# Nonclassical Problem for Ultraparabolic Equation in Abstract Spaces

Authors: Gia Avalishvili; Mariam Avalishvili
Journal: Journal of Function Spaces
 (2016)

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## Abstract

Nonclassical problem for ultraparabolic equation with nonlocal initial condition with respect to one time variable is studied in abstract Hilbert spaces. We define the space of square integrable vector-functions with values in Hilbert spaces corresponding to the variational formulation of the nonlocal problem for ultraparabolic equation and prove trace theorem, which allows one to interpret initial conditions of the nonlocal problem. We obtain suitable a priori estimates and prove the existence and uniqueness of solution of the nonclassical problem and continuous dependence upon the data of the solution to the nonlocal problem. We consider an application of the obtained abstract results to nonlocal problem for ultraparabolic partial differential equation with second-order elliptic operator and obtain well-posedness result in Sobolev spaces.


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## Body

## 1. Introduction

Evolution equations with several time-like variables are encountered in various models of science and technology, particularly, in mathematical models of multiparameter Brownian motion [1], theory of boundary layers [2], mathematical models of diffusion of pollutants in water flows [3], transport theory [4], mathematical models of age structured biological population dynamics [5, 6], mathematical finance [7], mechanics, physics, and cosmology [8, 9]. In these models one of time variables is usual time and others might denote various quantities, for example, coordinates, temperature, and age or size of individuals of biological population.Various mathematical problems are investigated for multitime evolution equations. Ultraparabolic equations with classical Cauchy conditions with respect to one time variable, problems with classical initial conditions with respect to all time variables, and classical or nonlocal boundary conditions with respect to space variables, inverse problems are investigated in spaces of classical smooth functions and in suitable spaces of generalized functions, and some papers are also devoted to numerical methods for ultraparabolic equations with classical initial conditions (see [10–26] and references given therein). Abstract rather general ultraparabolic equation was studied by Lions [27] in spaces of abstract vector-valued distributions. Applying methods of the theory of semigroups ultraparabolic integrodifferential equation with classical initial conditions with respect to time variables in the spaces of Hölder continuous vector-functions with values in a Banach space was investigated by Lorenzi [28] and ultraparabolic equations arising in age structured population dynamics with integral condition with respect to one of time variables were studied by many researchers; see [6, 29, 30] and their references.Mathematical models of populations incorporating age structure [6] are given by ultraparabolic equations and include nonlocal integral condition with respect to age variable, that is, nonlocal initial condition with respect to the second time variable. Nonclassical problem with nonlocal initial condition for abstract first-order evolution equation was investigated by Krejn and Laptev [31] and in the case of parabolic equation was studied by Kerefov [32]. Various initial-boundary value problems with nonlocal initial conditions are generalizations of initial-boundary value problems with classical and periodical initial conditions and they can be also considered as problems of controllability by initial conditions. Later on, initial-boundary value problems with nonlocal initial conditions were investigated for various time-dependent partial differential equations and systems (see [33–45] and references given therein). Nonlocal in time problems with discrete nonlocal initial conditions are used for investigation of radionuclides propagation in Stokes fluid and problems of predicting the state of a medium [44, 45].In this paper, we consider nonclassical problem for ultraparabolic equation:(1)∂u∂t1+∂u∂t2+Au=f,t1,t2∈0,T1×0,T2,with the following classical and nonlocal initial conditions: (2)u0,t2=φt2,0≤t2≤T2,ut1,0=αut1,ξ+ψt1,0≤t1≤T1,where α∈R,  0<ξ<T2,  φ,  ψ, and f are given vector-functions from suitable spaces, and A is a given linear continuous operator in corresponding abstract Hilbert spaces.As far as the authors know ultraparabolic equations with nonlocal initial condition (1) in abstract spaces have not been investigated yet. Moreover, in the case of classical initial conditions, that is, for α=0, the well-posedness results are obtained either in spaces of continuous functions or in spaces of distributions with respect to time variables, for which the trace operator on boundary of domain of time variables cannot be defined. So, there are no results on the existence and uniqueness of solution in the spaces with minimal regularity properties necessary to define the initial conditions (2). In the present paper we obtain well-posedness theorem for nonlocal problem (1), (2), which in the case of α=0 also gives new existence, uniqueness, and continuous dependence result for classical problem for the abstract ultraparabolic equation.The paper is organized as follows. In Section2, we introduce some notations and give properties of spaces of square integrable vector-functions defined on two-dimensional domain with values in Hilbert spaces. We prove new trace theorem and formulas for integration by parts. In addition, we show characteristic properties of some spaces of vector-valued functions, which are used to investigate the nonclassical problem for ultraparabolic equation. In Section 3, we give variational formulation of the problem (1), (2) and prove our main result on the existence and uniqueness of its solution. Finally, in Section 4, we consider an application of the obtained abstract result to nonlocal problem for ultraparabolic partial differential equation in Sobolev spaces.


