H1(Ω) ) . Since the inequality above holds for all v ∈ H˚1(Ω), ‖v‖H1(Ω) ≤ 1, we receive the following inequality ‖uNt ‖ 2 L2(Ω) ≤ C ( ‖f‖2L2(Ω) + ‖u N‖2H1(Ω) ) (3.10) Multiplying (3.10) by e−γt, γ > γ0+ ε, then integrating them with respect to t from 0 to T , and by using (3.9), we obtain ‖uNt ‖ 2 L2(QT ,γ) ≤ C‖f‖2L2(QT ,γ0). (3.11) 18 Dirichlet-Cauchy problem for second-order parabolic equations in domains with edges Combining (3.9) and (3.11), we arrive at ‖uN‖2H1,1(QT ,γ) ≤ C‖f‖ 2 L2(QT ,γ0), (3.12) here C is a absolute constant. According to inequality (3.12), by standard weakly convergent arguments, we can conclude that the sequence {uN}∞N=1 possesses a subsequence convergent to a func- tion u ∈ H˚1,1(QT , γ), which is a generalized solution of the problem (2.2)-(2.4). Moreover, it follows from (3.12) that inequality (3.1) holds. Finally, we will prove the uniqueness of generalized solution. It suffices to check that the only the generalized solution of problem (2.2)-(2.4) with f ≡ 0 is u ≡ 0. (3.13) By setting v = u(., t) in identity (2.6) (for f ≡ 0) and adding it to its complex conjugate, we get d dt (‖u(., t)‖2) + 2ReB(u, u; t) = 0. Since (2.5), we have d dt (‖u‖2L2(Ω)) + 2µ‖u‖ 2 H1(Ω) ≤ 0, for a.e. t ∈ [0, T ). From this inequality and Gronwall-Belmann’s inequality, it implies (3.13). By esti- mate (3.12), we also get that the solution u depends continuously on f . 4. The smoothness with respect to time variable of the generalized solution In this section, we will show that u is smooth with respect to time variable, provided the coefficients of the operator L and the right-hand side f of system (2.2), are smooth with respect to time variable. We also show that the smoothness with respect to time variable of u is independent of the base Ω of cylinders. To simplify notation, we write aijtk = ∂kaij ∂tk , bitk = ∂kbi ∂tk , ctk = ∂kc ∂tk and Btk(u, v; t) = n∑ i,j=1 ∫ Ω aijtk(., t) ∂u ∂xj ∂v ∂xi dx+ n∑ i=1 ∫ Ω bitk(., t) ∂u ∂xi vdx+ ∫ Ω c(., t)tkuvdx. 19 Do Van Loi Theorem 4.1. Let h ∈ N∗, and we assume that (i) sup { |aijtk |, |bitk |, |ctk | : i, j = 1, . . . , n; (x, t) ∈ QT , k ≤ h } ≤ µ, (ii) ftk ∈ L2(QT , γk), k ≤ h; ftk(x, 0) = 0, 0 ≤ k ≤ h− 1. Then for an arbitrary real number γ satisfying γ > γ0, the generalized solution u ∈ H˚1,1(QT , γ) of problem (2.2)-(2.4) has derivatives with respect to t up to order h with utk ∈ H˚ 1,1(QT , γ + ε), k = 0, ..., h, and we have the estimate ‖uth‖ 2 H1,1(QT ,γ+ε) ≤ C h∑ j=0 ‖ftj‖ 2 L2(QT ,γ0), (4.1) where C is a constant independent of u, f and ε > 0 is small. Proof. From