A note on the Goursat problem for a multidimensional hyperbolic equation. 1 Introduction. Formulation of the problem

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1 Contemporary Analysis and Applied M athematics Vol., No., 98-06, 03 A note on the oursat problem for a multidimensional hyperbolic equation Hadjimamed Soltanov Turkmen State Institute of Energy, , Bayramhan str., Mary, Turkmenistan ashyrmaral00@mail.ru Abstract. In the present paper, the oursat problem for a multidimensional hyperbolic equation is investigated. Uniqueness of the solution and weak solvability of the oursat problem are established. Key words. oursat problem, multidimensional hyperbolic equation, uniqueness, weak solvability. Introduction. Formulation of the problem The study of well-posedness of the Cauchy problem and boundary value problems for hyperbolic partial differential equations has been studied extensively in a large cycle of papers (see, for example ] and the references therein). In the paper 3], the oursat problem in a three dimensional space was studied. Uniqueness of the solution and weak solvability of the oursat problem were established. In present paper, we consider a multidimensional hyperbolic equation n n Lu k i (t) u xi x i u tt + a i u xi + bu t + cu = f (x, t), (.) where x = (x,..., x n ), k i (t), a i (x, t), b(x, t), c(x, t) and f(x, t) are given functions in a domain bounded by below with part of hyperplane t = 0 and by above with the characteristic of equation (.) involving apex O(x 0, t 0 )(t 0 > 0). Let S be a characteristic-cone surface of equation (.). Then a domain bounded by = S surface. Remark. For all (x, t) S we have the following identity n k i (t) vi vn+ = 0. (.) 98

2 Hadjimamed Soltanov Here, v i = cos ( n, x i ) (i =,..., n), v n+ = cos ( n, t), n be the outward normal to the boundary of the domain. Problem (oursat). satisfying conditions Obtain the solution u(x, t) of equation (.) in the domain u (x, t) C () ( ) C () (), (.3) u (x, t) S = 0. (.4) In the present paper, the oursat problem for a multidimensional hyperbolic equation is investigated. Uniqueness of the solution and weak solvability of the oursat problem are established. The paper is organized as follows. Section is introduction where we provide the formulation of the oursat problem. In Section, theorem on uniqueness of the solution of the oursat problem is established. In Section 3, theorem on weak solvability of the oursat problem is proved. Finally, Section 4 is conclusion. Uniqueness of the solution of the oursat problem Now, we will introduce some notations which are used throughout the paper. Let D be a set of all functions u(x, t) defined on and satisfying conditions (.3)-(.4). Let D be a set of all functions u(x, t) defined on and satisfying condition (.3) and initial conditions u (x, 0) = 0, u t (x, 0) = 0. (.) Theorem. Suppose that all coefficients of equation (.) are continuously differentiable functions on closed domain and satisfying conditions k i (t) 0, k i (0) > 0, k i (t) k 0 i > 0, (.) b (x, t) b 0 > 0, (.3) c (x, t) c 0 < 0, c t (x, t) 0. (.4) If there exists a solution of oursat problem in D, then it is unique. 99

3 A note on the oursat problem for a multidimensional hyperbolic equation Proof. Let λ be any negative number and u(x, t) D be any function. Then, multiplying both sides of equation (.) by u t (x, t) and taking the integral over the domain, we get the following identity u t L (u) d = u t f (x, t) d. (.5) Integrating by parts and using the reen formula, we obtain + n k i (t) λk i (t)] u x i + b λ] u t + λc c t] u d n a i u t u xi d + n ki (t) u xi u t v i k i (t) u ] x i v n+ u t v n+ ds Using identity (.) + = S = c (x, t) u v n+ ds = u t f (x, t) d. (.6) c (x, t) u v n+ ds c (x, t) u v n+ ds + c (x, t) u v n+ ds c (x, t) u v n+ ds. Since we have that Then, applying condition (.4), we get Since we have that v i = 0 (i =,, n), v n+ = on, (.7) c (x, t) u v n+ ds = c (x, t) u ds. c (x, t) u v n+ ds 0. (.8) u (x, t) = 0 on S, u xi = u n v i (i =,, n), u t = u n v n+. (.9) 00

