ON A PROBLEM OF DETERMINING THE PARAMETЕR OF A PARABOLIC EQUATION ПРО ЗАДАЧУ ВИЗНАЧЕННЯ ПАРАМЕТРА ПАРАБОЛIЧНОГО РIВНЯННЯ

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1 UDC 57.5 A. Ashyralyev (Fatih Univ., Istanbul, Turkey) ON A PROBLEM OF DETERMINING THE PARAMETЕR OF A PARABOLIC EQUATION ПРО ЗАДАЧУ ВИЗНАЧЕННЯ ПАРАМЕТРА ПАРАБОЛIЧНОГО РIВНЯННЯ The boundary-value problem of determining the parameter p of a parabolic equation v (t) + Av(t) = f(t) + p(0 t ), v(0) = ϕ, v() = ψ in arbitrary Banach space E with the strongly positive operator A is considered. The exact estimates in Hölder norms for the solution of this problem are established. In applications, exact estimates for the solution of the boundary-value problems for parabolic equations are obtained. Розглянуто крайову задачу визначення параметра p параболiчного рiвняння v (t) + Av(t) = f(t) + p(0 t ), v(0) = ϕ, v() = ψ у довiльному банаховому просторi E iз сильно додатним оператором A. Встановлено точнi за нормами Гельдера оцiнки для розв язку цiєї задачi. У застосуваннях одержано точнi оцiнки для розв язкiв крайових задач для параболiчних рiвнянь.. Introduction. Methods of solutions of the nonlocal boundary-value problems for evolution equations with a parameter have been studied extensively by many researchers (see, e.g., [ 2] and the references given therein). We consider the following local boundary-value problem for the differential equation v (t) + Av(t) = f(t) + p(0 t ), v(0) = ϕ, v() = ψ (.) in an arbitrary Banach space with linear (unbounded) operator A and an unknown parameter p. In the paper [] the solvability of the problem (.) in the space C(E) of the continuous E-valued functions ϕ(t) defined on [0, ], equipped with the norm ϕ C(E) = max 0t ϕ(t) E was studied under the necessary and sufficient conditions for the operator A. The solution depends continuously on the initial and boundary data. Namely: Theorem.. Assume that A is the generator of the analytic semigroup exp{ ta}(t 0) and all points 2πik, k Z, k 0 are not belongs to the spectrum σ(a). Let v(0) E, v() D(A) and f(t) C β (E), 0 < β. Then for the solution (v(t), p) of problem (.) in C(E) E the estimates [ p E v(0) E + v() E + Av() E + ] β f C β (E), v C(E) [ v(0) E + v() E + f C(E) ] hold, where M does not depend on β, v(0), v() and f(t). Here C β (E) is the space obtained by completion of the space of all smooth E-valued functions ϕ(t) on [0, ] in c A. ASHYRALYEV, ISSN Укр. мат. журн., 200, т. 62, 9

2 ON A PROBLEM OF DETERMINING THE PARAMETR OF A PARABOLIC EQUATION 20 the norm ϕ C β (E) = max ϕ(t) E + 0t sup 0t<t+τ ϕ(t + τ) ϕ(t) E τ β. We say (v(t), p) is the solution of the problem (.) in C 0 (E) E if the following conditions are satisfied: i) v (t), Av(t) C 0 (E), p E E, ii) (v(t), p) satisfies the equation and boundary conditions (.). Here C 0 (E), (0 γ β, 0 < β < ) is the Hölder space with weight obtained by completion of the space of all smooth E-valued functions ϕ(t) on [0, ] in the norm ϕ C 0 (E) = max ϕ(t) E + 0t sup 0t<t+τ (t + τ) γ ϕ(t + τ) ϕ(t) E τ β. In the present paper the exact estimates in Hölder norms for the solution of problem (.) are proved. In applications, exact estimates for the solution of the boundary-value problems for parabolic equations are obtained. 2. C 0 (E)-estimates for the solution of problem (.). We study the problem (.) in the spaces C 0 (E). To these spaces there correspond the spaces of traces E, which consist of the elements w E for which the following norm is finite: w = max 0z A exp{ za}w E+ (z + τ) γ A (exp{ (z + τ) A} exp{ za}) w E + sup 0z<z+τ τ β. Assume that A is the generator of the analytic semigroup exp{ ta}(t 0) with exponentially decreasing norm, when t +, i.e., the following estimates hold: exp{ ta} E E e δt, t A exp{ ta} E E, t > 0, M > 0, δ > 0. (2.) From (2.) it follows that Here T = (I exp{ A}). We have that T E E (δ). (2.2) t v(t) = exp{ ta}v(0) + p = T {Av() A exp{ A}v(0) 0 exp{ (t s)a}f(s)ds + (I exp{ ta})a p, 0 A exp{ ( s)a}f(s)ds} (2.3) for the solution of problem (.) in the space C 0 (E) (see, for example, [, 22]). ISSN Укр. мат. журн., 200, т. 62, 9

