# On Solvability of Third-Order Operator Differential Equation with Parabolic Principal Part in Weighted Space

Authors: Araz R. Aliev; Sabir S. Mirzoev; Mustafa A. Soylemezo
Journal: Journal of Function Spaces
 (2017)

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## Abstract

Sufficient conditions are found for the correct and unique solvability of a class of third-order parabolic operator differential equations, whose principal parts have multiple characteristics, in a Sobolev-type space with exponential weight. The estimates for the norms of intermediate derivative operators are obtained and the relationship between these estimates and solvability conditions is established. Besides, the connection is found between the order of exponential weight and the lower bound for the spectrum of abstract operator appearing in the principal part of the equation.


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## Body

## 1. Problem Statement

LetH be a separable Hilbert space with the scalar product x,y, x,y∈H, and let A be a self-adjoint positive definite operator in H (A=A∗≥cE, c>0, E is a unit operator). Denote by Hγ (γ≥0) a scale of Hilbert spaces generated by the operator A; that is, Hγ=Dom⁡(Aγ), (x,y)γ=(Aγx,Aγy), x,y∈Dom⁡Aγ. For γ=0, we assume that H0=H, (x,y)0=(x,y), x,y∈H.Denote byL2R;H (R=-∞,+∞) a space of measurable (see [1]) functions with the values in H equipped with the norm(1)gL2R;H=∫-∞+∞gt2dt1/2and by W23R;H a space of functions with the values in H such that d3vt/dt3, A3vt∈L2R;H, equipped with the norm(2)vW23R;H=d3vdt3L2R;H2+A3vL2R;H21/2.For more details on the space W23R;H see [2, Chapter 1].Note that throughout this paper all the derivatives are understood in the sense of the theory of distributions, and the operatorAγ is defined by the spectral decomposition of the operator A; that is, Aγ=∫c+∞σγdEσ,γ≥0, where Eσ is a decomposition of the unit of the operator A.Now, let us recall one fact related to the spaceW23R;H.It is known that ifvt∈W23R;H, then the inequalities(3)A3-jdjvdtjL2R;H≤cjvW23R;H,j=0,3¯,are valid, where cj, j=0,3¯, are constants independent of function v(t). This fact is referred to as the intermediate derivatives theorem (see [2, Chapter 1]). Also, these inequalities are usually referred to as Kolmogorov-type inequalities.Let-∞<κ<+∞. For the functions ut defined on R with the values in H, we introduce the following spaces with the weight e-κ/2t:(4)L2,κR;H=ut:uL2,κR;H=∫-∞+∞utH2e-κtdt1/2<+∞,W2,κ3R;H=ut:uW2,κ3R;H=∫-∞+∞d3utdt3H2+A3utH2e-κtdt1/2<+∞.Obviously, in case κ=0 we get the spaces L2,0R;H=L2R;H, W2,03R;H=W23R;H.In the sequel, byLX,Y we will mean a set of linear bounded operators from the Hilbert space X to another Hilbert space Y. If Y=X, we will write L(X) instead of L(X,Y). By σ(A) we will denote the spectrum of the operator A.Consider the operator differential equation(5)ddt+A3ut+A1d2utdt2+A2dutdt=ft,t∈R,where A=A∗≥cE, c>0, A1, A2 are linear and, in general, unbounded operators, ft∈L2,κR;H, ut∈W2,κ3R;H.Definition 1.
If for everyf(t)∈L2,κ(R;H) there exists a vector function u(t)∈W2,κ3(R;H) which satisfies (5) almost everywhere with(6)uW2,κ3R;H≤const⁡fL2,κR;H,then this vector function is called aregular solution of (5), and (5) is said to beregularly solvable.The principal part of (5) has multiple characteristics, so, according to the classification of [3], this equation belongs to the class of parabolic operator differential equations. The equations of form (5) characterize the problems of diffusion or heat conductivity in viscoelastic media [4]. Besides, such equations are also interesting in view of the fact that some classes of equations, which can be useful in modeling the problems of world population growth, can be reduced to them [5].Note that the solvability issues for the operator differential equations in the spaces without weights have been studied quite widely (see Krein [6], Lions and Magenes [2], V. I. Gorbachuk and M. L. Gorbachuk [7], Goldstein [8], Yakubov [9, 10], S. Yakubov and Y. Yakubov [11], and the references therein). Fundamental contribution to this field was made by Gasymov in [12–14]. These issues remain as relevant today as they were long before. A lot of researches appeared in this field over the last years; see, for example, Aliev [15], Aliev and Mirzoev [16], Favini et al. [17], Favini and Yakubov [18, 19], Favini et al. [20], Mirzoev et al. [21, 22], Aliev and Yakubov [23, 24], and Mirzoev et al. [25].Despite the number of works covering the solvability of operator differential equations in weighted spaces being relatively small (see, e.g., Dubinskiĭ [3], Shkalikov [26], Mirzoev [27], and Aliev [28]), recently a lot of works appeared dedicated to the issues of regular and normal solvability in weighted spaces for operator differential equations with multiple characteristics (see Aliev [29], Mirzoev and Humbataliev [30], Aliev and Elbably [31], and Aliev and Lachinova [32]). But the equations considered in the above-mentioned works, except for [32], belong to the class of quasi-elliptic operator differential equations [3]. In [32], the initial-boundary value problem was studied for parabolic equation on the half-axis R+=0,+∞.In this work, we find the regular solvability conditions for (5) on the entire axis. We also obtain the estimates for the norms of intermediate derivative operators in a Sobolev-type space through the norm of the operator generated by the principal part of (5) (it should be noted here that the estimates for the norms of intermediate derivatives for scalar functions have been obtained in [33, 34] and the references therein). Moreover, we establish the relationship between these estimates and the regular solvability conditions for (5). The found regular solvability conditions are expressed in terms of operator coefficients of (5), which makes them easily verifiable and convenient for use both in theoretical problems and in applications.


