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) ∈ Nn, |α| = α1 + .... + αn, the symbol Dαu = ∂|α|u/∂α1x1 .....∂αnxn = uxα11 ...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... Wm2 (G) is the space consisting of all functions u(x) ∈ L2(G), such that Dαu(x) ∈ L2(G) for almost |α| ≤ m with the norm ‖u‖Wm 2 (G) = ( m∑ |α|=0 ∫ G |Dαu|2dx )1/2 . 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 ‖u‖L2((0,T );X) = ( T∫ 0 ‖u(t)‖2Xdt ) 1 2 . W 12 ((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 W 12 ((0, T );X) is defined by ‖u‖W 1 2 ((0,T );X) = ( T∫ 0 ‖u(t)‖2X + ‖ut‖2Xdt ) 1 2 . W 12 ((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 W 12 ((0, T );X, Y ) is defined by ‖u‖W 1 2 ((0,T );X,Y ) = ‖u(t)‖2L2((0,T );X) + ‖ut‖2L2((0,T );Y )dt ) 1 2 . Now we introduce a differential operator of order 2m L = L(x, t,Dx) = m∑ |α|,|β|=0 (−1)|α|Dαx (aαβ(x, t)Dβx), in GT with smooth coefficients in GT , aαβ = aβα for |α| , |β| ≤ m. We introduce also a system of boundary operators Bj = Bj(x, t,Dx) = ∑ |α|≤µj bj,α(x, t)D α, j = 1, · · · , m, 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 HmB (G) = {u ∈ Wm2 (G) : Bju = 0 on S for j = 1, · · · , λ} with the same norm in Wm2 (G). By H−mB (G) we denote the dual space to H m B (G) We consider the following problem without an initial condition in the cylinder G∞ ut + Lu = f in G∞, (2.1) Bju = 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 ‖fh‖2L2((h,T );H−mB (G)) ≤ ‖f‖ 2 L2((h,T );H −m B (G)) . (2.4) Now we consider the following problem for an equation of parabolic type in the cylinder Gh ut + Lu = fh in Gh, (2.5) Bju = 0 on Sh, j = 1, · · · , m, (2.6) u|t=h = 0 on G. (2.7) Let fh ∈ L2((h, T );H−mB (G)). Using the results in [3], we get a unique gener- alized solution uh ∈ W12 ((h, T );HmB (G), H−mB (G)) of problem (2.5) - (2.7) satisfing ‖uh‖2W1 2 ((h,T );Hm B (G),H−m B (G)) ≤ C‖fh‖2L2((h,T );H−mB (G)), (2.8) where C is a constant independent of fh and uh. Let f ∈ L2((−∞, T );H−mB (G)). A function u ∈ W12 ((−∞, T );HmB (G), H−mB (G)) is called a generalized solution of problem (2.1) - (2.2) if uh −→ u in W12 ((−∞, T );HmB (G), H−mB (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 );H−mB (G)), then there exists a unique general- ized solution u ∈ W12 ((−∞, T );HmB (G), H−mB (G)) of the problem (2.1) - (2.2) which satisfies ‖u‖2 W1 2 ((−∞,T );Hm B (G),H−m B (G)) ≤ C‖f‖2 L2((−∞,T );H −m B (G)) , (3.1) 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 ∈ W12 ((h, T );HmB (G), H−mB (G)) which satisfies (2.8). From (2.4) and (2.8), we obtain ‖uh‖2W1 2 ((h,T );Hm B (G),H−m B (G)) ≤ C‖f‖2 L2((h,T );H −m B (G)) . (3.2) 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 ∈ W12 ((h, T );HmB (G), H−mB (G)) is a solution of the following problem, similar to (2.5) - (2.7) vt + Lv = fh − fh′ in Gh, (3.3) Bjv = 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 ‖v‖2 W1 2 ((h,T );Hm B (G),H−m B (G)) ≤ C‖fh − fh′‖2L2((h,T );H−mB (G)). (3.6) We have ‖fh − fh′‖2L2((h,T );H−mB (G)) = T∫ h ‖fh(t)− fh′ (t)‖2H−m B (G)) dt ≤ h+1∫ h′ ‖fh(t)− fh′ (t)‖2H−m B (G)) dt −→ 0 (h, h′ → −∞). From (3.6) we see that uh is a Cauchy sequence in W12 ((h, T );HmB (G), H−mB (G)). Thus, uh tends to a function u ∈ W12 ((−∞, T );HmB (G), H−mB (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 W12 ((−∞, T );HmB (G), H−mB (G)) as h→ −∞, where uˆh is a generalized solution of the problem (2.5) - (2.7) with fh is replaced by fˆh. We have ‖uh−uˆh‖2W1 2 ((h,T );Hm B (G),H−m B (G)) ≤ C‖fh−fˆh‖2L2((h,T );H−mB (G)) ≤ C‖f‖ 2 L2((h,h+1);H −m B (G)) , where C depends on χh and χˆh. We remark that ‖f‖L2((h,h+1);H−mB (G)) = h+1∫ h ‖f‖H−m B (G)dt→ 0 as h→ −∞. For simplicity we will write W :=W12 ((−∞, T );HmB (G), H−mB (G)). We have ‖u− uˆ‖W ≤ ‖u− uh‖W + ‖uh − uˆh‖W + ‖uˆh − uˆ‖W −→ 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. 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