## 2. Preliminaries

We denote byL(X;Y) the space of linear continuous operators from X to Y, where X and Y are Banach spaces. Let C([0,T];X) denote the space of continuous vector-functions on [0,T] with values in X. We denote by L2(Δ;X) the space of measurable vector-functions g:Δ→X such that gX∈L2(Δ) and the generalized derivatives of g are denoted by ∂g/∂τi∈D′(Δ;X)=L(D(Δ);X),i=1,…,s,  dg/dτ1=g′, for s=1 (cf. [46]), where D(Δ) is the space of infinitely differentiable functions with compact support in Δ, where Δ⊂Rs,  s∈N is a bounded domain.Throughout this paper, we usec to denote generic constants that are independent of the main parameters involved, but whose values may differ from line to line and may change even within a single chain of estimates.LetV and H be separable Hilbert spaces, such that V is dense in H and continuously embedded in it. We identify space H with its dual by using inner product (·,·) in H and hence V⊂H⊂V′ with continuous and dense embeddings, where V′ is the dual space of V [46]. We denote by ·,· the duality relation between V′ and V. To investigate problem (1), (2) let us consider the following space of vector-valued functions on ω=(0,T1)×(0,T2):(3)W=w∈L2ω;V;∂w∂t1,∂w∂t2∈L2ω;V′,which is a Hilbert space equipped with the norm (4)vW=vL2ω;V2+∂w∂t1L2ω;V′2+∂w∂t2L2ω;V′21/2.In the following three theorems we give some properties of the space W.Theorem 1.
For each vector-functionw∈W and 0≤τ1≤T1,  0≤τ2≤T2, there exist traces w(τ1,t2)=(trt1=τ1w)(t2) and w(t1,τ2)=(trt2=τ2w)(t1), such that the trace operators trt1=τ1:W→L2(0,T2;H) and trt2=τ2:W→L2(0,T1;H) are continuous.Proof.
We can identify each vector-functionv from space L2(ω;V) with vector-function v~ from L2(0,T1;L2(0,T2;V)). If w∈W, then ∂w/∂t1∈L2(ω;V′), and since the linear combinations of products θη,  θ∈D(0,T1), and η∈D(0,T2) are dense in D(ω), we have w~∈L2(0,T1;L2(0,T2;V)), dw~/dt1∈L2(0,T1;L2(0,T2;V′)). Note that L2(0,T2;V)⊂L2(0,T2;H)⊂L2(0,T2;V′) with continuous and dense embeddings and L2(0,T2;V′)=L20,T2;V′. Consequently, from embedding theorem [46] we have that w~∈C([0,T1];L2(0,T2;H)) and C([0,T1];L2(0,T2;H)) is continuously embedded in(5)W1=w1∈L20,T1;L20,T2;V;dwdt1∈L20,T1;L20,T2;V′;that is, (6)w~τ1L20,T2;H≤cw~L20,T1;L20,T2;V2+dw~dt1L20,T1;L20,T2;V′21/2≤cwW,for any τ1∈[0,T1]. Hence, we can define the trace operator w(τ1,t2)=(trt1=τ1w)(t2)=w~(τ1), which is a linear continuous operator from W to L2(0,T2;H). Likewise, for each 0≤τ2≤T2 we obtain the existence of the linear continuous trace operator w(t1,τ2)=(trt2=τ2w)(t1) from W to L2(0,T1;H).Theorem 2.
For vector-functionsw,w∗∈W the following formulas for integration by parts are valid: (7)∫0T2∫0T1∂w∂t1,w∗dt1dt2+∫0T2∫0T1∂w∗∂t1,wdt1dt2=∫0T2wT1,t2,w∗T1,t2dt2-∫0T2w0,t2,w∗0,t2dt2,∫0T1∫0T2∂w∂t2,w∗dt2dt1+∫0T1∫0T2∂w∗∂t2,wdt2dt1=∫0T1wt1,T2,w∗t1,T2dt1-∫0T1wt1,0,w∗t1,0dt1.Proof.
Forw,w∗∈W we consider the corresponding vector-functions w~,w~∗∈W1 defined in the proof of Theorem 1. For vector-functions w~,w~∗∈W1 the following formula for integration by parts is valid [46]:(8)∫0T1dw~dt1,w~∗T2dt1+∫0T1dw~∗dt1,w~T2dt1=w~T1,w~∗T1T2-w~0,w~∗0T2,where ·,·T2 denotes the duality relation between L2(0,T2;V′) and L2(0,T2;V) and (·,·)T2 denotes the inner product in L2(0,T2;H). The latter formula is equivalent to the first formula of the theorem. Likewise we obtain the second formula.Theorem 3.
For vector-functionsw∈W and v∈V the following equalities are valid: (9)∂w∂t1,v=∂∂t1w,v,∂w∂t2,v=∂∂t2w,vinD′ω.Proof.