## 2. Solvability of (5) in W2,κ3R;H When Aj=0, j=1,2

We first consider (5) in case A1=A2=0.Denote byP0 the operator acting from W2,κ3R;H to L2,κR;H as follows:(7)P0ut≡ddt+A3ut,ut∈W23R;H.The following theorem is true.Theorem 2.
LetA be a self-adjoint positive definite operator with the lower bound for its spectrum λ0(A=A∗≥λ0E,λ0>0) and κ>-2λ0. Then the operator P0 performs an isomorphism between the spaces W2,κ3(R;H) and L2,κ(R;H).Proof.
Let us make a changeu(t)=v(t)eκ/2t in the equation(8)P0ut=ft,u(t)∈W2,κ3(R;H), f(t)∈L2,κ(R;H). Then v(t)=u(t)e-κ/2t∈W23(R;H). As(9)ddt+A3ut=eκ/2tddt+κ2+A3vt=ft,we have(10)ddt+κ2+A3vt=fte-κ/2t.Taking into account the fact that g(t)=f(t)e-κ/2t∈L2R;H, we can rewrite (10) in the form(11)ddt+κ2+A3vt=gtin the space L2R;H; that is, v(t)∈W23(R;H), g(t)∈L2(R;H).
Denote(12)P0,κvt=ddt+κ2+A3vt,vt∈W23R;H.Then (11) can be rewritten as follows:(13)P0,κvt=gt,where vt∈W23R;H, gt∈L2R;H. To solve the last equation, we make use of the Fourier transform:(14)iξ+κ2E+A3v^ξ=g^ξ,where v^ξ, g^ξ are Fourier transforms of the vector functions v(t), g(t), respectively. Let us show that for κ>-2λ0 the operator pencil(15)P0,κiξ;A=iξ+κ2E+A3is invertible. In fact, let λ∈σ(A)λ≥λ0. Then the characteristic polynomial (15) has the form(16)P0,κiξ;λ=iξ+κ2+λ3.Hence we have(17)P0,κiξ;λ=iξ+κ2+λ3=λ+κ22+ξ23/2≥λ+κ23≥λ0+κ23>0,ξ∈R,that is, the spectral decomposition of the operator A implies the invertibility of the operator pencil P0,κ(iξ;A) for κ>-2λ0. Consequently, from (14) we can find v^(ξ):(18)v^ξ=iξ+κ2E+A-3g^ξ.Thus,(19)vt=12π∫-∞+∞iξ+κ2E+A-3g^ξeiξtdξ.
It is clear thatv(t) satisfies (11) almost everywhere.
Let us prove thatvt∈W23R;H. In fact, by virtue of well-known Plancherel theorem, it suffices to show that A3v^ξ∈L2R;H and -iξ3v^(ξ)∈L2R;H. Obviously,(20)vW23R;H2=d3vdt3L2R;H2+A3vL2R;H2=-iξ3v^ξL2R;H2+A3v^ξL2R;H2.As(21)A3v^ξL2R;H=A3iξ+κ2E+A-3g^ξL2R;H≤supξ∈R⁡A3iξ+κ2E+A-3H→Hg^ξL2R;H,let us estimate the norm A3iξ+κ/2E+A-3H→H  for ξ∈R. Spectral theory of self-adjoint operators