AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation10.1155/2015/703652703652Research ArticleThe Existence and Uniqueness of a New Boundary Value Problem (Type of Problem “E”) for Linear System Equations of the Mixed Hyperbolic-Elliptic Type in the Multivariate Dimension with the Changing Time Directionhttp://orcid.org/0000-0003-1619-472XNurmammadovMahammad A.1,2WongPatricia J. Y.1Department of MathematicsAzerbaijan Architecture and Building University5 A. Sultanova Street, 1073 BakuAzerbaijanazmiu.edu.az2Department of MathematicsAzerbaijan Teachers Institute104 H. Aliyev Street, 1115 BakuAzerbaijanami.az
Dedicated to Prof. V. N. Vragov on the occasion of his 70th birthday.
The existence and uniqueness of the boundary value problem for linear systems equations of the mixed hyperbolic-elliptic type in the multivariate domain with the changing time direction are studied. Applying methods of functional analysis, “ε-regularizing” continuation by the parameter and by means of prior estimates, the existence and uniqueness of generalized and regular solutions of a boundary problem are established in a weighted Sobolev space.
1. Introduction
Up to the present, various generalizations of mixed and composite type of equations have been investigated by many authors. Interest of investigations of nonclassical equations arises in applications in the field of hydrogas dynamics, modeling of physical processes (see, e.g., [1–6] and the references given therein).
Nonclassical model is defined as the model of mathematical physics, which is represented in the form of the equation or systems of partial differential equations that does not fit into one of the classical types: elliptic, parabolic, or hyperbolic. In particular, nonclassical models are described by equations of mixed type (the Tricomi equation), degenerate equations (the Keldysh equation) or the equations of Sobolev type (or the Barenblatt-Zsolt-Kachina equation), the equation of mixed type with the changing time direction, and forward-backward equations. In [7], a fundamental theory of the solvability of many different boundary value problems for nonclassical equations was presented.
In recent years the attention of many scholars has turned to the study of well-posed boundary value problems for nonclassical equations of mathematical physics, in particular, for forward-backward equations of the parabolic type (as the changing time direction) (see, e.g., [8, 9], etc.) and for equations of mixed type with several lines of change of type (see, e.g., [8–12] and the references given therein).
In the theory of boundary value problems for degenerate equations and equations of mixed type, it is a well-known fact that the well-posedness and the class of its correctness essentially depend on the coefficient of the first-order derivative (younger member) of equations (e.g., [13–16] and the references given therein).
In [14] the new called Fichera function was introduced, in order to identify subsets of the boundary of the domain where the boundary value problem for such kind of equations is posed, where it is necessary (or not) to specify the boundary condition. A namely, the boundary conditions depends on sign of function Fichera’s Ф(x).
In [13] (see, Chapter 1, pages 191–197, and Chapter 3, pages 239–245) and [15] new boundary conditions (so-called type of problem “E” in which some part of the boundary will be exempt from the boundary conditions) were studied.
In [16–18] various Dirichlet problems which can be formulated for equations of Keldysh type, one of the two main classes of linear elliptic-hyperbolic equations, were investigated. Open boundary conditions (in which data are prescribed on only part of the boundary) and closed boundary conditions (in which data are prescribed on the entire boundary) were both considered. Emphasis is on the formulation of boundary conditions for which solutions can be shown to exist in an appropriate function space.
Boundary value problems for equations of mixed hyperbolic-elliptic type with changing time direction had been studied in detail in [10–12]. Great difficulties come into being in the investigation of systems of degenerate elliptic and hyperbolic equations.
Note that many authors studied hyperbolic-parabolic equations and system equations of degenerating elliptic type, system equations of hyperbolic-elliptic type and system equations of hyperbolic (see, e.g., [3, 5, 6, 19] and the references given therein).
In mathematical modeling, partial differential equations of the mixed type are used together with boundary conditions specifying the solution on the boundary of the domain. In some cases, classical boundary conditions cannot describe process or phenomenon precisely. Therefore, mathematical models of various physical, chemical, biological, or environmental processes often involve nonclassical conditions. Such conditions usually are identified as nonlocal boundary conditions and reflect situations when the data on the domain boundary cannot be measured directly or when the data on the boundary depend on the data inside the domain. In numerical methods for solving these equations, the problem of stability has received a great deal of importance and attention. In particular, a suitable model for analyzing the stability is provided by a proper unconditionally absolutely stable difference scheme with an unbounded operator. The method of operators as a tool for the investigation of the stability of the solution of nonlocal problems for partial differential equations of the mixed type in Hilbert and Banach spaces and of difference schemes for approximate solution of these problems has been systematically developed (see, e.g., [20–22] and the references given therein).
