2_759_karimov.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 6, 2010, 948-957 ISSN 1307-5543 – www.ejpam.com SPECIAL ISSUE ON COMPLEX ANALYSIS: THEORY AND APPLICATIONS DEDICATED TO PROFESSOR HARI M. SRIVASTAVA, ON THE OCCASION OF HIS 70TH BIRTHDAY Boundary-value Problems with Non-Local Initial Condition for Parabolic Equations with Parameter John Michael Rassias1, Erkinjon Tulkinovich Karimov 2,∗ 1 Pedagogical department, Mathematics and Informatics section, National and Capodistrian Uni- versity of Athens, Athens, Greece 2 Institute of Mathematics and Information Technologies, Uzbek Academy of Sciences, Tashkent, Uzbekistan Abstract. In 2002, J.M.Rassias [14] imposed and investigated the bi-parabolic elliptic bi-hyperbolic mixed type partial differential equation of second order. In the present paper some boundary-value problems with non-local initial condition for model and degenerate parabolic equations with parameter were considered. Also uniqueness theorems are proved and non-trivial solutions of certain non-local problems for forward-backward parabolic equation with parameter are investigated at specific val- ues of this parameter by employing the classical "a-b-c" method. Classical references in this field of mixed type partial differential equations are given by: J.M.Rassias [16] and M.M.Smirnov [25]. Other investigations are achieved by G.C.Wen et al. (in period 1990-2007). 2000 Mathematics Subject Classifications: 35K20, 35K65 Key Words and Phrases: Degenerate parabolic equation, forward-backward parabolic equation with a parameter, boundary-value problems with non-local initial conditions, classical "a-b-c" method. 1. Introduction Degenerate partial differential equations have numerous applications in Aerodynamics and Hydrodynamics. For example, problems for mixed subsonic and supersonic flows were considered by F.I.Frankl [2]. Reviews of interesting results on degenerated elliptic and hy- perbolic equations up to 1965, one can find in the book by M.M.Smirnov [24]. Among other ∗Corresponding author. Email addresses: jrassias�primedu.uoa.gr (J. Rassias), erkinjon�gmail. om (E. Karimov) http://www.ejpam.com 948 c© 2010 EJPAM All rights reserved. J. Rassias, E. Karimov / Eur. J. Pure Appl. Math, 3 (2010), 948-957 949 research results on this kind of equations were investigated by J.M.Rassias [14, 15, 16, 17, 18, 19, 20, 21], G.C.Wen [26, 27, 28, 7, 29], A.Hasanov [6] and references therein. Also works by M.Gevrey [4], A.Friedman [3], Yu.Gorkov [5] are well-known on construction fundamental solutions for degenerated parabolic equations. Boundary-value problems with initial non-local condition for model parabolic equations were studied by N.N.Shopolov, for example see [23]. Various non-local problems for mixed type equations containing parabolic type equation were studied by many authors, for instance, see works by Kerefov [12], Sabytov [22], Berdyshev [1], Karimov [10, 11]. However, in 2002, J.M.Rassias [14] imposed and investigated the bi-parabolic elliptic bi-hyperbolic mixed type partial differential equation of second order. In the present paper some boundary-value problems with non-local initial condition for model and degenerate parabolic equations with parameter were considered. Also unique- ness theorems are proved and non-trivial solutions of certain non-local problems for forward- backward parabolic equation with parameter are investigated at specific values of this param- eter by employing the classical "a-b-c" method. Classical references in this field of mixed type partial differential equations are given by: J.M.Rassias [16] and M.M.Smirnov [25]. Other investigations are achieved by G.C.Wen et al. (in period 1990-2007). 