implies(22)A3iξ+κ2E+A-3H→H=supλ∈σA⁡λ3iξ+κ2+λ-3=supλ∈σA⁡λ3λ+κ/22+ξ23/2≤supλ∈σA⁡λ3λ+κ/23≤max⁡λ03λ0+κ/23,1.Consequently,(23)A3iξ+κ2E+A-3g^ξL2R;H≤max⁡λ03λ0+κ/23,1g^ξL2R;H.Similarly we have(24)-iξ3v^ξL2R;H=-iξ3iξ+κ2E+A-3g^ξL2R;H≤supξ∈R-iξ3iξ+κ2E+A-3H→Hg^ξL2R;H.Hence, for ξ∈R and κ>-2λ0 we obtain(25)-iξ3iξ+κ2E+A-3H→H=supλ∈σA⁡-iξ3iξ+κ2+λ-3=supλ∈σA⁡ξ3λ+κ/22+ξ23/2≤ξ3λ0+κ/22+ξ23/2≤1.Consequently,(26)-iξ3v^ξL2R;H≤g^ξL2R;H.
Thus,vt∈W23(R;H).
It is clear that the vector functionv(t)eκ/2t belongs to W2,κ3R;H and is a regular solution of (8).
It is also clear that the equationP0ut=0 has only trivial solution in the space W2,κ3R;H.
Now let us show thatP0 is a bounded operator from the space W2,κ3(R;H) to the space L2,κ(R;H). In fact, by virtue of Cauchy-Schwarz and Young inequalities, for ut∈W2,κ3R;H, we have(27)Red3udt3,Ad2udt2L2,κR;H≤d3udt3L2,κR;HAd2udt2L2,κR;H≤12d3udt3L2,κR;H2+12Ad2udt2L2,κR;H2,Red3udt3,A2dudtL2,κR;H≤d3udt3L2,κR;HA2dudtL2,κR;H≤12d3udt3L2,κR;H2+12A2dudtL2,κR;H2,Red3udt3,A3uL2,κR;H≤d3udt3L2,κR;HA3uL2,κR;H≤12d3udt3L2,κR;H2+12A3uL2,κR;H2,ReAd2udt2,A2dudtL2,κR;H≤Ad2udt2L2,κR;HA2dudtL2,κR;H≤12Ad2udt2L2,κR;H2+12A2dudtL2,κR;H2,ReAd2udt2,A3uL2,κR;H≤Ad2udt2L2,κR;HA3uL2,κR;H≤12Ad2udt2L2,κR;H2+12A3uL2,κR;H2,ReA2dudt,A3uL2,κR;H≤A2dudtL2,κR;HA3uL2,κR;H≤12A2dudtL2,κR;H2+12A3uL2,κR;H2.Then, taking into account intermediate derivatives theorem [2, Chapter 1], we get(28)P0uL2,κR;H2=d3udt3L2,κR;H2+9Ad2udt2L2,κR;H2+9A2dudtL2,κR;H2+A3uL2,κR;H2+6Red3udt3,Ad2udt2L2,κR;H+6Red3udt3,A2dudtL2,κR;H+2Red3udt3,A3uL2,κR;H+18ReAd2udt2,A2dudtL2,κR;H+6ReAd2udt2,A3uL2,κR;H+6ReA2dudt,A3uL2,κR;H≤8d3udt3L2,κR;H2+24Ad2udt2L2,κR;H2+24A2dudtL2,κR;H2+8A3uL2,κR;H2=8uW2,κ3R;H2+3Ad2udt2L2,κR;H2+3A2dudtL2,κR;H2≤const⁡uW2,κ3R;H2;that is,(29)P0uL2,κR;H≤const⁡uW2,κ3R;H.
So we obtain that the operatorP0:W2,κ3R;H→L2,κ(R;H) is one-to-one and bounded. Then, by the Banach theorem on inverse operator, the operator P0-1:L2,κR;H→W2,κ3(R;H) is bounded.
Thus,P0 is an isomorphism between the spaces W2,κ3(R;H) and L2,κ(R;H). Theorem is proved.Theorem2 has the following.Corollary 3.