the assumptions on the coefficients of operator L and the function f , it implies that the solution {CNk } N k=1 of problem (3.2)-(3.3) has derivatives with respect to t up to order h + 1. We will prove by induction that ‖uNth‖ 2 H1,0(QT ,γ+ε) ≤ C h∑ j=0 ‖ftj‖ 2 L2(QT ,γ0) . (4.2) Firstly, we differential h times both sides of (3.2) with respect to t to finding the following equality: (uNth+1 , ωk) + h∑ l=0 ( h l ) Bth−l(u N tl , ωk; t) = (fth , ωk), k = 1, ..., N (4.3) From these equalities together with the initial (3.3) and the assumption (ii), we can show by induction on h that uNtk |t=0 = 0 for k = 0, . . . , h. (4.4) We multiply equality (4.3) by dhCNk (t) dth and sum k = 1, . . . , N, to discover (uNth+1, u N th) + h∑ j=0 ( h j ) Bth−j (u N tj , u N th; t) = (fth , u N th). Adding this equality to its complex conjugate, we get d dt ( ‖uN(., t)‖2L2(Ω) ) + 2Re h∑ j=0 ( h j ) Bth−j (u N tj , u N th ; t) = 2Re(fth , u N th). (4.5) 20 Dirichlet-Cauchy problem for second-order parabolic equations in domains with edges According to Theorem 3.1, it implies that inequalities (4.2) holds for h = 0. Assume now inequalities (4.2) is valid for k = h − 1, we will show it to be true for k = h. Returning to equality (4.5), the second term in the left-hand side of (4.5) is written in the following form: 2Re h∑ j=0 ( h j ) Bth−j (u N tj , u N th; t) = = 2ReB(uNth , u N th; t) + 2Re h−1∑ j=0 ( h j ) Bth−j (u N tj , u N th; t). Hence, from (4.5) we have d dt ( ‖uN(., t)‖2L2(Ω) ) +2ReB(uNth, u N th ; t) = 2Re(fth , u N th)−2Re h−1∑ j=0 ( h j ) Bth−j (u N tj , u N th; t). (4.6) For shortly, we denote by I, II, the terms from the first to the second, respectively, of the right-hand side of (4.6). By using the asumption (i) and the Cauchy inequality, we obtain the following estimates: (I) ≤C(ε1)‖fth‖ 2 L2(Ω) + ε1‖uth‖ 2 L2(Ω). (II) ≤C(ε2) h−1∑ j=0 ‖uNtj ‖ 2 H1(Ω) + ε2µnh‖u N th‖ 2 H1(Ω). Employing the estimates above, we get from (4.6) that d dt ( ‖uN(., t)‖2L2(Ω) ) + 2ReB(uNth , u N th; t) ≤ C(ε1)‖fth‖ 2 L2(Ω) + ε1‖uth‖ 2 L2(Ω) + C(ε2) h−1∑ j=0 ‖uNtj ‖ 2 H1(Ω) + ε2µnh‖u N th‖ 2 H1(Ω). (4.7) By using again (2.5), we obtain from (4.7) the estimate d dt ( ‖uN(., t)‖2L2(Ω) ) + (2µ0 − ε2µnh)‖u N th‖ 2 H1(Ω) ≤ ε1‖uth‖ 2 L2(Ω) + C(ε1)‖fth‖ 2 L2(Ω) + C(ε2) h−1∑ j=0 ‖uNtj ‖ 2 H1(Ω). (4.8) 21 Do Van Loi By let ε2 > 0 such that (2µ0 − ε2µnh) < µ0, and let ε = ε1 min{µ0;1} . From (4.8), we get d dt ( ‖uN(., t)‖2L2(Ω) ) + ‖uNth‖ 2 H1(Ω) ≤ ε‖uth‖ 2 L2(Ω) + C [ ‖fth‖ 2 L2(Ω) + h−1∑ j=0 ‖uNtj ‖ 2 H1(Ω) ] . (4.9) Now write η(t) := ‖uNth(., t)‖ 2 L2(Ω) ; ξ(t) := [ ‖fth‖ 2 L2(Ω) + h−1∑ j=0 ‖uNtj ‖ 2 H1(Ω) ] , t ∈ [0, T ). Then (4.9) implies η′(t) ≤ ε.