2. Non-local Problems for Degenerate Parabolic Equations with Parameter. Let us consider a parabolic equation ymux x − xnuy −λxn ymu = 0, (1) with two lines of degeneration in the domain Φ = �� x , y � : 0< x < 1, 0< y < 1 , where m, n> 0, λ ∈ C . The problem 1. To find a regular solution of the equation (1) satisfying boundary conditions u � 0, y � = 0, u � 1, y � = 0, 0≤ y ≤ 1, (2) and non-local initial condition u (x , 0) = αu (x , 1) , 0≤ x ≤ 1, (3) where α is non-zero real number. The following statements are true: Theorem 1. Let α ∈ [−1,0)∪ (0,1] , Reλ ≥ 0. If there exists a solution of the problem 1, then it is unique. Corollary 1. The problem 1 can have non-trivial solutions only when parameter λ lies outside of the sector ∆= {λ : Reλ ≥ 0}. These non-trivial solutions represented by upk � x , y � = Cpk � 2 n+ 2 � 1 n+2 µ 1 2(n+2) k x 1 2 I 1 n+2 � 2 p µk n+ 2 x n+2 2 � e(− ln |α|−ipπ)ym+1 , (4) J. Rassias, E. Karimov / Eur. J. Pure Appl. Math, 3 (2010), 948-957 950 where Cpk are constants, p, k are natural numbers. Eigenvalues defined as λpk = µk + (m+ 1) ln |α|+ i (m+ 1) pπ. Here µk are roots of the equation I 1 n+2 � 2 p µ n+ 2 � = 0, where Is () is the first kind modified Bessel function of s-th order. We will omit the proof, because further we consider similar problem in three-dimensional domain in a full detail. Let Ω be a simple-connected bounded domain in R3 with boundaries Si (i = 1,6). Here S1 = �� x , y, t � : t = 0, 0< x < 1, 0< y < 1 , S2 = �� x , y, t � : x = 1, 0< y < 1, 0< t < 1 , S3 = �� x , y, t � : y = 0, 0< x < 1, 0< t < 1 , S4 = �� x , y, t � : x = 0, 0< y < 1, 0< t < 1 , S5 = �� x , y, t � : y = 1, 0< x < 1, 0< t < 1 , S6 = �� x , y, t � : t = 1, 0< x < 1, 0< y < 1 . We consider the following degenerate parabolic equation xn ymut = ymux x + xnuy y −λxn ymu (5) in the domain Ω. Here m> 0, n> 0, λ= λ1 + iλ2, λ1,λ2 ∈ R. The problem 2. To find a function u � x , y, t � satisfying the following conditions: 1. u � x , y, t � ∈ C � Ω � ∩ C 2,2,1 x ,y,t (Ω); 2. u � x , y, t � satisfies the equation 5 in Ω; 3. u � x , y, t � satisfies boundary conditions u � x , y, t � ���S2∪S3∪S4∪S5 = 0 ; (6) 4. and non-local initial condition u � x , y, 0 � = αu � x , y, 1 � . (7) Here α= α1 + iα2, α1,α2 are real numbers, moreover α2 1 +α 2 2 6= 0. Theorem 2. If α2 1 +α 2 2 < 1, λ1 ≥ 0 and exists a solution of the problem 2, then it is unique. J. Rassias, E. Karimov / Eur. J. Pure Appl. Math, 3 (2010), 948-957 951 Proof. Let us suppose that the problem 2 has two u1, u2 solutions. Denoting u = u1 − u2 we claim that u≡ 0 in Ω. First we multiply equation (5) to the function u � x , y, t � , which is complex conjugate function of u � x , y, t � . Then integrate it along the domain Ωǫ with boundaries S1ǫ = �� x , y, t � : t = ǫ, ǫ < x < 1− ǫ, ǫ < y < 1− ǫ , S2ǫ = �� x , y, t � : x = 1− ǫ, ǫ < y < 1− ǫ, ǫ < t < 1− ǫ , S3ǫ = �� x , y, t � : y = ǫ, ǫ < x < 1− ǫ, ǫ < t < 1− ǫ , S4ǫ = �� x , y, t � : x = ǫ, ǫ < y < 1− ǫ, ǫ < t < 1− ǫ , S5ǫ = �� x , y, t � : y = 1− ǫ, ǫ < x < 1− ǫ, ǫ < t < 1− ǫ , S6ǫ = �� x , y, t � : t = 1− ǫ, ǫ < x < 1− ǫ, ǫ < y < 1− ǫ . Then taking real part of the obtained equality and considering Re � ymuux x � = Re � ymuux � x − ym ��ux ��2 , Re � xnuuy y � = Re � xnuuy � y − xn ��uy ��2 , Re � xn ymuut � = � 1 2 xn ym |u|2 � t , after using Green’s formula we pass to the limit at ǫ→ 0. Then we get ∫ ∂Ω ∫ Re � ymuux cos (ν , x) + xnuuy cos � ν , y �− 1 2 xn ym |u|2 cos (ν , t) � dτ = ∫ ∫ Ω ∫ � ym ��ux ��2 + xn ��uy ��2 +λ1 xn ym |u| � dσ where ν is outer normal. Considering Re � uux � = Re � uux � , Re � uuy � = Re � uuy � we obtain Re ∫ ∫ S1 1 2 xn ym |u|2 dτ1 + ∫ ∫ S2 ymRe � uux � dτ2− ∫ ∫ S3 xnRe � uuy � dτ3− ∫ ∫ S4 ymRe � uux � dτ4 + ∫ ∫ S5 xnRe � uuy � dτ5−Re ∫ ∫ S6 1 2 xn ym |u|2 dτ6 (8) = ∫ ∫ Ω ∫ � ym ��ux ��2 + xn ��uy ��2 +λ1 xn ym |u| � dσ. From (8) and by using conditions (6), (8), we find 1 2 � 1− � α2 1 +α 2 2 �� 1∫ 0 1∫ 0 xn ym ��u�x , y, 1 ���d xd y + ∫ ∫ Ω ∫ � ym ��ux ��2 + xn ��uy ��2 +λ1 xn ym |u| � dσ= 0. (9) J. Rassias, E. Karimov / Eur. J. Pure Appl. Math, 3 (2010), 948-957 952 Setting α2 1 +α 2 2 < 1, λ1 ≥ 0, from (9) we have u � x , y, t �≡ 0 in Ω. We find below non-trivial solutions of the problem 2 at some values of parameter λ for which the uniqueness condition Reλ= λ1 ≥ 0 is not fulfilled. We search the solution of Problem 2 as follows u � x , y, t � = X (x) · Y �y� · T (t) . (10) After some evaluations we obtain the following eigenvalue problems: ¨ X ′′ (x)+µ1 xnX (x) = 0 X (0) = 0, X (1) = 0; (11) ¨ Y ′′ � y � +µ2 ymY � y � = 0 Y (0) = 0, Y (1) = 0; (12) ¨ T ′ (t) + � λ+µ � T (t) = 0 T (0) = αT (1) . (13) Here µ = µ1+µ2 is a Fourier constant. Solving eigenvalue problems (11), (12) we find µ1k = � n+ 2 2 gµ1k �2 , µ2p = � m+ 2 2 gµ2p �2 , (14) Xk (x) = Ak � 2 n+ 2 � 1 n+2 µ 1 2(n+2) 1k x 1 2 J 1 n+2 � 2 p µ1k n+ 2 x n+2 2 � , (15) Yp � y � = Bp � 2 m+ 2 � 1 m+2 µ 1 2(m+2) 2p y 1 2 J 1 m+2   2 p µ2p m+ 2 x m+2 2   , (16) where k, p = 1,2, . . ., gµ1k and gµ2p are roots of equations J 1 n+2 (x) = 0 and J 1 m+2 � y � = 0, respectively. The eigenvalue problem (13) has non-trivial solution only when ¨ α1 = eλ1+µkp cosλ2 α2 = eλ1+µkp sinλ2. Here λ= λ1 + iλ2, α = α1 + iα2, µkp = µ1k +µ2p. After elementary calculations, we get λ1 = −µkp + ln Æ α2 1 +α2 2 , λ2 = arctan α2 α1 + sπ, s ∈ Z+ (17) Corresponding eigenfunctions have the form Tkp (t) = Ckpe � µkp−ln p α2 1+α 2 2−i � arctan α2 α1 +sπ �� t . (18) J. Rassias, E. Karimov / Eur. J. Pure Appl. Math, 3 (2010), 948-957 953 Considering (10), (15), (16) and (18) we can write non-trivial solutions of the problem 2 in the following form: ukp � x , y, t � = Dkp � 2 n+ 2 � 1 n+2 � 2 m+ 2 � 1 m+2 µ 1 2(n+2) 1k µ 1 2(m+2) 2p p x yJ 1 n+2 � 2 p µ1k n+ 2 x n+2 2 � × J 1 m+2   2 p µ2p m+ 2 y m+2 2   e � µkp−ln p α2 1+α 2 2−i � arctan α2 α1 +sπ �� t , where Dkp = Ak · Bp · Ckp are constants. Remark 1. One can easily see that λ1 < 0 in (17), which contradicts to condition Reλ= λ1 ≥ 0 of the theorem 2. Remark 2. The following problems can be studied by similar way. Instead of condition (6) we put conditions as follows: Problem’s name P3 P4 P5 P6 P7 P8 P9 P10 S2 ux u u ux u u ux u S3 uy u uy u uy u u u S4 u ux u ux u ux u u S5 u uy uy u u u u uy 3. Non-local Problem for "Forward-backward" Parabolic Equation with Parameter. In this section we prove the uniqueness of solution of a non-local problem for "forward- backward" parabolic equation with parameter. We have to note work by C.D.Pagani and G.Talenti [13], where boundary-value problems for equation sgn(x)uy − ux x + ku = f (x , y) were investigated. Existence theorems are proved, with an integral equations technique with the developing of Wiener-Hopf integral equations of the first kind with solutions belonging