Whenκ>-2λ0, the norms P0uL2,κ(R;H) and uW2,κ3(R;H) are equivalent in the space W2,κ3(R;H).Remark 4.
Forκ=-2λ0, the operator P0 is noninvertible.
As is known, the intermediate derivative operators(30)Ajd3-jdt3-j:W2,κ3R;H⟶L2,κR;H,j=1,2,are continuous [2]. By virtue of this fact and Corollary 3, the norms of the operators (30) can be estimated through P0uL2,κ(R;H).Theorem 5.
LetA be a self-adjoint positive definite operator with the lower bound for its spectrum λ0(A=A∗≥λ0E,λ0>0) and κ>-2λ0. Then, for every ut∈W2,κ3(R;H) the following inequalities are true:(31)Ajd3-judt3-jL2,κR;H≤cjκ;λ0P0uL2,κR;H,j=1,2,where(32)c1κ;λ0=2λ0κ22λ0+κ3,-2λ0<κ≤-λ0,233/21+κ/λ01/2,κ>-λ0,c2κ;λ0=4λ02κ2λ0+κ3,-2λ0<κ≤-λ0,233/21+κ/λ0,κ>-λ0.Proof.
As the mappingvt→ute-κ/2t is an isomorphism between the spaces W23R;H and W2,κ3(R;H), to prove inequalities (31) it suffices to estimate the norms Ajd/dt+κ/23-jvL2(R;H), j=1,2, by P0,κvL2(R;H). Then, making a change P0,κv(t)=g(t) and applying Fourier transform, we obtain(33)Ajiξ+κ23-jP0,κ-1iξ;Ag^ξL2R;H≤supξ∈RAjiξ+κ23-jiξ+κ2E+A-3H→Hg^ξL2R;H,j=1,2.Consequently, for ξ∈R and κ>-2λ0, we need to estimate the norms(34)Ajiξ+κ23-jiξ+κ2E+A-3H→H=supλ∈σA⁡λjiξ+κ/23-jiξ+κ/2+λ3=supλ∈σA⁡λjξ2+κ2/43-j/2λ+κ/22+ξ23/2,j=1,2.
Consider the casej=1. Solving extremal problem, we have(35)supλ∈σA⁡λξ2+κ2/4λ+κ/22+ξ23/2≤supλ≥λ0,ξ∈R⁡ξ2/λ2+κ2/4λ021+κ/2λ02+ξ2/λ23/2=supξ2/λ2≥0⁡ξ2/λ2+κ2/4λ021+κ/2λ02+ξ2/λ23/2=c1κ;λ0,where c1κ;λ0=2λ0κ2/2λ0+κ3 if -2λ0<κ≤-λ0, and c1κ;λ0=2/33/21+κ/λ01/2 if κ>-λ0.
Forj=2, we have(36)supλ∈σA⁡λ2ξ2+κ2/41/2λ+κ/22+ξ23/2≤supλ≥λ0,ξ∈R⁡ξ2/λ2+κ2/4λ021/21+κ/2λ02+ξ2/λ23/2=supξ2/λ2≥0⁡ξ2/λ2+κ2/4λ021/21+κ/2λ02+ξ2/λ23/2=c2κ;λ0,where c2κ;λ0=4λ02κ/2λ0+κ3 if -2λ0<κ≤-λ0, and c2(κ;λ0)=2/33/21+κ/λ0 if κ>-λ0.
Thus, considering the obtained estimates in inequalities (33), we have(37)Ajiξ+κ23-jP0,κ-1iξ;Ag^ξL2R;H≤cjκ;λ0g^ξL2R;H,j=1,2.
The latter inequalities, in turn, are equivalent to the following ones:(38)Ajddt+κ23-jvL2R;H≤cjκ;λ0P0,κvL2R;H,j=1,2.Theorem is proved.