4 Hadjimamed Soltanov Then, using identity (.), condition (.), (.7), and (.9) we get = = S + S + n ki (t) u xi u t v i k i (t) u ] x i v n+ u t v n+ ds n ki (t) u xi u t v i k i (t) u ] x i v n+ u t v n+ ds n k i (t) u xi u t v i ds + n n k i (t) u x i v n+ u t v n+ ds k i (t) vi vi ] v n+ v n+ u nds n ] k i (t) u x i + u t ds + Applying (.6), (.8), (.0) we get n ] k i (t) u x i + u t ds 0. (.0) n k i (t) λk i (t)] u x i + b λ] u t + λc c t] u d + Applying the well-known inequality n a i u t u xi d u t f (x, t) d. (.) ab ε a + ε b for a, b > 0 and ε > 0, we obtain n k i (t) λk i (t)] u x i + b λ] u t + λc c t] u d n a i u x i + u t d ε f (x, t) d (.) for any negative λ. Since a i is a bounded function, we can choose λ such that there exists α > 0 where n ( n ) α u x i + u t + u ] ] k i(t) λk i (t) u x i + b λ ε] u t + λc c t u 0 n a i u x i + u t. (.3)

5 A note on the oursat problem for a multidimensional hyperbolic equation Applying estimates (.) and (.3), we can write n u x i + u t + u d εα f (x, t)d. (.4) From estimate (.4) it follows that if there exists a solution of oursat problem in D, then it is unique. Moreover, from estimate (.4) it follows the stability of the solution of oursat problem. Theorem. is proved. 3 Solvability of the solution of the oursat problem Let u(x, t) D and v(x, t) D be any given functions. Then, we have the following identity vl(u)d = vfd. (3.) Integrating by parts and using the reen formula, we obtain n k i (t)u xi v xi u t v t + n a i uv xi u xi v] d (3.) Since + we have that b uv t u t v] + n ] a i + b x i t c uv d n a i uvv i + buvv n+ ds = n k i (t)u xi vv i u t vv n+ + S v(x, t) = 0 on, n k i (t)u xi vv i u t vv n+ + n = k i (t)u xi vv i u t vv n+ + Applying (3), we get n k i (t)u xi vv i u t vv n+ + 0 n a i uvv i + buvv n+ ds n a i uvv i + buvv n+ ds. vfd. n a i uvv i + buvv n+ ds (3.3)

6 Hadjimamed Soltanov n = k i (t)u xi vv i u t vv n+ ds = Applying formulas (3.) and (3.3), we can write n k i (t)vi vn+ u n vds = 0. Lu, v = W u, v (3.4) n k i (t)u xi v xi u t v t + n a i uv xi u xi v] d b uv t u t v] + n ] a i + b x i t c uv d = vfd. Definition 3. A function u 0 (x, t) D is called a weak solution of the oursat problem if it satisfies identity (3.4) for all v(x, t) D. For k =,, let us introduce a Euclidean space E k of all elements u D k with the inner product n u, v = u xi v xi + u t v t + uv d. If we complete E k in the norm u Ek = u, u, then we obtain a Hilbert space H k. Theorem 3. Suppose that for all coefficients of equation (.), the assumptions of Theorem. hold. Then, there exists a unique weak solution of oursat problem in H. Proof. Note that for all u(x, t) H and v(x, t) H identity (3.4) holds. Using formula n W u, v = k i (t)u xi v xi u t v t + n a i uv xi u xi v] d and Cauchy-Schwarz inequality, we get b uv t u t v] + n ] a i + b x i t c uv d (3.5) W u, v N u H v H. (3.6) Here, N is a positive constant does not depend on u and v. From inequality (3.6) it follows boundness of linearly expression W u, v with respect to u and v. Therefore, the isomorphism of Hilbert spaces H and H is based on identity (3.5). Moreover, for the fixed function u 0 (x, t) H the expression W u 0, v is a linearly bounded functional with respect v H. Exactly 03