Finally, the problem for the system of equations of hyperbolic-elliptic type, including property of changing time direction, has not been extensively investigated. Therefore in the present paper we will study this problem.
2. Problem Statement, Notation, and Preliminaries
Let G be a bounded domain in the Euclidean space Rn of the point x=x1,…,xn, including a part of hyperplane xn=0 and with smooth boundary ∂G∈C2, G+=G∩xn>0, G-=G∩xn<0. The boundary of G+ consists of a part of hyperplane xn=0 for xn>0 and smooth surface ∂G+. Analogically, the boundary G- consists of a part of hyperplane xn=0 for xn<0 and smooth surface ∂G-. Assume that D=G×-T,T, T>0; S=∂G×-T,T, where Γ=∂D is a boundary of domain D. In the domain D consider the system of equations(1)L1u,υ=k11tutt+k2xΔxu+∑i=1nai11x,tuxillllllllllllllll+∑i=1nai21x,tυxi+b11x,tut+b12x,tυtlllllllllllllll+c11x,tu+c12x,tυ=f1x,tL2u,υ=k12tυtt+Δxυ+∑i=1nai12x,tuxillllllllllllllll+∑i=1nai22x,tυxi+b21x,tutllllllllllllllll+b22x,tυt+c21x,tu+c22x,tυllllllllllll=f2x,t,where the Δx is Laplace operator Δx=∂2/∂x12+⋯+∂2/∂xn2.
Everywhere we will assume that the coefficients of the system of (1) are sufficiently smooth. Moreover, the conditions (2)tk1it>0fort≠0,t∈-T,T,i=1,2;xnk2x<0forxn≠0,x=x1,…,xn∈G∈Rnare satisfied. As far as is known that quadratic form of the equations of system (1) changes, then this system contains portions degenerating elliptic, degenerating hyperbolic, mixed, and composite type differential equations at the same time including changing direction time of variable in the domain D. Since, couples form such view system equations in first time considered by author.
2.1. The Boundary Value Problem (So-Called Problem “E”)
Find the solution of system equations (1) in the domain D-, satisfying the conditions(3)us=0,uΓ-T+¯=0,uΓT-¯=0,(4)υS=0,υΓ-T+¯=0,υΓT-¯=0.
Remark 1.
In this situation, the Γ-T+, Γ-T-, ΓT+, and ΓT- set are carriers as boundary conditions which depending on the sign b22x,t, k1it, i=1,2, b11x,t when the 2b22-k1t(2)(t)≥δ1>0, 2b11-k1t(1)(t)≥δ>0 conditions must be satisfied everywhere in D. Thus indicated boundary value problems for the system of (1) are put in the forms (3) and (4), where some of the parts of boundary Γ=∂D will be exempt from the boundary conditions. Consequently, the use of the term type of problem “E” and setting the boundary conditions (3) and (4) correspond to and are consistent with the approach cited above.
By the symbol CL we denote a class of twice continuously differentiable functions in the closed domain D, satisfying the boundary conditions (3) and (4), by H1,L(D), H2,L(D) in Sobolev’s space with weighted spaces obtained by the class CL which is closed by the norms(5)uH1,L(D)2=∫Dut2+k2x∑i=1nuxi2+u2dD,uH2,L(D)2=∫Dutt2+k22(x)∑i=1nuxixi2+k2(x)∑i=1nuxit2llllllllllllllllllllllllll+k2x∑i=1nuxi2+ut2+u2dD,respectively. Since k2(x)≠0 for xn≠0, by Sobolev’s embedding theorems [23] the functions from the spaces H2,L(D) will satisfy the boundary conditions (3), (4).
Lemma 2.
Let the following conditions
2b11(x,t)-k1t(1)(t)≥δ>0 for t=0, x∈G,
2b22(x,t)-k1t(2)(t)≥δ1>0 for t=0, x∈G,
ai11x,t2≤M1k2(x), ai12x,t2≤M2k2(x),
∑i=1nai11-k2xi2≤M3k2(x), where M1, M2, M3, and M are sufficiently large constants,
2c22(x,t)-∑i=1nai2(2)-b22(x,t)≤0, (x,t)∈D,
ai212≤Mk2(x),
-μc22-c22t>0; λc11-c11t>0 (the constants λ>0, μ>0 are chosen)
be satisfied. Then there exist constants λ≥λ0>0, μ≥μ0>0 such that for all functions u(x,t), υ(x,t) from the class cL the following inequality holds true:(6)∫D+L1(u,υ)e-λtutdD++∫D+e-λtυtL2(u,υ)dD++∫D-eμtutL1(u,υ)dD-+∫D-eμtυtL2(u,υ)dD-≥muH1,LD2+υH1,LD2,where the constant m is not dependent on function of u(x,t) and υ(x,t).