to Sobolev spaces. In the domain D = D1 ∪ D2 ∪ I0, D1 = �� x , y � :−1≤ x ≤ 0, 0≤ y ≤ 1 , I0 = �� x , y � : x = 0, 0≤ y ≤ 1 , D2 = �� x , y � : 0≤ x ≤ 1, 0≤ y ≤ 1 let us consider equa- tion Lu= λu, (19) where λ ∈ R, Lu= ux x − si gn (x)uy . The problem 3. To find a regular solution of the equation (19) from the class of functions u � x , y � ∈ C � D � ∩ C1 � D ∪ I1 ∪ I2 � , satisfying non-local conditions k1ux (−1, y) + k2u(−1, y) = k3ux (1, y), 0≤ y ≤ 1, (20) J. Rassias, E. Karimov / Eur. J. Pure Appl. Math, 3 (2010), 948-957 954 k4ux(1, y) + k5u(1, y) = k6ux(−1, y), 0≤ y ≤ 1; (21) u (x , 0) = αu (x , 1) , −1≤ x ≤ 1. (22) Here ki � i = 1,6 � , α is given non-zero constant, I1 = �� x , y � : x = −1, 0≤ y ≤ 1 , I2 = �� x , y � : x = 1, 0≤ y ≤ 1 . Note, non-local conditions (20), (21) were used for the first time by N.I.Ionkin and E.I.Moiseev [9, 8]. Theorem 3. If |α|= 1, λ > 0, k3k5 = k2k6, k1k2 < 0, k4k5 > 0 (23) and exists a solution of the problem 3, then it is unique. Proof. We multiply equation (19) to the function u � x , y � and integrate along the domains D1 and D2. Using Green’s formula and condition (22), we get 1∫ 0 u �−0, y � ux �−0, y � d y = 0∫ −1 α2 − 1 2 u2 (x , 1) d x + 1∫ 0 u �−1, y � ux �−1, y � d y + ∫ ∫ D1 � u2 x +λu2 � d xd y, 1∫ 0 u � +0, y � ux � +0, y � d y = 1∫ 0 α2 − 1 2 u2 (x , 1)d x + 1∫ 0 u � 1, y � ux � 1, y � d y − ∫ ∫ D2 � u2 x +λu2 � d xd y. From conditions (20), (21), we find u � 1, y � ux � 1, y � = k6 k5 ux �−1, y � ux � 1, y �− k4 k5 u2 x � 1, y � , u �−1, y � ux �−1, y � = k3 k2 ux �−1, y � ux � 1, y �− k1 k2 u2 x �−1, y � . Taking above identities into account, we establish 0∫ −1 α2−1 2 u2 (x , 1) d x + 1∫ 0 1−α2 2 u2 (x , 1) d x + 1∫ 0 h k4 k5 u2 x � 1, y �− k1 k2 u2 x �−1, y �i d y+ + 1∫ 0 h k3 k2 − k6 k5 i u �−1, y � ux � 1, y � d y + ∫ ∫ D1 � u2 x +λu2 � d xd y + ∫ ∫ D2 � u2 x +λu2 � d xd y. Considering condition (23), we get u � x , y �≡ 0 in D and the proof of Theorem 3 is complete. REFERENCES 955 Remark 3. By similar method one can prove the uniqueness of solution of boundary-value prob- lem with non-local initial condition for equation 0= ¨ ymux x + (−x)n uy −λ (−x)n ymu= 0, x < 0 ymux x − xnuy −λxn ymu= 0, x > 0. Open question. A question is still open, on the unique solvability of boundary value problems for the following equation: 0= ¨ ym1 (−x)n2 ux x + (−x)n1 ym2uy −λ1u= 0, x < 0 ym1 xn2ux x − xn1 ym2uy −λ2u = 0, x > 0, where λ1, λ2 are given complex numbers and mi, ni = const > 0 (i = 1,2). References [1] A.S.Berdyshev and E.T.Karimov. Some non-local problems for the parabolic-hyperbolic type equation with non-characteristic line of changing type. CEJM, 4(2): 183-193, 2006. [2] F.I.Frankl. On the problems of Chaplygin for mixed subsonic and supersonic flows. Izv.Akad. Nauk SSSR Ser. Mat., 9:121-143, 1945. [3] A.Friedman. Fundamental solutions for degenerate parabolic equations. Acta Mathemat- ica, 133:171-217, 1975. [4] M.Gevrey. Sur les equations aux derivees partielles du type parabolique. J.Math.Appl., 4:105-137, 1914. [5] Yu.P.Gorkov. Construction of a fundamental solution of parabolic equation with degen- eration. Calcul. methods and programming, 6:66-70, 2005. [6] A.Hasanov. Fundamental solutions of generalized bi-axially symmetric Helmholtz equa- tion. Complex Variables and Elliptic Equations, 52(8):673-683, 2007. [7] S.Huang, Y.Y.Qiao and G.C.Wen. Real and complex Clifford analysis. Advances in Complex Analysis and its Applications, 5. Springer, New York, 2006. [8] N.I.Ionkin. The