η(t) + Cξ(t), for a.e. t ∈ [0, T ). Thus the differential form of Gronwall-Belmann’s inequality yields the estimate η(t) ≤ Ceεt t∫ 0 ξ(s)ds), t ∈ [0, T ). (4.10) We obtain from (4.10) the estimate ‖uNth(., t)‖ 2 L2(Ω) ≤ Ce εt t∫ 0 [ ‖fth(., s)‖ 2 L2(Ω) + h−1∑ j=0 ‖uNtj (., s)‖ 2 H1(Ω) ] ds ≤ Ce(ε+γ0)t‖fth‖ 2 L2(QT ,γ0) + Ce(ε+γ)t h−1∑ j=0 ‖uNtj ‖ 2 H1,0(QT ,γ+ε) Multiplying both sides of the inequality above by e−(γ+ε)t, then integrating them with respect to t from 0 to T and using the induction assumptions, we arrive at ‖uNth‖ 2 L2(QT ,γ+ε) ≤ C h∑ j=0 ‖ftj‖ 2 L2(QT ,γ0) . (4.11) Returning once more to inequality (4.9), we multiply both sides of the inequality above by e−(γ+ε)t, then integrating them with respect to t from 0 to τ, τ ∈ (0, T ), we obtain τ∫ 0 e−(γ+ε)t ( d dt ‖uNth‖ 2 L2(Ω) ) dt+ 2µ0 τ∫ 0 e−(γ+ε)t‖uNth‖ 2 H1(Ω)dt ≤ C h∑ j=0 ‖ftj‖ 2 L2(QT ,γ0) . 22 Dirichlet-Cauchy problem for second-order parabolic equations in domains with edges Noting that τ∫ 0 e−(γ+ε)t ( d dt ‖uNth‖ 2 L2(Ω) ) dt = τ∫ 0 d dt ( e−(γ+ε)t‖uNth‖ 2 L2(Ω) ) dt+ (γ + ε) τ∫ 0 e−γt‖uNth‖ 2 L2(Ω)dt = e−(γ+ε)τ‖uNth(x, τ)‖ 2 L2(Ω) + (γ + ε) τ∫ 0 e−γt‖uNth‖ 2 L2(Ω)dt ≥ 0. We employ the inequalities above to find τ∫ 0 e−(γ+ε)t‖uNth‖ 2 H1(Ω)dt ≤ C h∑ j=0 ‖ftj‖ 2 L2(QT ,γ0) , ∀τ ∈ (0, T ). (4.12) Since the right-hand side of (4.12) is independent of τ , we get ‖uNth‖ 2 H1,0(QT ,γ+ε) ≤ C h∑ j=0 ‖ftj‖ 2 L2(QT ,γ0), (4.13) where C is a constant independent of u, f and N . By the arguments analogous to the proof of Theorem 3.1, and using the asumption (i) again, we obtain the estimate ‖uNtk‖ 2 L2(QT ,γ+ε) ≤ C h∑ j=0 ‖ftj‖ 2 L2(QT ,γ0) , (4.14) for k = 1, . . . , h+ 1. Combining (4.13) and (4.14), we conclude that ‖uNth‖ 2 H1,1(QT ,γ+ε) ≤ C h∑ j=0 ‖ftj‖ 2 L2(QT ,γ0) . (4.15) From this inequality, by again standard weakly convergent arguments, we can con- clude that the sequence {uN tk }∞N=1 possesses a subsequence convergent to a function u(k) ∈ H˚1,1(QT , γ+ ε); moreover, u (k) is kth generalized derivative in t of generalized solution u of problem (2.2) - (2.4). The estimate (4.1) follows from (4.15) by passing the weak convergences. Acknowledgement. This work was supported by National Foundation for Science and Technology Development (NAFOSTED), Vietnam, under project No. 101.01.58.09. 23 Do Van Loi REFERENCES [1] R. A. Adams. Sobolev Spaces, 1975. Academic Press. [2] A. Kokatov and B. A. Plamenevssky, 2005. 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