Proof.
Let the functions u(x,t), υ(x,t)∈CL and consider the following integrals:(7)J1=∫D+L1(u,υ)e-λtutdD++∫D-eμtutL1(u,υ)dD-J2=∫D+L2(u,υ)e-λtυtdD++∫D-L2(u,υ)eμtυtdD-.After integration by parts and allowing for boundary conditions of (3), (4) and taking into account nonnegative boundary integrals we get(8)J1=∫D+L1u,υe-λtutdD++∫D-eμtutL1(u,υ)dD-≥12∫D+e-λt-λ∑i=1nk2(x)uxi2(2b11-k1t(1)(t)+λk1(1)(t))ut2llllllllllllllllllllllllllll+∑i=1n(ai1(1)uxiut+ai2(1)υxiut)llllllllllllllllllllllllllll+b12υtut-k2xi′uxiut+(λc11-c11t)u2llllllllllllllllllllllllllll+c12υut+-λ∑i=1nk2(x)uxi2dD++∫D-eμt2∑i=1n(ai1(1)uxiut+ai2(1)υxiut)(2b11-k1t(1)(t)-μk1(1)(t))ut2lllllllllllllllllllllllllll+2∑i=1n(ai1(1)uxiut+ai2(1)υxiut)lllllllllllllllllllllllllll+μ∑i=1nk2(x)υxi2+2b12υtutlllllllllllllllllllllllllll+(-μc11-c11t)u2+c12υut∑i=1n(ai1(1)uxiut+ai2(1)υxiut)dD-+∫G+k1(1)(T)e-λTut2(x,T)dx+∫G-k1(1)(-T)eμtut2(x,-T)dx-∫S+k2xe-λtux2x,tdx+∫S-k2(x)eμtux2(x,t)dx-∫G-c11(x,-T)e-μTu2(x,-T)dx+∫G+c11x,TeλTu2x,Tdx,J2=∫D+e-λtυtL2(u,υ)dD++∫D-eμtυtL2(u,υ)dD-≥12∫D+e-λt-λ∑i=1nk2(x)υxi2(2b22-k1t(2)(t)+λk1(2)(t))υt2llllllllllllllllllllllllllll+-λ∑i=1nk2(x)υxi2+(λc22+c22t)υ2llllllllllllllllllllllllllll+2∑i=1n(ai1(2)uxiυt+ai2(2)υxiυt)llllllllllllllllllllllllllll+2c21uυt+2b21utυt-λ∑i=1nk2(x)υxi2dD++∫D-eμt∑i=1n(ai1(2)uxiυt+ai2(2)υxiυt)(2b22-k1(2)(t)-μk1(2)(t))υt2lllllllllllllllllllllllllll+∑i=1n(ai1(2)uxiυt+ai2(2)υxiυt)lllllllllllllllllllllllllll+μ∑i=1nυxi2+2c21uυtlllllllllllllllllllllllllll+2b21utυt+(-μc22-c22t)υ2∑i=1n(ai1(2)uxiυt+ai2(2)υxiυt)dD-+∫G+k1(2)(T)e-λTυt2(x,T)dx-∫G-k1(2)(-T)eμTtυt2(x,-T)dx-∫G-c22(x,-T)e-μTυ2(x,-T)dx+∫G+c22x,Te-λTυ2x,Tdx.Now, using inequality of Cauchy-Bunyakovskiy, inequality of Poincare, and conditions of Lemma 2 for coefficients of system equations (1) and taking into account chosen constants λ=λ0>0, μ=μ0>0 with the fact that the coefficients k1it, i=1,2, are homogeneous on the boundaries and after summarizing estimates for J1 and J2, then obtain validity of inequality (6).
Definition 3.
One says that u(x,t) and υ(x,t) are regular solution of problem ((1)–(4)), if the functions ux,t,υ(x,t)∈H2,L(D) satisfy equation of (1) almost everywhere in domain D.
Note that for solvability of problem ((1)–(4)) the standard methods ([3, 5–9, 13, 15–19, 24], etc.) are not applicable, because the condition of Somigliana [25] is not fulfilled. We need to seek other structure steps of proof or nonclassical method for solvability of problem ((1)–(4)). For this reason, first of all, begin to formulate the theory of existence; first take the decaying system equations in the following form:(9)L1u=k11(t)utt+k2(x)Δu+∑i=1nai11uxi+b11ut+c11u=f1x,t,(10)L2υ=k12(t)υtt+Δυ+∑i=1nai22υxi+b22υt+c22υ=f2x,t.For proving solvability of problem ((9), (3)) we use the method of “ε-regularization” and it is the fact that the hyperplane xn=0 is characteristic for (9). Therefore, we can consider the boundary value problem ((9), (3)) in the following form.
