INVERSE PROBLEM OF A HYPERBOLIC EQUATION WITH AN INTEGRAL OVERDETERMINATION CONDITION

Electronic Journal of Differential Equations, Vol. 216 (216), No. 138, pp. 1 7. ISSN: 172-6691. URL: http://eje.math.txstate.eu or http://eje.math.unt.eu INVERSE PROBLEM OF A HYPERBOLIC EQUATION WITH AN INTEGRAL OVERDETERMINATION CONDITION TAKI-EDDINE OUSSAEIF, ABDELFATAH BOUZIANI Abstract. In this article we stuy the inverse problem of a hyperbolic equation with an integral overetermination conition. The existence, uniqueness an continuous epenence of the solution of the solution upon the ata are establishe. 1. Introuction In this article we stuy the unique solvability of the inverse problem of etermining a pair of functions {u, f} satisfying the equation with the initial conitions the bounary conition u tt u + βu t = f(t)g(x, t), x, t [, T ], (1.1) u(x, ) = ϕ(x), x, (1.2) u t (x, ) = ψ(x), x, (1.3) u(x, t) =, (x, t) [, T ], (1.4) an the nonlocal conition v(x)u(x, t)x = θ(t), t [, T ]. (1.5) Here is a boune omain in R n with smooth bounary. The functions g, ϕ, ψ, θ are known functions an β is a positive constant. Inverse problems for hyperbolic PDEs arise naturally in geophysics, oil prospecting, in the esign of optical evices, an in many other areas the interior of an object is to be image using the response of the object to acoustic waves (satisfying hyperbolic PDEs). Aitional information about the solution to the inverse problem is given in the form of integral observation conition (1.5). The parameter ientification in a partial ifferential equation from the ata of integral overetermination conition plays an important role in engineering an physics [4, 5, 6, 8, 9, 13]. From the physical point of view, these conitions may be interprete as measurements of the temperature u(x, t) by a evice averaging over the omain of spatial variables. 21 Mathematics Subject Classification. 35R3, 49K2. Key wors an phrases. Inverse problem; hyperbolic equation; integral conition. c 216 Texas State University. Submitte October 27, 215. Publishe June 8, 216. 1

2 T.-E. OUSSAEIF, A. BOUZIANI EJDE-216/138 Note that inverse problems with integral overetermination are closely relate to nonlocal problems [2, 11]. Stuies have shown that classical methos often o not work when we eal with nonlocal problems [1, 7]. To ate, several methos have been propose for overcoming the ifficulties arising from nonlocal conitions. The choice of metho epens on the kin of nonlocal conitions. We note that the inverse problem for a parabolic equation with integral conition (1.5) an its unique solubility have been stuie by many authors (see for example [3, 1, 14, 15]). Also, there are some papers evote to the stuy of existence an uniqueness of solutions of inverse problems for various parabolic equations with unknown source functions. Inverse problems of etermining the right-han sie of a parabolic equation uner a final overetermination conition were stuie in papers [11, 12, 15, 16]. In the present work, new stuies are presente for the inverse problem for a hyperbolic equation. The existence an uniqueness of the classical solution of the problem (1.1)-(1.5) is reuce to a fixe point problem. 2. Functional space Let us introuce certain notation use below. We set g (t) = v(x)g(x, t)x, Q T = [, T ]. (2.1) The spaces W 1 2 (), C((, T ), L 2 ()) an W 2,1 2 (Q T ) with corresponing norms are unerstoo as follows: The Banach space W 1 2 () consists of all functions from L 2 () having all weak erivatives of the first orer that are square integrable over with norm u W 1 2 () = ( u 2 L 2() + u x 2 L 2() )1/2, We enote by C((, T ), L 2 ()) the space comprises of all continuous functions on (, T ) with values in L 2 (). The corresponing norm is given by u C((,T ),L2()) = max (,T ) u L 2() <. Let us also introuce the Sobolev space W 2,1 2 (Q T ) of functions u(x, t) with finite norm u W 2,1 2 (Q T ) = ( u 2 L 2(Q T ) + u x 2 L 2(Q T ) + D tu 2 L 2(Q T ) u u L2(), We note that the weighte arithmetic-geometric mean inequality (Cauchy s ε- inequality) is: 2 ab εa 2 + 1 ε b2, for ε >. Also, we use the notation ( n 1/2 ( u = u 2 x i x) an u = i=1 i,j=1 ) 1/2 n 1/2. u 2 x i,x j x)