stability of a problem in the theory of heat condition with non-classical boundary conditions. (Russian). Differencial’nye Uravnenija, 15(7):1279-1283, 1979. [9] N.I.Ionkin and E.I.Moiseev. A problem for a heat equation with two-point boundary conditions. (Russian). Differencial’nye Uravnenija 15(7):1284-1295, 1979. [10] E.T.Karimov. About the Tricomi problem for the mixed parabolic-hyperbolic type equation with complex spectral parameter. Complex Variables and Elliptic Equations, 56(6):433-440, 2005. REFERENCES 956 [11] E.T.Karimov. Some non-local problems for the parabolic-hyperbolic type equation with complex spectral parameter. Mathematische Nachrichten, 281(7):959-970, 2008. [12] A.A.Kerefov. The Gevrey problem for a certain mixed-parabolic equation. (Russian) Dif- ferencial’nye Uravnenija ,13(1):76Ű83, 1977. [13] C.D.Pagani and G.Talenti. On a forward-backward parabolic equation. Annali di Matem- atica Pura ed Applicata, 90(1):1-57, 1971. [14] J.M.Rassias. Uniqueness of Quasi-Regular Solutions for a Bi-Parabolic Elliptic Bi- Hyperbolic Tricomi Problem. Complex Variables and Elliptic Equations, 47(8):707-718, 2002. [15] J.M.Rassias. Mixed Type Partial Differential Equations in Rn, Ph.D. Thesis, University of California, Berkeley, USA, 1977. [16] J.M.Rassias. Lecture Notes on Mixed Type Partial Differential Equations. World Scientific, 1990. [17] J.M.Rassias. Mixed type partial differential equations with initial and boundary values in fluid mechanics. Int.J.Appl.Math.Stat., 13(J08):77-107, 2008. [18] J.M.Rassias, A.Hasanov. Fundamental solutions of two degenerated elliptic equa- tions and solutions of boundary value problems in infinite area. Int.J.Appl.Math.Stat., 8(M07):87-95, 2007. [19] J.M.Rassias. Tricomi-Protter problem of nD mixed type equations. Int.J.Appl.Math.Stat. 8(M07):76-86, 2007. [20] J.M.Rassias. Existence of weak solutions for a parabolic elliptic-hyperbolic Tricomi prob- lem. Tsukuba J.Math., 23(1):37-54, 1999. [21] J.M.Rassias. Uniqueness of quasi-regular solutions for a parabolic elliptic-hyperbolic Tri- comi problem. Bull.Inst.Math.Acad.Sinica, 25(4):277-287, 1997. [22] K.B.Sabitov. To the theory of mixed parabolic-hyperbolic type equations with spectral parameter. Differencial’nye Uravnenija, 25(1):117-126, 1989. [23] N.N.Shopolov. Mixed problem with non-local initial condition for a heat conduction equation. Reports of Bulgarian Academy of Sciences, 3(7):935-936, 1981. [24] M.M.Smirnov. Degenerate elliptic and hyperbolic equations. Nauka, Moscow, 1966. [25] M.M.Smirnov Equations of Mixed Type, Translations of Mathematical Monographies, 51, American Mathematical Society, Providence, R.I. pp.1-232. 1978. [26] G.C.Wen. The Exterior Tricomi Problem for Generalized Mixed Equations with Parabolic Degeneracy. Acta Mathematics Sinica, English Series, 22(5):1385-1398, 2006. REFERENCES 957 [27] G.C.Wen and D.Chen. Discontinuous Riemann-Hilbert problems for quasilinear degen- erate elliptic complex equations of first order. Complex Variables and Elliptic Equations 50(7-11):707-718, 2005. [28] G.C.Wen. The mixed boundary-value problem for second order elliptic equations with degenerate curve on the sides of an angle. Mathematische Nachrichten, 279(13- 14):1602-1613, 2006. [29] G.C.Wen and H.G.W.Begehr. Boundary value problems for elliptic equations and systems. Pitman Monographs and Surveys in Pure and Applied Mathematics, 46. Longman Scientific and Tech., Harlow; John Wiley and Sons, Inc.,N.Y., 1990.