2.2. Boundary Value Problem 1
Find the solution of (9) in the domain D+, satisfying the boundary conditions(11)uΓ-T+¯=0,us+=0.
2.3. Boundary Value Problem 2
Find the solution of (9) in the domain D-, satisfying the boundary conditions(12)uΓT-¯=0,us-=0.By CL′D+, CL′D- we denote a class of infinitely differentiable functions in the closed domains D+, D- satisfying the boundary conditions (11) and (12), respectively.
3. Uniqueness Solution of Problem ((1)–(4)) in Space H2,LDTheorem 4.
Let the conditions of Lemma 2 be fulfilled. Then the regular solution of the problem ((1)–(4)) is unique.
Proof.
Indeed, let u1,υ1 and u2,υ2 be two solutions of problem ((1)–(4)) which is satisfying the systems equations (1). Let u=u1-u2, υ=υ1-υ2. Then the functions u,υ will be satisfying equations L1u,υ=0 and L2u,υ=0 in the domain D. Suppose that u≠0, υ≠0 are satisfied. Let us take sequence, functions un,υn∈CL, n=1,2,…, and so forth, such that un→u in H2,LD for n→∞, υn→υ in H2,LD for n→∞. By the inequality of (6) we have(13)L1u,υL2(D)+L2u,υL2(D)≥m1uH2,LD+υH2,LD,where the constant m1 is independent of the functions ux,t and υx,t. Therefore we can assert that L1un,υn→L1u,υ, L2un,υn→L2u,υ for n→∞. By the virtue of inequality of (6) we have (14)L1un,υn,e-λtuntL2(D+)+L1un,υn,eμtuntL2(D-)+L2un,υn,e-λtυntL2(D+)+L2un,υn,eμtυntL2(D-)≥m1unH1,LD2+υnH1,LD2.Hence, passing to limit as n→∞ in last inequality, we get un→0, υn→0 in space H1,LD. On the other sides we have (15)un-uH1,L(D)≤un-uH2,L(D)⟶0,υn-υH1,L(D)≤υn-υH2,L(D)⟶0for n→∞. Hence, u≡0, υ≡0. That is proof of Theorem 4. Now, we need the proof of solvability problem ((1)–(4)).
4. The Existence of Weak (Regular) Solution of Problems ((9), (11)) and ((9), (12))Lemma 5.
Let the conditions (a)–(c), (e), and (f) of Lemma 2 be satisfied. Then for any functions ux,t∈CL′D+ux,t∈CL′D- the following inequalities (16)L1u,e-λtutL2(D+)≥m1uH1,LD+2,L1u,eμtutL2(D-)≥m2uH1,L(D-)2are valid.
Proof.
Let us consider the integrals(17)∫D+L1ue-λtutdD+=∫D+f1e-λtutdD+,∫D-L1ueμtutdD-=∫D-f1eμtutdD-.After integration by parts, allowing for boundary conditions and taking into account nonnegative boundary integrals, we get(18)L1(u),e-λtutL2(D+)≥12∫D+-λ∑i=1nk2(x)uxi2e-λt-λ∑i=1nk2(x)uxi2(2b11(x,t)-k1t(1)(t)+λk1(1)(t))ut2llllllllllllllllllllllllllll+2∑i=1nai1(1)uxiut+(λc11-c11t)u2llllllllllllllllllllllllllll+-λ∑i=1nk2(x)uxi2dD++∫G+c11(x,T)eλTu2(x,T)dx+∫G+k12Te-λTut2x,Tdx,∀ux,t∈cL′D+,L1(u),eμtutL2(D-)≥12∫D-eμt∑i=1nk2(x)uxi2(2b11(x,t)-k1t(1)(t)-μk1(1)(t))ut2lllllllllllllllllllllllllll+2∑i=1nai1(1)uxiut+(μc11-c11t)u2llllllllllllllllllllllllll+μ∑i=1nk2(x)uxi2dD--∫G-c11(x,-T)e-μTu2(x,-T)dx-∫G-k12-TeμTut2x,-Tdx,∀ux,t∈cL′D-.Hence, using Cauchy-Bunyakovskiy and Poincare inequalities and taking into account conditions (a)–(c), (e), and (f) of Lemma 2 and chosen constants λ=λ0>0, μ=μ0>0 with the fact that the coefficients k11t, k121t are homogeneous on the boundaries, then we get the truth of inequalities (16). Moreover, using Holder’s inequality we have (19)f1L2(D+)≥m1uH1,LD+2,f1L2D-≥m2uH1,LD-2,where the constants m1 and m2 are independent of the function ux,t.