3 202 A. ASHYRALYEV Theorem 2.. Let v(0) A f(0), v() A f() E and f(t) C 0 (E), 0 γ β, 0 < β <. Then for the solution (v(t), p) of problem (.) in C 0 (E) E the estimates 0 (E) + Av p C 0 (E) + max 0t v(t) A f(t) [ v(0) A f(0) + v() A f() ] + β ( β) f C 0 (E), (2.4) A p [ v(0) A f(0) + + v() A f() ] + β ( β) f C 0 (E) hold, where M does not depend on γ, β, v(0), v() and f(t). Proof. Using formula (2.3), we can write (2.5) Av(t) f(t) = exp{ ta} (Av(0) f(0)) + exp{ ta} (f(0) f(t)) + 0 t + 0 A exp{ (t s)a} (f(s) f(t)) ds + (I exp{ ta}) p = = exp{ ta} (Av(0) f(0)) + (I exp{ ta}) p + J(t), (2.6) p = T {Av() f() exp{ A} (Av(0) f(0)) A exp{ ( s)a} (f(s) f()) ds + exp{ A} (f() f(0))} = = T {Av() f() exp{ A} (Av(0) f(0)) J()}, (2.7) where t J(t) = exp{ ta} (f(0) f(t)) + By [22], Theorems 5. and 5.2 in Chapter, 0 A exp{ (t s)a} (f(s) f(t)) ds. J C 0 (E) β ( β) f C 0 (E), (2.8) A J(t) β ( β) f C 0 (E). (2.9) From the definition of the space E and the estimate (2.) it follows that exp{ ta} ( v(0) A f(0) ) + (I exp{ ta}) A p ISSN Укр. мат. журн., 200, т. 62, 9

4 ON A PROBLEM OF DETERMINING THE PARAMETR OF A PARABOLIC EQUATION 203 exp{ ta} E E v(0) A f(0) + I exp{ ta} E E A p [ v(0) A f(0) + A p ] for all t, t [0, ]. By (2.2), (2.9), (2.0) and the triangle inequality max v(t) A f(t) 0t (2.0) [ v(0) A f(0) + A p ] + β ( β) f C 0 (E), (2.) A p T E E [ v() A f() + + exp{ A} v(0) E E A f(0) + A J() ] [ v() A f() + v(0) A f(0) + A J() ]. (2.2) Estimate for max v(t) A f(t) and estimate (2.5) are proved. Using formula 0t (2.6) and equation (.), we can write v (t) = exp{ ta} (Av(0) f(0) p) + J(t). (2.3) Applying the triangle inequality and the definition of the space C 0 (E), we obtain 0 (E) v(0) A f(0) + A p + J C 0 (E). From this estimate and (2.2), (2.2) it follows that 0 (E) [ v(0) A f(0) + + v() A f() ] + β ( β) f C 0 (E). The estimate for Av(t) p in the norm C 0 (E) follows from this estimate and the triangle inequality. Theorem 2. is proved. Remark 2.. The spaces C 0 (E) in which exact estimates has been established, depend on parameters β and γ. However, the constants in these inequalities depend only on β. Hence, we can be choose the parameter γ freely, which increases the number of spaces. With the help of A we introduce the fractional space E α (E, A), 0 < α <, consisting of all v E for which the following norms are finite: v α = sup λ α A exp{ λa}v E + v E. λ>0 3. C -estimates for the solution of problem (.). We study the problem (.) in the spaces C. To these spaces there correspond the spaces of traces ISSN Укр. мат. журн., 200, т. 62, 9

