A non-local problem with integral conditions for hyperbolic equations

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1 Electronic Journal of Differential Equations, Vol. 1999(1999), No. 45, pp ISSN: URL: or ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login: ftp) A non-local problem with integral conditions for hyperbolic equations L. S. Pulkina Abstract A linear second-order hyperbolic equation with forcing and integral constraints on the solution is converted to a non-local hyperbolic problem. Using the Riesz representation theorem and the Schauder fixed point theorem, we prove the existence and uniqueness of a generalized solution. 1 Introduction Certain problems arising in: plasma physics [1], heat conduction [, 3], dynamics of ground waters [4, 5], thermo-elasticity [6], can be reduced to the non-local problems with integral conditions. The above-mentioned papers consider problems with parabolic equations. However, some problems concerning the dynamics of ground waters are described in terms of hyperbolic equations [4]. Motivated by this, we study the equation Lu u xy + A(x, y)u x + B(x, y)u y + C(x, y)u = f(x, y) (1) with smooth coefficients in the rectangular domain D = {(x, y) :<x<a,<y<b}, bounded by the characteristics of equation (1), with the conditions u(x, y) dx = ψ(y), u(x, y) dy = φ(x). () where φ(x), ψ(y) are given functions and <<a, <<b. The special case = a, = b is considered by author in [7]. The consistency condition assumes the form φ(x) dx = ψ(y) dy Mathematics Subject Classifications: 35L99, 35D5. Key words and phrases: Non-local problem, generalized solution. c 1999 Southwest Texas State University and University of North Texas. Submitted July 9, Published November 15,

2 A non-local problem with integral conditions EJDE 1999/45 A problem for a loaded equation Since the integral conditions () are not homogeneous, we construct a function K(x, y) = 1 ψ(y)+ 1 φ(x) 1 φ(x) dx, satisfying the conditions (), and introduce a new unknown function ū(x, y) = u(x, y) K(x, y). Then (1) is converted into a similar equation Lū = f, where f=f LK, while the corresponding integral data are now homogeneous. Now we construct another function M(x, y) = 1 ū(x, y) dx + 1 a b which satisfies the conditions M(x, y) dx = ū(x, y) dx, ū(x, y) dy 1 ab M(x, y) dy = ū(x, y) dx dy, ū(x, y) dy. Let ū(x, y) =w(x, y) +M(x, y), where w(x, y) satisfies a differential equation to be determined. To find the form of this equation, we consider the previous equality as an integral equation with respect to ū ū(x, y) 1 ū(x, y) dx 1 ū(x, y) dy + 1 ū(x, y) dx dy = w(x, y). a b ab (3) It is not difficult to show that ū(x, y) =w(x, y)+ 1 w(x, y) dx+ 1 w(x, y) dy+ 1 w(x, y) dx dy. (4) If we substitute (4) into the left-hand side of the equation Lū = f, thenwe obtain the so called loaded equation with respect to w(x, y), Lw w xy + A(w + 1 and integral conditions w(x, y) dy) x + B(w + 1 +C(w + 1 w(x, y) dx =, 3 Generalized solution Define the function S by Sw = A(w + 1 w(x, y) dx wdy) x +B(w+ 1 w(x, y) dx) y w(x, y) dy (5) w(x, y) dx dy) = f(x, y) w(x, y) dy =. (6) wdx) y

3 EJDE 1999/45 L. S. Pulkina 3 +C(w + 1 wdx+ 1 wdy+ 1 wdxdy) and F (x, y, Sw) = f(x, y) Sw. Then (5) can be assumed to have the form We introduce the function space w xy = F (x, y, Sw). V = {w : w C 1 ( D), w xy C( D), wdx= The completion of this space, with respect to the norm w 1 = (w + wx + w y ) dx dy is denoted by H 1 (D). Notice that H 1 (D) is Hilbert space with (w, v) 1 = For v H 1 define the operator l by lv y v x (x, τ)dτ + x (wv + w x v x + w y v y ) dx dy. v y (t, y)dt y x wdy=}. v(t, τ) dt dτ. Consider the scalar product (w xy,lv) L. Employing integration by parts and taking account of w V,v H 1,wecanseethat(w xy,v) L =(w, v) 1. Definition. A function w H 1 (D) is called a generalized solution of the problem (5)-(6), if (w, v) 1 =(F(x, y, Sw),lv) L for every v H 1 (D). 4 Subsidiary problem Consider the problem with integral conditions (6) for the equation w xy = F (x, y). Theorem 1 Let F (x, y) L (D). Then there exists one and only one generalized solution w of the problem w xy = F (x, y) wdx=, where for some positive constant c 1, wdy =, c 1 w 1 F L. (7)