Definition 6.
The function ux,t∈H2,LD+ux,t∈H2,LD- is said to be regular solution of problems ((9), (11)), and ((9), (12)) if it is generalized solution and satisfies almost everywhere (9) in domain D+ (D-).
Lemma 7.
Let the conditions of Lemma 5 be fulfilled. Then regular solution of problems ((9), (11)), and ((9), (12)) is unique.
Proof.
Lemma 7 is proved by a way similar to Lemmas 2 and 5. Since the equation of (9) is also degenerating then due to regularizing effect for (9).
In the domain D+, we have “ε-regularized” equation of mixed type(20)L1εuε=k1(1)(t)uεtt+(k2-ε)Δuε+b11uεt+∑i=1nai1(1)uεxi+c11uε=f1x,tand we state for it the boundary value problem(21)uεxn=0=0,uεS+=0,uεΓ-T+¯=0.Analogically, we will consider the following boundary value problem:(22)L1εuε=k1(1)(t)uεtt+(k2+ε)Δuε+b11uεt+∑i=1nai1(1)uεxi+c11uε=f1x,t(23)uεxn=0=0,uεS-=0,uεΓT-¯=0.Proceeding from the results of [12] we can affirm the following Remark.
Remark 8.
If the conditions of Lemmas 5 and 7 and 2b11x,t-k1t1≥δ>0x,t∈D are satisfied, then for any right-hand side f1x,t,f1tx,t∈L2D+f1x,t,f1tx,t∈L2D- there exists a unique solution of boundary value problems (20), (21), ((22), (23)) from the space W22D+W22D- and this solution allows following estimates(24)f1L2D+2+f1tL2D+2≥m3uεW22D+2,f1L2D-2+f1tL2D-2≥m4uεW22D-2,where the constants m3 and m4 are independent of the function ux,t.
Proof.
This Remark can be proves similarly to Lemmas 2 and 5 and Theorem 4.
Theorem 9 (on the solvability of problem ((9), (11))).
Let the conditions of Lemma 5 and k2xik2xj≤M1k2x, f1x,t,f1tx,t∈L2D+, and 2b11x,t-k1t1t≥δ>0, forx,t∈D+, i,j=1,2,…,n, be satisfied. Then there exists a unique regular solution of problem ((9), (11)) from the space H2,LD+.
Theorem 10 (on the solvability of problem ((9), (12))).
Let the conditions of Lemma 2 and k2xik2xj≤M1k2x,f1x,t,f1tx,t∈L2(D-), 2b11x,t-k1t1t≥δ>0, and fx,t∈D-, i,j=1,2,…,n, be satisfied. Then there exists a unique regular solution of problem ((9), (12)) from the space H2,LD-.
Proof of Theorems 9 and 10.
The following a priori estimates(25)f1L2D+≥m5∫D+uεt2+k2-ε∑i=1nuεxi2+uε2dD+f1L2D-≥m6∫D-uεt2+k2+ε∑i=1nuεxi2+uε2dD-hold for the functions uεx,t∈W22D+uεx,t∈W22D- and being the solution of boundary value problems ((20), (21)), and ((22), (23)), respectively, where the constants m5 and m6 are independent of ε and ux,t. The proof of these statements is easily obtained by integration by parts and using the Cauchy inequality. Further for obtaining the second a priori estimation we take the function ξ1t(26)ξ1t=≡1fort∈-T,-η,T2>η>0,≤1fort∈-η,-η2,≡0fort∈-η2,T.Then, we consider the function wεx,t=ξ1tuεx,t. Obviously, the function wεx,t will satisfy the equation(27)L1εwε=ξ1f1+2k11tξ1′tuεt+k11tξ1′′tuε=Fε.Hence, by virtue of inequalities (24) and (25), the set of functions Fεx,t is uniformly bounded in space L2D+. On the other side, in domain (28)Dη/2+=x∈D,-T<t<-η2,the equation L1εwε=Fε belongs to elliptical type of equation; therefore, multiplying equation of (27) by -wεtt and after integrating by parts in the domain D+, allowing boundary conditions, use the Cauchy-Bunyakovskiy inequality we get(29)FεL2D+≥m7∫D+wεtt2+wε2+wεt2+k2-ε∑i=1nwεxi2llllllllllllllllllll+k2-ε∑i=1nwεxixi2dD+,where constant m7 is independent of ε, ux,t. Now, let us consider the function