## 3. Solvability of (5) in W2,κ3(R;H) When Aj≠0, j=1,2

Denote byP1 the operator acting from W2,κ3(R;H) to L2,κ(R;H) as follows:(39)P1ut≡A1d2utdt2+A2dutdt,ut∈W2,κ3R;H.The following lemma is true.Lemma 6.
LetA be a self-adjoint positive definite operator in H and let AjA-j∈LH, j=1,2. Then the operator P1 is bounded from W2,κ3R;H to L2,κ(R;H).Proof.
By the conditions of lemma, forut∈W2,κ3R;H, we have(40)P1uL2,κR;H=A1d2udt2+A2dudtL2,κR;H≤A1d2udt2L2,κR;H+A2dudtL2,κR;H≤A1A-1H→HAd2udt2L2,κR;H+A2A-2H→HA2dudtL2,κR;H.Using intermediate derivatives theorem [2, Chapter 1], we get(41)P1uL2,κR;H≤const⁡uW2,κ3R;H.Lemma is proved.The obtained results allow studying (5) in case Aj≠0, j=1,2.Denote byP the operator which acts from W2,κ3R;H to L2,κR;H as follows:(42)Put=P0ut+P1ut,ut∈W2,κ3R;H.The lemma below can be proved with the help of Theorem2 and Lemma 6.Lemma 7.
LetA be a self-adjoint positive definite operator in H and let AjA-j∈L(H), j=1,2. Then the operator P acts boundedly from W2,κ3(R;H) to L2,κ(R;H).Now let us state the main result of this work.Theorem 8.
LetA=A∗≥λ0E, λ0>0, κ>-2λ0, and AjA-j∈L(H), j=1,2, and let the inequality(43)c1κ;λ0A1A-1H→H+c2κ;λ0A2A-2H→H<1be true, where the numbers c1(κ;λ0) and c2(κ;λ0) are defined as in Theorem 5. Then (5) is regularly solvable.Proof.
Let us write (5) in the form of operator equation:(44)P0ut+P1ut=ft,where f(t)∈L2,κ(R;H), ut∈W2,κ3R;H.
By Theorem2, (8) is regularly solvable. Make a change P0u(t)=w(t) and rewrite (44) as follows:(45)E+P1P0-1wt=ft.By Theorem 5, for every w(t)∈L2,κ(R;H), we have(46)P1P0-1wL2,κR;H=P1uL2,κR;H≤A1A-1H→HAd2udt2L2,κR;H+A2A-2H→HA2dudtL2,κR;H≤A1A-1H→Hc1κ;λ0P0uL2,κR;H+A2A-2H→Hc2κ;λ0P0uL2,κR;H=c1κ;λ0A1A-1H→H+c2κ;λ0A2A-2H→HwL2,κR;H.As the operator E+P1P0-1 is invertible in the space L2,κ(R;H), when(47)c1κ;λ0A1A-1H→H+c2κ;λ0A2A-2H→H<1,we can define u(t) by the formula(48)ut=P0-1E+P1P0-1-1ft,and we have(49)uW2,κ3R;H≤P0-1L2,κR;H→W2,κ3R;HE+P1P0-1-1L2,κR;H→L2,κR;HfL2,κR;H≤const⁡fL2,κR;H.Theorem is proved.Theorem8 has the following.Corollary 9.
Under the conditions of Theorem8, the operator P is an isomorphism between the spaces W2,κ3(R;H) and L2,κ(R;H).In conclusion, let us illustrate our main result by the example of a problem for a partial differential equation. Consider the following problem on the stripR×[0,π]:(50)∂∂t-∂2∂x23ut,x+px∂4ut,x∂x2∂t2+qx∂5ut,x∂x4∂t=ft,x,∂2sut,0∂x2s=∂2sut,π∂x2s=0,s=0,1,2,where ft,x∈L2,κR;L20,π, and px, qx are the functions bounded on 0,π. The problem (50) can be easily reduced to (5), with H=L20,π, A1=px∂2/∂x2, A2=qx∂4/∂x4, and the operator A being defined in L20,π by the equality Au=-d2u/dx2 and the conditions u0=uπ=0. In this case, λ0=1. Applying Theorem 8, we obtain that if κ>-2 and(51)c1κ;1supx∈0,π⁡px+c2κ;1supx∈0,π⁡qx<1,where(52)c1κ;1=2κ22+κ3,-2<κ≤-1,233/21+κ1/2,κ>-1,c2κ;λ0=4κ2+κ3,-2<κ≤-1,233/21+κ,κ>-1,then problem (50) has a unique solution in the space Wt,x,2,κ3,6R;L20,π.



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