On the solvability of the boundary problem for second order parabolic equations without an initial condition in cylinders with non smooth base

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The goal of this paper is to establish the unique existence of generalized solutions of boundary problem for second-order parabolic equations without an initial condition in cylinders with non-smooth base.

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Nội dung Text: On the solvability of the boundary problem for second order parabolic equations without an initial condition in cylinders with non smooth base

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2011, Vol. 56, No. 7, pp. 18-22 ON THE SOLVABILITY OF THE BOUNDARY PROBLEM FOR SECOND-ORDER PARABOLIC EQUATIONS WITHOUT AN INITIAL CONDITION IN CYLINDERS WITH NON-SMOOTH BASE Nguyen Manh Hung and Le Thi Duyen(∗) Hanoi National University of Education (∗) E-mail: Linhlinh041@gmail.com Abstract. The goal of this paper is to establish the unique existence of gen- eralized solutions of boundary problem for second-order parabolic equations without an initial condition in cylinders with non-smooth base. Keywords: Generalized solutions, without an initial condition. 1. Introduction The initial-boundary value problems for parabolic equations in domains with conical points were considered in [3, 4], where some important results on the unique existence of solutions for these problem were given. The problem without initial condition for second-order parabolic equations in cylinders with smooth base was considered in [1, 2]. In this paper, we will prove the unique solvability of bound- ary problem for second-order parabolic equations without an initial condition in cylinders with non-smooth base. 2. Formulation of the problem Let G be a bounded domain in Rn , n ≥ 2, with the boundary ∂G. We suppose that S = ∂G \ 0 is an infinitely differentiable surface everywhere except the origin. Denote G∞ = G × (−∞, T ), S∞ = S × (−∞, T ), Gh = G × (h, T ), Sh = S × (h, T ), for each T, 0 < T ≤ ∞. We use the following notation: for each multi-index α = (α1 , ....αn ) ∈ N n , |α| = α1 + .... + αn , the symbol D α u = ∂ |α| u/∂ αx11 .....∂ αxnn = uxα1 1 ...xαnn denotes the generalized derivative of order α with respect to x = (x1 , ..., xn ). We begin with recalling some functional spaces which will be used frequently in this paper. 18
On the solvability of the boundary problem... W2m (G) is the space consisting of all functions u(x) ∈ L2 (G), such that D α u(x) ∈ L2 (G) for almost |α| ≤ m with the norm m Z X 1/2 kukW2m(G) = |D αu|2 dx . |α|=0 G Let X, Y be Banach spaces. L2 ((0, T ); X) is the space consisting of all measurable functions u : (0, T ) → X with the norm ZT 21 kukL2 ((0,T );X) = ku(t)k2X dt . 0 W21 ((0, T ); X) is the space consisting of all u ∈ L2 ((0, T ); X) such that the ′ generalized derivative ut = u exists and belongs to L2 ((0, T ); X). The norm in W21 ((0, T ); X) is defined by ZT 21 kukW21((0,T );X) = ku(t)k2X + kutk2X dt . 0 W21 ((0, T ); X, Y ) is the space consisting of all u ∈ L2 ((0, T ); X) such that the ′ generalized derivative ut = u exists and belongs to L2 ((0, T ); Y ). The norm in W21 ((0, T ); X, Y ) is defined by 21 kukW21((0,T );X,Y ) = ku(t)k2L2 ((0,T );X) + kut k2L2 ((0,T );Y ) dt . Now we introduce a differential operator of order 2m m X L = L(x, t, Dx ) = (−1)|α| Dxα (aαβ (x, t)Dxβ ), |α|,|β|=0 in GT with smooth coefficients in GT , aαβ = aβα for |α| , |β| ≤ m. We introduce also a system of boundary operators X Bj = Bj (x, t, Dx ) = bj,α (x, t)D α , j = 1, · · · , m, |α|≤µj on ST with smooth coefficients in GT . Suppose that ordBj = µj ≤ m − 1 for j = 1, · · · , λ, 19