ξ2t∈C∞-T,T such that ξ2t≡0 for -T<t<-2η, ξ2t≡1, -η<t<T. Since 0≤ξ2t≤1 and ξ1t+ξ2t≡1, then φεx,t=ξ2tuεx,t. It is easy to see that the functions φεx,t satisfy the equation(30)L1εφε=ξ2tf1x,t+2k1t1tξ2′tuε(x,t)+k11tξ2′′tuε=Φεx,t.Hence include the fact that the functions Фεx,t,Фεtx,t are uniformly bounded with respect to ε in the space L2D+. Therefore, we can take finite difference (31)φεh=φεx,t+h-φεx,th.It is easy to see that the functions φε(x,t) satisfy the equation (32)L1εφεh=ξ2tf1+2k11tξ2′tuεt+k11(t)ξ2′′tuε=Φεhx,t.Using the results on smoothness of the solution of problem ((20), (21)) and a priori estimates (25), (29) and passing to limit as h→0 in the obtained inequalities (33)ϕεhL2D+≥m8∫D+∑i=1nφεxhi2+k2-ε∑i=1nφεxixi2φεhh2+φε2+φεh2+k2-εllllllllllllllllll·∑i=1nφεxhi2+k2-ε∑i=1nφεxixi2dD+and establishing relation between the functions f1(x,t) and Фε(x,t) we get(34)f1L2D++f1tL2D+≥m9∫D+uεtt2+k2-ε∑i=1nuεtxi2lllllllllllllllllllllllll+uεt2+k2-ε∑i=1nuεxi2+uε2dD+,lllllllllllllllll∀uεx,t∈CL′D+∫D+uεtt2+k2-ε∑i=1nuεtxi2.From the representations of function φεx,t and from (20) by standard estimation method, we get k2-ε∑i=1nuεxixi∈L2D+. Consequently, from the uniformly bounded sequences uε can be chosen such subsequences uεk which is weakly converges to the function ux,t∈H2,L(D+). Then passing to limit as εk→0, k→∞ in the identity(35)uεk,L1εk*wεL2D+=f1,wεL2D+,lllllllllllllllllllll∀wεx,t∈C0∞D+,where L1* is an operator conjugated to the operator L1.
Hence,(36)u,L1*wL2D+=f1,wL2D+,∀wx,t∈C0∞D+.
Consequently, the function ux,t∈H2,L(D+) is generalized solution of problem ((9), (11)) whilst by virtue of conditions 2b11(x,0)-k1t(1)(0)≥δ>0, ∀x∈G this solution satisfies (9) almost everywhere for D+ and by Sobolev’s embedding theorems [23] the functions from the spaces H2,L(D) will satisfy the boundary conditions (11) almost everywhere.
In a similar way, repeating all the steps carried out for the domain D+ for D- also we can establish that problem ((9), (12)) has a generalized solution and belongs to the space H2,LD- and satisfies (9) almost everywhere for D- and boundary conditions (12).
ai11x,t2≤M1k2(x), ai12x,t2≤M2k2x, ai212≤Mk2(x)∀(x,t)∈D, ∑i=1nai11-k2xi2≤M3k2(x), where M1, M2, M3, and M are sufficiently large constants,
λc11-c11t>0 (the constants λ>0 are chosen)
be satisfied. Then for ∀f1(x,t),f1t(x,t)∈L2(D) there exists generalized solution ux,t of problems ((9), (11)) and ((9), (12)) and it belongs to the space H2,LD+(H2,LD-) and for ε=0.
5. The Theorem of Existence and Uniqueness of Strong (Regular) Solution of Problems ((9), (11)) and ((9), (12))Definition 12 (following [26]).
The function ux,t∈H1,LD+ux,t∈H1,LD- is said to be a strong solution of boundary value problems (20), (21), ((22), (23)), if there exists a sequence of functions uεn∈CL′D+uεn∈CL′D- such that equality(37)limn→∞L1un-f1x,tL2D+=limn→∞un-uH1,LD+=0is fulfilled in the domain D- as well if taken instead of the domain D+.
The following theorem on the existence of strong solution holds.
Theorem 13.
Let the conditions of Lemma 2 and (38)k2xik2xj≤M1k2x,i,j=1,…,n,2b11-k11t≥δ>0,x,t∈Dbe satisfied. Then for any function f1∈L2D+f1∈L2D- there exists a unique strong solution of boundary value problem ((9), (11)) from the space H1,LD+ (for the problem ((9), (12)) from H1,LD-).
Proof.