5 204 A. ASHYRALYEV E +α β, which consist of the elements w E for which the following norm is finite: w +α β = max 0z A exp{ za}w α β + (z + τ) γ A (exp{ (z + τ) A} exp{ za}) w + sup α β 0z<z+τ τ β. Theorem 3.. Let v(0) A f(0), v() A f() E +α β and f(t) C, 0 γ β α, 0 < α <. Then for the solution (v(t), p) of problem (.) in C E +α β the estimates + Av p C + max 0t [ v(0) A f(0) +α β + v(t) A f(t) + v() A f() ] + +α β α ( α) f C, (3.) A p +α β [ v(0) A f(0) +α β + + v() A f() ] + +α β α ( α) f C hold, where M does not depend on γ, β, α, v(0), v() and f(t). Proof. By [22], Theorem 5.3 in Chapter, (3.2) J C α ( α) f C, (3.3) A J(t) +α β α ( α) f C. (3.4) From the definition of the space E +α β and the estimate (2.) it follows that exp{ ta} ( v(0) A f(0) ) + (I exp{ ta}) A p +α β exp{ ta} E E v(0) A f(0) +α β + + exp{ ta} E E A p +α β [ v(0) A f(0) +α β + A p ] +α β (3.5) for all t, t [0, ]. By (2.2), (3.4), (3.5) and the triangle inequality max v(t) A f(t) 0t +α β ISSN Укр. мат. журн., 200, т. 62, 9

6 ON A PROBLEM OF DETERMINING THE PARAMETR OF A PARABOLIC EQUATION 205 [ v(0) A f(0) + A p ] + +α β +α β α ( α) f C, (3.6) A p [ v() +α β T E E A f() +α β + + exp{ A} E E v(0) A f(0) +α β + A J() ] +α β [ v() A f() +α β + v(0) A f(0) +α β + A J() ]. +α β (3.7) Estimate for max 0t v(t) A f(t) +α β and estimate (3.2) are proved. Applying (2.3), the triangle inequality and the definition of the space C, we obtain v(0) A f(0) + A p + J C. From this estimate and (3.7), (3.7) it follows that [ v(0) A f(0) +α β + + v() A f() ] + +α β α ( α) f C. The estimate for Av(t) p in the norm C follows from this estimate and the triangle inequality. Theorem 3. is proved. Note that applying the definition of the space E +α β, we can obtain w +α β Aw E α γ, Aw E α γ. We have not been able to establish the opposite inequality necessary for the equivalence of norms. Nevertheless, we have the following result. Theorem 3.2. Let v(0) A f(0), v() A f() E α γ and f(t) C, 0 γ β α, 0 < α <. Then for the solution (v(t), p) of problem (.) in C E α γ the estimates + Av p C + max 0t Av(t) f(t) α γ [ ] Av(0) f(0) α γ + Av() f() α γ + α ( α) f C, (3.8) p α γ [ Av(0) f(0) α γ + + Av() f() α γ + α ( α) f C hold, where M does not depend on γ, β, α, v(0), v() and f(t). ] (3.9) ISSN Укр. мат. журн., 200, т. 62, 9