Nguyen Manh Hung and Le Thi Duyen m ≤ ordBj = µj ≤ 2m − 1 for j = λ + 1, · · · , m, and coefficients of Bj are independent of t if ordBj < m. Let HBm (G) = {u ∈ W2m (G) : Bj u = 0 on S for j = 1, · · · , λ} with the same norm in W2m (G). By HB−m (G) we denote the dual space to HBm (G) We consider the following problem without an initial condition in the cylinder G∞ ut + Lu = f in G∞ , (2.1) Bj u = 0 on S∞ , j = 1, · · · , m, (2.2) where f is defined in G∞ . Let χ ∈ C ∞ (R) such that   χ(t) = 0,  t≤0 χ(t) = 1, t ≥ 1   0 ≤ χ(t) ≤ 1, ∀t ∈ R. Let h be a nonnegative integer. We set χh (t) = χ(t−h). We define the function fh (x, t) by the equality fh = χh (t)f. (2.3) For any h, we have kfh k2L2 ((h,T );H −m (G)) ≤ kf k2L2 ((h,T );H −m (G)) . (2.4) B B Now we consider the following problem for an equation of parabolic type in the cylinder Gh ut + Lu = fh in Gh , (2.5) Bj u = 0 on Sh , j = 1, · · · , m, (2.6) u|t=h = 0 on G. (2.7) Let fh ∈ L2 ((h, T ); HB−m (G)). Using the results in [3], we get a unique gener- alized solution uh ∈ W21 ((h, T ); HBm(G), HB−m (G)) of problem (2.5) - (2.7) satisfing kuh k2W 1 ((h,T );H m (G),H −m (G)) ≤ Ckfh k2L2 ((h,T );H −m (G)) , (2.8) 2 B B B where C is a constant independent of fh and uh . Let f ∈ L2 ((−∞, T ); HB−m(G)). A function u ∈ W21 ((−∞, T ); HBm(G), HB−m(G)) is called a generalized solution of problem (2.1) - (2.2) if uh −→ u in W21 ((−∞, T ); HBm(G), HB−m(G)) as h → −∞, here uh is a generalized solution of problem (2.5) - (2.7). 20
On the solvability of the boundary problem... 3. The unique solvability of the generalized solution Theorem 3.1. If f ∈ L2 ((−∞, T ); HB−m(G)), then there exists a unique general- ized solution u ∈ W21 ((−∞, T ); HBm(G), HB−m(G)) of the problem (2.1) - (2.2) which satisfies kuk2W 1((−∞,T );H m (G),H −m (G)) ≤ Ckf k2L2 ((−∞,T );H −m (G)) , (3.1) 2 B B B where C is the constant independent of f and u. Proof. Let fh be defined by (2.3). Ref. [3] shows that the problem (2.5) - (2.7) has a unique generalized solution uh ∈ W21 ((h, T ); HBm(G), HB−m(G)) which satisfies (2.8). From (2.4) and (2.8), we obtain kuh k2W 1 ((h,T );H m (G),H −m (G)) ≤ Ckf k2L2 ((h,T );H −m (G)) . (3.2) 2 B B B ′ ′ Suppose that h < h . We define uh′ in the cylinder Gh for h ≤ t < h by setting ′ uh′ (x, t) = 0 for h ≤ t < h . We set v = uh − uh′ , then v ∈ W21 ((h, T ); HBm(G), HB−m (G)) is a solution of the following problem, similar to (2.5) - (2.7) vt + Lv = fh − fh′ in Gh , (3.3) Bj v = 0 on Sh , j = 1, · · · , m, (3.4) v|t=h = uh (h) − uh′ (h) = 0 on G. (3.5) Using also these results in [3] for the problem (3.3) - (3.5), we get kvk2W 1 ((h,T );H m (G),H −m (G)) ≤ Ckfh − fh′ k2L2 ((h,T );H −m (G)) . (3.6) 2 B B B We have ZT kfh − fh′ k2L2 ((h,T );H −m (G)) = kfh (t) − fh′ (t)k2H −m (G)) dt B B h Zh+1 ′ ≤ kfh (t) − fh′ (t)k2H −m (G)) dt −→ 0 (h, h → −∞). B h′ From (3.6) we see that uh is a Cauchy sequence in W21 ((h, T ); HBm(G), HB−m (G)). Thus, uh tends to a function u ∈ W21 ((−∞, T ); HBm(G), HB−m (G)) which is a gener- alized solution of the problem (2.1) - (2.2). 21
Nguyen Manh Hung and Le Thi Duyen Now, we will prove the uniqueness of the solution. Let uˆ be also a generalized solution of the problem (2.1) - (2.2). This means that exists the function uˆh such that uˆh −→ uˆ in W21 ((−∞, T ); HBm(G), HB−m(G)) as h → −∞, where uˆh is a generalized solution of the problem (2.5) - (2.7) with fh is replaced by fˆh . We have kuh − uˆh k2W 1 ((h,T );H m (G),H −m (G)) ≤ Ckfh − fˆh k2L2 ((h,T );H −m (G)) ≤ Ckf k2L2((h,h+1);H −m (G)) , 2 B B B B where C depends on χh and χ ˆh . We remark that Zh+1 kf kL2((h,h+1);H −m (G)) = kf kH −m (G) dt → 0 as h → −∞. B B h For simplicity we will write W := W21 ((−∞, T ); HBm(G), HB−m (G)). We have ku − uˆkW ≤ ku − uh kW + kuh − uˆh kW + kˆ uh − uˆkW −→ 0 as h → −∞. This implies u = uˆ in W. The proof is completed. Acknowledgement. This work was supported by the National Foundation for Science and Technology Development (NAFOSTED:101.01.58.09), Vietnam. REFERENCES [1] YU. P. Krasovskii, 1992. An estimate of solutions of parabolic problems without an intial condition, Math. USSR Izvestiya, Vol. 38, No. 2, pp. 429-433. [2] N. M. Bokalo, 1990. Problem without initial conditions for some classes of non- linear parabolic equations, UDC 517. 95, pp. 2291-2322. [3] Nguyen Manh Hung, Nguyen Thanh Anh, 2008. The initial-boundary value prob- lems for parabolic equations in domains with conical points. Advances in Mathe- matics Research, Vol. 10, pp. 1-30. [4] Nguyen Manh Hung and Nguyen Thanh Anh, 2008. Regularity of solutions of initial-boundary value problems for parabolic equations in domains with conical points. Journal of Differential Equations, Vol. 245, Issue 7, pp. 1801-1818. 22