From Theorems 4, 9, and 10 there exists u+x,t solution of problem ((9), (11)) and u-x,t solution of problem ((9), (12)) in the domains D+ and D-, respectively, belonging, respectively, to the spaces H2,LD+ and H2,LD-. Then by the construction of such spaces there exist sequences un∈CL′D+un∈CL′D- such that(39)limn→∞un+-u+H2,L(D+)=limn→∞un--u-H2,L(D-)=0.From the obvious inequality(40)un+H2,LD+≥mL1un+L2D+,un-H2,L(D-)≥mL1un-L2D-it follows that L1un+→f1+ in L2(D+), for n→∞. L1un-→f1- in L2(D-), for n→∞. Thus, suppose that f1t+∈L2D+, f1t-∈L2D-; then regular solutions u+ and u- are strong solution. We are constructing the sequences of functions f1n+∈W21D+, f1n-∈W21D- such that f1n+→f1+ in L2(D+), f1n-→f1- in L2(D-), for n→∞. Then for the functions f1+ and f1- there exists strong solution problem of ((9), (11)) and ((9), (12)) from the spaces H2,LD+ and H2,LD-, respectively. So, by inequality of Lemma 2 we have (41)f1n+L2D+≥mun+H1,LD+,f1n-L2D-≥mun-H1,LD-.Hence, we can include that un+→u+ in H1,LD+, un-→u- in H1,LD-, for n→∞ and these functions are strong solutions of problems ((9), (11)) and ((9), (12)), respectively.
6. The Solvability of Problem ((9), (3))Remark 14 (gluing solutions in the spaces).
Let the functions u+∈Hi,LD+, u-∈Hi,LD-, i=1,2. Then the constructed function (42)ux,t=u+x,t,x,t∈D+,u-x,t,x,t∈D-will also be from the class ux,t∈Hi,LD, i=1,2.
Proof.
Consider for i=1. Let us take the functions un∈C∞D, un∈CL′D+, and un∈CL′D- such that ux,t∈L2D and also un→u+ in H1,LD+, un→u- in H1,LD-, n→∞. Then we can write (43)∫Dk2unφxndD=∫D-k2unφxndD-+∫D+k2unφxndD+,∀φ∈C0∞D.Consequently, we have (44)∫D+k2unφxndD+=-∫D+k2unxnφdD+-∫D-k2unxnφdD-and it is fulfilled in the domain D- as well if taken instead of the domain D+.
Then from the inequality of k2xn2≤Mk2 we include that the function k2unxn uniformly bounded in space L2D. Hence, it follows that u∈H1,LD. Analogically, in case of i=2 we need to consider the integral ∫Dk2unxnφxndD. Consequently, Remark 14 is proved.
Thus, we have the proof of the following theorem according essentially to a combination of the proofs of Theorems 4, 9, and 10 and Lemmas 2, 5, and 7 and Remark 14.
Note. Compare the classical conditions of “gluing solutions” in [5, 7, 9, 13, 16, 17] which belong to the boundary conditions, but in this case although the conditions of “gluing solutions” are not written together with conditions (3), (4), finally, the gluing solutions determined by (42), of problem ((1)–(4)), belong to the spaces Hi,LD, i=1,2, which are proved.
Now, we can prove the main theorem of solvability of problem ((9), (3)).
Theorem 15 (on the solvability of problem ((9), (3))).
Let any conditions of Lemmas 2 and 5 and Theorems 4, 9, 10, and 13 be satisfied. Then for any functions f1,f1t∈L2D there exists a unique generalized solution of problem ((9), (3)) from the space H2,LD.
Proof.
Since on the base of Theorems 9, 10, and 13 there exists a unique solution u+x,t, u-x,t of problems ((9), (11)) and ((9), (12)) from the spaces H2,LD+ and H2,LD-, respectively, then function ux,t which is constructed by formula (42) will also be from the class ux,t∈H2,LD and at the same time it is generalized solution of (9); moreover, the functions u+x,t and u-x,t are strong generalized solution of problem ((9), (3)).
Consequently, it means that the strong and weak solutions of corresponding problems are identity. It follows that problem ((9), (3)) is solvability. The uniqueness of problem ((9), (3)) follows by means of inequality of Lemma 2. That is proof of Theorem 4. Analogically, the existence of strong solution of problem ((9), (3)) from the space H1,LD can be proved.
7. On the Solvability of Problem ((1)–(4))
For proving the solvability of problem ((1)–(4)) we use the method of “continuation by parameter.” It holds.
Theorem 16 (on the solvability problem of ((10), (4))).
Let the conditions(45)2c22x,t-∑i=1nai22x,t-b22x,t≤-δ1<0,x,t∈D,(46)ai12x,t2≤M1k2(x)be satisfied. Then for any functions of f2(x,t)∈L2D there exists unique solution of problem ((10), (4)) in the space H2,LD (in case, instead of condition of (46), if taken smallest of coefficient ai12x,t, then there exists unique solution of problem ((10), (4)) in space W22D).