4 4 A non-local problem with integral conditions EJDE 1999/45 Proof. For F (x, y) L (D), Ψ(v) =(F, lv) L is a bounded linear functional on H 1 (D). Indeed, (F, lv) F L lv L 3max{a,b,a b } F L v 1. Thus by the Riesz-representation theorem there exists a unique w H 1 (D) such that Ψ(v) = (F, lv) L = (w,v) 1. Hence (w, v) 1 = (w,v) 1 for every v H 1 (D), i.e., w is generalized solution. Letting 1 c 1 =3max{a,b,a b },we obtain inequality (7). Lemma 1 Operator S : H1 L is bounded, that is, there exists a positive constant c such that Sw L c w 1. Proof. Let A(x, y) A, B(x, y) B,and C(x, y) C. Then Sw = Aū x + Bū y + Cū, and Sw L = (Aū x + Bū y + Cū) dx dy 3(A ū x L + B ū y L + C ū L ). Now by straightforward calculation, using the inequality ab a + b,and Hölder s inequality, we find that ū L c 3 w L, ( with c 3 =4 1+ (a )a + (b )b ) (b )(a )ab + ; ( ū x L c 4 w x L, with c 4 = 1+ (b )b ) ; ( ū y L c 5 w y L, with c 5 = 1+ (a )a ). Hence Sw L c w 1,wherec =3max{A c 4,B c 5,C c 3}. Indeed, Sw L 3(A c 4 w x L + Bc 5 w y L + C c 3 w L ) c ( w x L + w y L + w L ) = c w 1. As S is linear S( λw) = λs(w) for arbitrary λ. Let λ> 1 c 1, and let S λ (w) =S( λw). Theorem If f(x, y) L (D) and f(x, y) P, then there exists at least one generalized solution w H 1 (D) to problem (5)-(6), where w 1 P η, with η = c 1 1 λ. Furthermore, the solution is uniquely determined, if c <c 1.

5 EJDE 1999/45 L. S. Pulkina 5 Proof. Consider the closed ball W = {S λ ω : S λ ω L (D), S λ ω L P ab η }. Then c F (x, y, Sω) f(x, y) + 1 η S λ ω, and for all S λ ω W we have F (x, y, Sω) c 1P ab η. From Theorem 1 there exists a unique generalized solution of the problem w xy = F (x, y, Sω), w(x, y) dx =, so that (w, v) 1 =(F, lv) L and w 1 1 c 1 w(x, y) dy = F P ab η. Define an operator T : Sω W w = TSω H 1 (D), T (W ) W. Notice that T is a continuous operator. To see this, let (Sω) n, (Sω) W and (Sω) n (Sω) as n.thenforw n =T(Sω) n,w =T(Sω) we have (w n w,v)=(f(x, y, (Sω) n ) F(x, y, (Sω) ),lv) L =((Sω) n (Sω),lv) L. Now from Theorem 1 w n w 1 1 c 1 (Sω) n (Sω) L, n. Furthermore, T is a compact operator. In order to show this, we take a sequence {(Sω) n } W,thatis (Sω) n L P ab η. For w n = T(Sω) n we have w n P ab η, so a sequence {w n } is bounded in H 1 (D), therefore there exists a subsequence weakly convergent in H 1 (D). Since any bounded set in H 1 is compact in L, then there exists a subsequence, which we again denote by {w n }, strongly convergent in L (D) tow,asn.noww satisfies the inequality w L P ab/η. As S is a bounded operator, T is completely continuous and so TS is completely continuous. Thus from Schauder s fixed-point theorem there exists at least one w W such that w = TSw and (w,v) 1 =(F(x, y, Sw ),lv) L for allv H 1 (D). Assume that w 1, w are distinct generalized solutions, then (w 1 w,v) 1 =(F(x, y, Sw 1 ) F (x, y, Sw ),lv) L. From (7) and Lemma 1 we have w 1 w 1 1 Sw 1 Sw L c w 1 w 1. c 1 c 1 Thus, if c <c 1 then it gives a contradiction; therefore, w 1 = w.

6 6 A non-local problem with integral conditions EJDE 1999/45 References [1] Samarskii A.A., Some problems in the modern theory of differential equations, Differentsialnie Uravnenia, 16 (198), [] Cannon J.R., The solution of the heat equation subject to the specification of energy, Quart. Appl. Math., 1 (1963), [3] Ionkin N.I., Solution of boundary-value problem in heat-conduction theory with nonclassical boundary condition, Differentsialnie Uravnenia, 13 (1977), [4] Nakhushev A.M., On certain approximate method for boundary-value problems for differential equations and its applications in ground waters dynamics, Differentsialnie Uravnenia, 18 (198), [5] Vodakhova V.A., A boundary-value problem with Nakhushev non-local condition for certain pseudo-parabolic water-transfer equation, Differentsialnie Uravnenia, 18 (198), [6] Muravei L.A., Philinovskii A.V., On certain non-local boundary- value problem for hyperbolic equation, Matem. Zametki, 54 (1993), [7] Pulkina L., A non-local problem for hyperbolic equation, Abstracts of Short Communications and Poster Sessions, ICM-1998, Berlin, p.17. Ludmila S. Pulkina Department of Mathematics Samara State University 44311, 1, Ac.Pavlov st. Samara, Russia. pulkina@ssu.samara.ru & louise@valhalla.hippo.ru