7 206 A. ASHYRALYEV Remark 3.. The spaces C in which exact estimates has been established, depend on parameters α, β and γ. However, the constants in these inequalities depend only on α. Hence, we can be choose parameters β, γ freely, which increases the number of spaces. In particular, Theorems 3. and 3.2 imply the well-posedness theorem in C(E α ). Remark 3.2. Theorems 2. and 3., 3.2 hold for the following boundary-value problems: v (t) + Av(t) = f(t) + p(0 t ), v(0) = ϕ, v(λ) = ψ, 0 < λ, v (t) Av(t) = f(t) + p(0 t ), v() = ϕ, v(λ) = ψ, 0 λ < in an arbitrary Banach space with positive operator A and an unknown parameter p. Applications. First, the boundary-value problem on the range {0 t, x R n } for the 2m-order multidimensional parabolic equation is considered: v(t, x) t + r =2m r v(t, x) a r (x) x r... + σv(t, x) = f(t, x) + p(x), 0 < t <, xrn n r =2m r =2m a r (x) r v(0, x) x r n... xrn a r (x) r v(, x) x r r = r r n, n... xrn + σv(0, x) = f(0, x), x R n, + σv(, x) = f(, x), x R n, (4.) where a r (x) and f(t, x) are given as sufficiently smooth functions. Here, σ is a sufficiently large positive constant. It is assumed that the symbol B x (ξ) = r =2m of the differential operator of the form a r (x) (iξ ) r... (iξ n ) rn, ξ = (ξ,..., ξ n ) R n B x = r =2m a r (x) r x r... xrn n (4.2) acting on functions defined on the space R n, satisfies the inequalities 0 < M ξ 2m ( ) m B x (ξ) 2 ξ 2m < for ξ 0. The problem (4.) has a unique smooth solution. This allows us to reduce the problem (4.) to the problem (.) in a Banach space E = C µ (R n ) of all continuous bounded functions defined on R n satisfying a Hölder condition with the indicator µ (0, ) with a strongly positive operator A = B x + σi defined by (4.2) (see [25] and [26]). ISSN Укр. мат. журн., 200, т. 62, 9

8 ON A PROBLEM OF DETERMINING THE PARAMETR OF A PARABOLIC EQUATION 207 For the solution of the boundary problem (4.) the following esti- Theorem 4.. mates are satisfied: v C + 0 (C µ (R n )) (µ) β( β) f C 0 (C µ (R n )), 0 γ β <, 0 < µ <, v C + 0 (C 2m(α β) (R n )) (α, β) f C 0 (C 2m(α β) (R n )), p C 2m(α γ) (R n ) (α, β, γ) f C 0 (C 2m(α β) (R n )), 0 γ β, 0 < 2m(α β) <, where M(µ), M(α, β) and M(α, β, γ) does not depend on f(t, x). The proof of Theorem 4. is based on the abstract Theorems 2., 3.2 and on the following theorem on the structure of the fractional spaces E α (A, C µ (R n )). Theorem 4.2. E α (A, C µ (R n )) = C 2mα+µ (R n ) for all 0 < α < 2m, 0 < µ < < [22]. Second, let Ω be the unit open cube in the n-dimensional Euclidean space R n, 0 < x k <, k n, with boundary S, Ω = Ω S. In [0, ] Ω we consider the mixed boundary-value problem for the multidimensional parabolic equation v(t, x) t n r= α r (x) 2 v(t, x) x 2 r x = (x,..., x n ) Ω, 0 < t <, + σv(t, x) = f(t, x) + p(x), n r= α r (x) 2 v(0, x) x 2 r + σv(0, x) = f(0, x), x Ω, (4.3) n r= α r (x) 2 v(, x) x 2 r v(t, x) = 0, x S, + σv(, x) = f(, x), x Ω, where α r (x) (x Ω) and f(t, x) (t (0, ), x Ω) are given smooth functions and α r (x) a > 0. Here, σ is a sufficiently large positive constant. We introduce the Banach spaces C β 0 (Ω), β = (β,..., β n ), 0 < x k <, k = =,..., n, of all continuous functions satisfying a Hölder condition with the indicator β = (β,..., β n ), β k (0, ), k n and with weight x β k k ( x k h k ) β k, 0 x k < x k + h k, k n, which equipped with the norm f C β 0 (Ω) = f C(Ω) + sup f(x,..., x n ) 0x k <x k +h k, kn f(x + h,..., x n + h n ) n k= h β k k x β k k ( x k h k ) β k, ISSN Укр. мат. журн., 200, т. 62, 9