EJDE-216/138 INVERSE PROBLEM OF A HYPERBOLIC EQUATION 3 3. Existence an uniqueness of the solution to the inverse problem We seek a solution of the original inverse problem in the form {u, f} = {z, f} + {y, } y is the solution of the irect problem y tt y + βy t =, (x, t) Q T, (3.1) y(x, ) = ϕ(x), x, (3.2) y t (x, ) = ψ(x), x, (3.3) y(x, t) =, (x, t) [, T ], (3.4) while the pair {z, f} is the solution of the inverse problem z tt z + βz t = f(t)g(x, t), (x, t) Q, (3.5) z(x, ) =, x, (3.6) z t (x, ) =, x, (3.7) z(x, t) =, (x, t) [, T ], (3.8) v(x)z(x, t) x = E(t), t [, T ], (3.9) E(t) = θ(t) v(x)y(x, t) x. We shall assume that the functions appearing in the ata for the problem are measurable an satisfy the following conitions: (H1) g C((, T ), L 2 ()), v W2 1 (), E W2 2 (, T ), g(x, t) m, g (t) p >, for p R, (x, t) Q T, ϕ(x), ψ(x) W2 1 () The corresponence between f an z may be viewe as one possible way of specifying the linear operator with the values A : L 2 (, T ) L 2 (, T, L 2 ()). (3.1) (Af)(t) = 1 { } g z vx. (3.11) In this view, it is reasonable to refer to the linear equation of the secon kin for the function f over the space L 2 (, T ): f = Af + W, (3.12) W = E + βe g. Remark 3.1. As {u, f} = {z, f} + {y, } y is the solution of the irect problem (3.1) (3.4). Obviously, y exists an is unique, so instea of proving the solvability of the original problem (1.1) (1.5), we prove the existence an uniqueness of the solution of the inverse problem (3.5) (3.9). Theorem 3.2. Suppose the input ata of the inverse problem (3.5)-(3.9) satisfies (H1). Then the following assertions are vali: (i) if the inverse problem (3.5) (3.9) is solvable, then so is equation (3.12). (ii) if equation (3.12) possesses a solution an the compatibility conition E() =, E () =, (3.13)

4 T.-E. OUSSAEIF, A. BOUZIANI EJDE-216/138 hols, then there exists a solution of the inverse problem (3.5)-(3.9). Proof. (i) Assume that the inverse problem (3.5) (3.9) is solvable. We enote its solution by {z, f}. Multiplying both sies of (3.5) by the function v scalarly in L 2 (), we obtain the relation t z t vx + z vx + β z t vx = f(t)g (x, t). (3.14) With (3.9) an (3.11), it follows from (3.14) that f = Af + E +βe g. This means that f solves equation (3.12). (ii) By the assumption, equation (3.12) has a solution in the space L 2 (, T ), say f. When inserting this function in (3.5), the resulting relations (3.5) (3.8) can be treate as a irect problem having a unique solution z W 2,1 2, (Q T ). Let us show that the function z satisfies also the integral overetermination conition (3.9). Equation (3.14) yiels z t vx + t z vx + β z t vx = f(t)g (x, t). (3.15) On the other han, being a solution of (3.12), the function z is subject to relation E + βe + z v x = f(t)g (x, t). (3.16) Subtracting (3.15) from (3.16), we obtain z t vx + β z t vx = E + βe. t Integrating the preceing ifferential equation an taking into account the compatibility conition (3.13), we fin that the function z satisfies the overetermination conition (3.9) an the pair of functions {z, f} is a solution of the inverse problem (3.5) (3.9). This completes the proof. Now, we state some properties of the operator A. Lemma 3.3. Let conition (H1) hol. Then there exist a positive ε for which A is a contracting operator in L 2 (, T ). Proof. Obviously, (3.11) yiels the estimate Af L2(,t) k ( t ) 1/2, z(., τ) 2 L p τ 2() (3.17) k = v L2(). Multiplying both sies of (3.5) by z t scalarly in L 2 () an integrating the resulting expressions by parts, we obtain the ientity 1 2 t z t(, t) 2 L + 1 2() 2 t z(, t) 2 L + β z 2() t(, t) 2 L = f(t) 2() g(x, t)z t x. So, by using the Cauchy s ε-inequality, we obtain the relation 1 2 t z t(, t) 2 L + 1 2() 2 t z(, t) 2 L + β z 2() t(, t) 2 L 2() m2 2ε f(t) 2 + ε 2 z t(, t) 2 L 2(), (3.18)