7 A note on the oursat problem for a multidimensional hyperbolic equation same manner for the fixed function v 0 H the expression W u, v 0 is the linearly bounded functional with respect to u H. From the isomorphism of Hilbert spaces H and H it follows one-to-one relation u(x, t) v(x, t) and u H = v H. Moreover, if u (x, t) v (x, t) and u (x, t) v (x, t), then for all α and β numbers it follows that αu (x, t) + βu (x, t) αv (x, t) βv (x, t). For the fixed function u 0 (x, t) H identity (3.4) is a linearly bounded functional in H. Since f(x, t) is the given function, we have that the right side expression vfd of identity (3.4) is a linearly bounded functional with respect to v H. Therefore, by the Riesz theorem 4] there exists unique function v 0 H such that the following identity W u, v = vfd = v 0, v holds. That means W u, v = v 0, v From the isomorphism of Hilbert spaces H and H it follows that for such v 0 (x, t) function there exists unique function u 0 (x, t) and v 0 (x, t) u 0 (x, t). From that it follows identity W u, v = vfd or L(u 0, v) = W u 0, v = vfd. So, by the definition of a weak solution of the oursat problem it follows that the function u 0 (x, t) H is the unique weak solution of the oursat problem. Theorem 3. is proved. 4 Conclusion In this paper we investigated the oursat problem for a multidimensional hyperbolic equation. Uniqueness of the solution and weak solvability of the oursat problem are established. Of course, the strong solvability of the oursat problem can been established under the smooth assumptions for f(x, t) and for all coefficients of equation (.). 04

8 Hadjimamed Soltanov 5 Acknowledgement The author would like to thank Prof. M. Meredov (International Turkmen-Turkish University, Ashgabat, Turkmenistan) and Prof. A. Ashyralyev (Fatih University, Istanbul, Turkey) for their helpful suggestions to improve this paper. References ] M. Nagumo, Lectures on Modern Theory of Partial Differential Equations, Mir, Moscow, 967 (in Russian). ] S. Mizohata, Theory of Partial Differential Equations, Mir, Moscow, 977 (in Russian). 3] A.I. Kozhanov, On boundary value problems for some class higher order equations, Sibirskiy Matematicheskiy Jurnal, 35(5) (994) (in Russian). 4] A.A. Dezin, eneral Questions of Boundary Value Problems Theory, Nauka, Moscow, 978 (in Russian). 5] T.Sh. Kalmenov, Boundary Value Problems for Linear Partial Differential Equations of Hyperbolic Type, ylym, Shymkent, 993 (in Russian). 6] A. Ashyralyev, P.E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Operator Theory Advances and Applications, Birkhauser Verlag, Basel, Boston, Berlin, ] D. Amanov, A. Ashyralyev, Well-posedness of boundary value problems for partial differential equations of even order, in: A. Ashyralyev, A. Lukashov (Eds), First international conference on analysis and applied mathematics: ICAAM 0, volume 470 of AIP Conference Proceedings, pp ] S.A. Aldashev, The well-posedness of the Dirichlet problem in the cylindric domain for the multidimensional wave equation, Mathematical Problems Engineering 00 (00) Article ID ] S.A. Aldashev, Criterion for the uniqueness of the solution of the Darboux problems for the three dimensional degenerate hyperbolic equation, Mathematical Journal 9(3(33)) (in Russian). 05

9 A note on the oursat problem for a multidimensional hyperbolic equation 0] E.I. Moiseev, On solution of a degenerate equations with the use of biorthogonal series, Differential Equations, 7() (99) ] M. Meredov, On the weak solution of a Cauchy problem for the hyperbolic equation, Proceedings of International Conference, Ashgabat, Ylym, pp.68-69, 0. ] N. Aggez, M. Ashyralyyewa, Numerical solution of stochastic hyperbolic equations, Abstract and Applied Analysis 0 (0) Article ID ] H. Soltanov, On the oursat problem in a three dimensional space, Scientific-Theoretical Journal of Supreme Council on Science and Technology under the President of Turkmenistan (03) ] A.N. Kolmogorov, S.V. Fomin, Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, 976 (in Russian). 06