9 208 A. ASHYRALYEV where C(Ω)-is the space of the all continuous functions defined on Ω, equipped with the norm f C(Ω) = max f(x). x Ω It is known that the differential expression [24] Av = n r= α r (t, x) 2 v(t, x) x 2 + δv(t, x) defines a positive operator A acting on C β 0 (Ω) with domain D(A) C2+β 0 (Ω) and satisfying the condition v = 0 on S. Therefore, we can replace the mixed problem (4.3) by the abstract boundary problem (.). Using the results of Theorem 2., we can obtain that the following theorem. Theorem 4.3. For the solution of the mixed boundary-value problem (4.3) the following estimate is valid: M(µ) v C + 0 (C µ 0 (Ω))) β( β) f C 0 (C µ (Ω))), 0 0 γ β <, µ = {µ,..., µ n }, 0 < µ k <, k n, where M(µ) is independent of β, γ and f(t, x). Third, we consider the mixed boundary-value problem for parabolic equation v(t, x) t a(x) 2 v(t, x) x 2 + σv(t, x) = f(t, x) + p(x), 0 < t <, 0 < x <, a(x) 2 v(0, x) x 2 + σv(0, x) = f(0, x), 0 x, a(x) 2 v(, x) x 2 + σv(, x) = f(, x), 0 x, (4.4) u(t, 0) = u(t, ), u x (t, 0) = u x (t, ), 0 t, where a(t, x) and f(t, x) are given sufficiently smooth functions and a(t, x) a > 0. Here, σ is a sufficiently large positive constant. We introduce the Banach spaces C β [0, ], 0 < β < of all continuous functions ϕ(x) satisfying a Hölder condition for which the following norms are finite ϕ Cβ [0,] = ϕ C[0,] + sup 0x<x+τ ϕ(x + τ) ϕ(x) τ β, where C[0, ] is the space of the all continuous functions ϕ(x) defined on [0,] with the usual norm ϕ C[0,] = max 0x ϕ(x). It is known that the differential expression [23] ISSN Укр. мат. журн., 200, т. 62, 9