EJDE-216/138 INVERSE PROBLEM OF A HYPERBOLIC EQUATION 5 Choosing < ε < 2β, an integrating (3.18) over (, t), an using (3.6) an (3.7)), we obtain 1 2 z t(, t) 2 L + 1 2() 2 z(, t) 2 L + (β ε 2() 2 ) m2 2ε t f(τ) 2. t z t (., τ) 2 L 2() (3.19) Omitting some terms on the left-han sie (3.19) an integrating over (, t), an using (3.6), leas to t z(., τ) 2 m2 L 2() τ ε t f(τ) 2 L 2(,T ) τ. (3.2) So, accoring to (3.17) an (3.2), we obtain the estimate ( t ) 1/2, Af L2(,t) δ f(τ) 2 L 2(,T ) τ t T, (3.21) So, we obtain δ = km p ε. Af L2(,T ) δ f L2(,T,L 2(,T )). (3.22) It follows from the foregoing that there exists a positive ε such that δ < 1. (3.23) Inequality (3.23) shows that the linear operator A is a contracting mapping on L 2 (, T, L 2 (, T )) an completes the proof. Regaring the unique solvability of the inverse problem, the following result will be useful. Theorem 3.4. Let conition (H1) an the compatibility conition (3.13) hol. Then the following assertions are vali: (i) a solution {z, f} of the inverse problem (3.5)-(3.9) exists an is unique, an (ii) with any initial iteration f L 2 (, T, L 2 (, T )) the successive approximations f n+1 = Ãf n. (3.24) converge to f in the L 2 (, T, L 2 (, T ))-norm (for à see below). Proof. (ii) We use the nonlinear operator à : L 2(, T ) L 2 (, T, L 2 (, T )) acting in accorance to the rule Ãf = Af + E + βe g, (3.25) the operator A an the function g arise from (3.11). From (3.24) it follows that (3.12) can be written as Hence it is sufficient to show that operator L 2 (, T, L 2 (, T )). By the relations f = Ãf. (3.26) Ãf 1 Ãf 2 = Af 1 Af 2 = A(f 1 f 2 ), à has a fixe point in the space

6 T.-E. OUSSAEIF, A. BOUZIANI EJDE-216/138 from estimate (3.22) we euce that Ãf 1 Ãf 2 L2(,T ) = A(f 1 f 2 ) L2(,T ) δ (f 1 f 2 ) L2(,T,L 2(,T )). (3.27) From (3.23) an (3.26), we fin that à is a contracting mapping on L 2(, T, L 2 (, T )). Therefore à has a unique fixe point f in L 2(, T, L 2 (, T )) an the successive approximations (3.24) converge to f in the L 2 (, T, L 2 (, T ))-norm irrespective of the initial iteration f L 2 (, T, L 2 (, T )). (i) This shows that, equation (3.26) an, in turn, equation (3.12) have a unique solution f in L 2 (, T, L 2 (, T )). Accoring to Theorem 3.2, this confirms the existence of solution to the inverse problem (3.5) (3.9). It remains to prove the uniqueness of this solution. Assume to the contrary that there were two istinct solutions {z 1, f 1 } an {z 2, f 2 } of the inverse problem uner consieration. We claim that in that case f 1 f 2 almost every on (, T ). If f 1 = f 2, then applying the uniqueness theorem to the corresponing irect problem (3.1) (3.4) we woul have z 1 = z 2 almost every in Q T. Since both pairs satisfy ientity (3.14), the functions f 1 an f 2 give two istinct solutions to equation (3.26). But this contraicts the uniqueness of the solution to equation (3.26) just establishe an proves the theorem. Corollary 3.5. Uner the conitions of Theorem 3.4, the solution f to equation (3.12) epens continuously upon the ata W. Proof. Let W an V be two sets of ata, which satisfy the assumptions of Theorem 3.4. Let f an g be solutions of the equation (3.12) corresponing to the ata W an V, respectively. Accoring to (3.12), we have f = Af + W, g = Ag + V. First, let us estimate the ifference f g. It is easy to see by using (3.22), that f g L2(,T,L 2(,T )) = (Af + W ) (Ag + V ) L2(,T ) so, we obtain The proof is complete. f g L2(,T,L 2(,T )) = A(f g) + (W V ) L2(,T ) δ f g L2(,T,L 2(,T )) + (W V ) L2(,T ); 1 (1 δ) (W V ) L 2(,T ). References [1] Bouziani, A.; Solution forte un problem mixte avec conition non locales pour uneclasse equations hyperboliques; Bull. e la Classe es Sciences, Acaemie Royale e Belgique. 1997. V. 8, pp. 53-7. [2] Cannon, J. R.; Lin, Y.; An inverse problem of fining a parameter in a semi-linear heat equation; J. Math. Anal. Appl. 199. V. 145. pp. 47-484. [3] Cannon, J. R.; Lin, Y.; Determination of a parameter p(t) in some quasilinear parabolic ifferential equations, Inverse Problems 4:1 (1988), 35-45.

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