Proof.
By virtue of condition (45) and 2b22-k1t(2)≥δ1>0b22(x,0)-k1t(2)(0)≥δ1>0∀x∈G; the operator(47)L2(υ)=k1(2)(t)υtt+Δυ+∑i=1nai2(2)υxi+b22υt+c22υis coercive. Since the coefficient of k1(2)(t) is sign fixed (according to [10]), then there exists unique solution of problem ((10), (4)) in space W21D. If υ(x,t)∈W21D then (according to [23, 24]) any solution of problem ((10), (4)) will be element of space W22D. Analogically, repeating all the steps carried out for the solution υ(x,t)∈H2,LD we can also establish that problem ((10), (4)) has generalized solution if condition (45) is satisfied. Therefore Theorem 16 is proved. Now we must prove solvability of problem ((1)–(4)).
Let(48)Mu¯=Ku¯tt+∑i=1nAiu¯xi+Bu¯t+Cu¯,Nu¯=∑i=1nPiu¯xi+Qu¯t+Ru¯,whereK=k1100k12,Ai=ai1100ai22,B=b1100b22,C=k2Δ+c1100Δ+c22,Pi=0ai21ai120,Q=0b12b210,R=0c12c210,u¯=uυ,f¯=f1f2.Then the system equations (1) can be written in the form(49)Lu¯=Mu¯+Nu¯=f¯.
Theorem 17.
Let the conditions of Lemmas 2, 5, and 7 and Theorems 4, 9, 10, 13, 15, and 16 are satisfies; moreover f1,f2,f1t∈L2D, f2(x,-T)=0, ai212≤Mk2(x) are fulfilled. Then there exists a unique solution of problem ((1)–(4)) in space H2,LD. In case that ai21 is smallest then there exists a unique solution of problem ((1)–(4)) from the space H2,L(D)∩W22(D).
Proof.
Multiplying (49), by the vector η1¯=e-λtut,-υ in domain D+, η2¯=eμtut,-υ in domain D-, after integration by parts and using the Cauchy inequality, allowing for boundary condition (by analogical action to Lemmas 2, 5, and 7), we get the following estimates: (50)Lu¯L2(D)≥mu¯H1,LDorLu¯L2D≥mu¯H1,LD∩W21D.Now, let Ht,0 be the space of vector function φ¯=(φ1,φ2) such that φ1,φ1t,φ2∈L2D and φ1(x,-T)=0. The norm of space Ht,S is defined by φt,02=φ1t02+φ202.
From the results of Theorems 15 and 16, follows a prior estimates (51)u¯H2,L(D)≤pMu¯t,0oru¯H2,LD∩W22D≤pMu¯t,0,where p constant is not dependent on u¯(x,t). We must show that analogical estimates (50) and (51) can be expressed for operator Lu. Indeed, we may write Mu¯=Lu¯-Nu¯; then (52)u¯H2,L(D)≤pLu¯t,0+Nu¯t,0oru¯H2,L(D)∩W22(D)≤pLu¯t,0+Nu¯t,0is valid. Now, we consider the set of equations Lτu¯=Mu¯+τNu¯ where 0≤τ≤1. Obviously, the following a prior estimate is uniformly bounded with respect to parameter of τ:(53)u¯H2,L(D)≤c1Lτu¯t,0,where c1 is independent of parameter τ and u¯(x,t). On the other hand for τ=0 we have L0u¯=Mu¯. In this case we considered problem is solvable. Notice that if τ=1 then L1=L. Then as well as known method of continuation by parameter we can prove solvability of problem ((1)–(4)).
8. Conclusion
The existence and uniqueness of the boundary value problem for linear systems equations of the mixed hyperbolic-elliptic type in the multivariate domain with the changing time direction are studied. The existence and uniqueness of generalized and regular solutions of a boundary problem are established in a weighted Sobolev space. Moreover, applying the result of the monograph [22], difference schemes for the numerical solution of the boundary value problem for linear systems equations of the mixed hyperbolic-elliptic type in the multivariate domain with the changing time direction can be presented. Of course, the stability estimates for the solution of these difference schemes have been established without any assumptions about the grid steps.
Conflict of Interests
The author declares that there is no conflict of interests.
Acknowledgments
The author would like to thank Prof. Allaberen Ashyralyev (Fatih University, Turkey), for his helpful suggestions for the improvement of this paper. The author would like to thank the reviewer who has read the paper very attentively and gave important and essential comments which helped in improvement of this paper.
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