10 ON A PROBLEM OF DETERMINING THE PARAMETR OF A PARABOLIC EQUATION 209 Av = a(x)v (x) + δv(x) define a positive operator A acting in C β [0, ] with domain C β+2 [0, ] and satisfying the conditions v(0) = v(), v x (0) = v x (). Therefore, we can replace the mixed problem (4.4) by the abstract boundary value problem (4.4). Using the results of Theorems 2., 3.2, we can obtain that Theorem 4.4. For the solution of the mixed problem (4.4) the following estimates are valid: v C + 0 (C µ [0,]) (µ) β( β) f C 0 (C µ [0,]), 0 γ β <, 0 < µ <, v C + 0 (C 2(α β) [0,]) (α, β) f C 0 (C 2(α β) [0,]), p C 2(α γ) [0,] (α, β, γ) f C 0 (C 2(α β) [0,]), 0 γ β, 0 < 2m(α β) <, where M(µ), M(α, β) and M(α, β, γ) does not depend on f(t, x). The proof of Theorem 4.4 is based on the abstract Theorems 2., 3.2 and on the following theorem on the structure of the fractional spaces E α (A, C[0, ]). Theorem 4.5 [23]. E α (A, C[0, ]) = C 2α [0, ] for all 0 < α < 2. Acknowledgement. The author would like to thank Prof. Pavel Sobolevskii (Jerusalem, Israel), for his helpful suggestions to the improvement of this paper.. Eidelman Yu. S. Boundary value problems for differential equations with parameters: PhD Thesis (in Russian). Voronezh, Eidelman Yu. S. The boundary value problem for differential equations with a parameter // Differents. Uravneniya P Eidelman Yu. S. Two-point boundary value problem for differential equations with a parameter // Dop. Akad. Nauk Ukr. RSR. Ser. A. 983, 4. S Prilepko A. I. Inverse problems of potential theory // Mat. Zametki S Iskenderov A. D., Tagiev R. G. The inverse problem of determining the right-hand sides of evolution equations in Banach space // Nauchn. Trudy Azerbaidzhan. Gos. Univ S Rundell W. Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data // Applicable Anal P Prilepko A. I., Vasin I. A. Some time-dependent inverse problems of hydrodynamics with final observation // Dokl. Akad. Nauk SSSR S Prilepko A. I., Kostin A. B. On certain inverse problems for parabolic equations with final and integral observation // Mat. Sb , Prilepko A. I. and Tikhonov I. V. Uniqueness of the solution of an inverse problem for an evolution equation and applications to the transfer equation // Mat. Zametki , 2. S D. G. Orlovski On a problem of determining the parameter of an evolution equation // Differents. Uravneniya S Choulli M. and Yamamoto M. Generic well-posedness of an inverse parabolic problem- the Hölder-space approach // Inverse Problems P Choulli M. and Yamamoto M. Generic well-posedness of a linear inverse parabolic problem with diffusion parameters // J. Inv. Ill-Posed Problems , 3. P Eidelman Yu. S. Conditions for the solvability of inverse problems for evolution equations // Dokl. Akad. Nauk Ukr. SSR Ser. A S ISSN Укр. мат. журн., 200, т. 62, 9

11 20 A. ASHYRALYEV 4. Clement Ph., Guerre-Delabrire S. On the regularity of abstract Cauchy problems and boundary value problems // Atti Accad. Naz. Lincei. Cl. sci. Fis., mate. natur. Rend Sez. 9. 9, 4. P Agarwal R., Bohner M., Shakhmurov V. B. Maximal regular boundary value problems in Banach-valued weighted spaces // Boundary-Value Problems P Gorbachuk V. I., Gorbachuk M. L. Boundary value problems for differential-operator equations (in Russian). Kiev: Naukova Dumka, Lunardi A. Analytic semigroups and optimal regularity in parabolic problems // Oper. Theory: Adv. and Appl. Basel etc.: Birkhäuser, Ashyralyev A. Well-posedness of the boundary value problem for parabolic equations in difference analogues of spaces of smooth functions // Math. Probl. Eng. 6 p., Ashyralyev A. O., Sobolevskii P. E. The linear operator interpolation theory and the stability of differenceschemes // Dokl. Akad. Nauk SSSR , 6. S Ashyralyev A., Karatay I., Sobolevskii P. E. Well-posedness of the nonlocal boundary value problem for parabolic difference equations // Discrete Dynam. Nature and Soc , 2. P Ashyralyev A., Hanalyev A., Sobolevskii P. E. Coercive solvability of nonlocal boundary value problem for parabolic equations // Abstract and Applied Analysis ,. P Ashyralyev A., Sobolevskii P. E. Well-posedness of parabolic difference equations // Oper. Theory: Adv. and Appl. Basel etc.: Birkhäuser, Ashyralyev A. Fractional spaces generated by the positivite differential and difference operator in a Banach space // Proc. Conf. Math. Methods and Engineering, Kenan Taş et al. Netherlands: Springer, P Sobolevskii P. E. The coercive solvability of difference equations // Dokl. Acad. Nauk SSSR , 5. S Smirnitskii Yu. A., Sobolevskii P. E. Positivity of multidimensional difference operators in the C-norm // Uspechi. Mat. Nauk , 4. S Smirnitskii Yu. A. Fractional powers of elliptic difference operators: PhD Thesis (in Russian). Voronezh, 983). Received ISSN Укр. мат. журн., 200, т. 62, 9