Applying Theorem2 for w∈W and w∗=ϕv,  ϕ∈D(ω), v∈V, we obtain(10)∫0T2∫0T1∂w∂ti,vϕt1,t2dt1dt2=-∫0T2∫0T1w,v∂ϕ∂tidt1dt2=0,for i=1,2. Since w(t1,t2)∈V,  (t1,t2)∈ω, we have w,v=(w,v), which proves the required equalities.LetA∈L(V;V′) be a self-adjoint coercive operator, which means that the bilinear form a(v,w)=〈Av,w〉 satisfies the conditions (11)av,w≤cavVwV,av,w=aw,v,av,v≥c^avV2,∀v,w∈V,where ca,c^a=const>0. We also suppose that there exists a system of eigenvectors {vk}k∈N of the operator A corresponding to the eigenvalues {λk2}k∈N, that is, a(vk,v)=λk2(vk,v), for all v∈V, which is complete in V and orthonormal in H. From (11) it follows that the operator A∈L(V;V′) is invertible and A-1∈L(V′;V) [47]. Note that a(vk,v)=Avk,v=λk2(vk,v)=λk2vk,v and hence Avk=λk2vk∈V, vk/λk2=A-1vk for all k∈N.Let us define spaceV1={v∈V;Av∈H}, which is a Hilbert space equipped with the scalar product (v,w)1=(Av,Aw) for all v,w∈V1. Space V1 is continuously embedded in V, since(12)vV=A-1AvV≤A-1AvV′≤cA-1AvH=cA-1v1,where v1=(v,v)1. Space V1 is dense in V. Indeed, if v∈V, then Av∈V′. Since H is dense in V′, there exists sequence (hn)n≥1,  hn∈H,  n∈N, such that hn→Av in V′, as n→∞. Consequently, A-1hn→A-1Av=v in V as n→∞ and A-1hn∈V1,  n∈N.So,V1⊂V⊂H⊂V′⊂V1′ with continuous and dense embeddings, where V1′ is the dual space of V1. The duality relation between V1′ and V1 is denoted by ·,·1. Note that for all v,w∈V1 we have(13)Av,w≤AvHwH=v1wH,Av,w1=Av,w=av,w=aw,v=Aw,v≤AwHvH=vHw1.Hence, AvH=v1 and AvV1′≤cvH. Therefore, the restriction of operator A on V1 is a linear continuous operator from V1 to H, and since V1 is dense in H, there exists unique continuous continuation of A on H, which we denote by A1:H→V1′,  A1v=Av for all v∈V1. From the definition of the operator A1 it follows that (14)A1h,z1=h,Az∀h∈H,z∈V1.Indeed, since V1 is dense in H, there exists sequence (zn)n≥1,  zn∈V1,  n∈N, such that zn→h in H as n→∞. Consequently, Azn→A1h in V1′ as n→∞, and (15)A1h,z1=limn→∞⁡Azn,z=limn→∞⁡azn,z=limn→∞⁡az,zn=limn→∞⁡zn,Az=h,Az.In the following lemmas we give some properties of spacesL2(Δ;H),  L2(Δ;V),  L2(Δ;V′),  L2(Δ;V1), and L2(Δ;V1′), which will be used to prove the main theorem.Lemma 4.
(a) Ifg∈L2(Δ;H), then (16)∑n=1∞gnL2Δ2≤gL2Δ;H2,gn=g,vn,n∈N.
(b) Suppose thatgn∈L2(Δ),  n∈N. The series ∑n=1∞gnL2(Δ)2 converges if and only if the series ∑n=1∞gnvn converges in the space L2(Δ;H).Proof.
(a) Since the system of vector{vk}k∈N is orthonormal in H we have(17)∫Δg-∑n=1NgnvnH2dΔ=gL2Δ;H2-∑n=1NgnL2Δ2≥0for all N∈N, and tending N→∞ we obtain the inequality.
(b) The equivalence of the convergences of the series directly follows from the following equalities:(18)∑n=NMgnvnL2Δ;H2=∫Δ∑n=NMgnvnH2dΔ=∫Δ∑n=NMgn2dΔ=∑n=NMgnL2Δ2,∀N,M∈N,N≤M.Lemma 5.
(a) Ifg∈L2(Δ;V), then (19)∑n=1∞λn2gnL2Δ2≤cagL2Δ;V2,gn=g,vn,n∈N.
(b) Suppose thatgn∈L2(Δ),  n∈N. The series ∑n=1∞λn2gnL2(Δ)2 converges if and only if the series ∑n=1∞gnvn converges in the space L2(Δ;V).Proof.
(a) Since{vk}k∈N are eigenvectors of the operator A and {vk}k∈N is orthonormal in H we have(20)∫Δag-∑n=1Ngnvn,g-∑k=1NgkvkdΔ=∫Δag,gdΔ-∫Δ∑n=1Nλn2gn2dΔ≥0for all N∈N, and tending N→∞ from (11) we obtain the required inequality.
(b) The equivalence of the convergences of the series directly follows from the following inequalities for allN,M∈N,  N≤M,(21)∑n=NMgnvnL2Δ;V2≤1c^a∫Δa∑n=NMgnvn,∑k=NMgkvkdΔ=1c^a∑n=NMλn2gnL2Δ2≤cac^a∑n=NMgnvnL2Δ;V2.Lemma 6.
(a) Ifg∈L2(Δ;V′), then (22)∑n=1∞gnL2Δ2λn2≤caA-12gL2Δ;V′2,gn=g,vn,n∈N.
(b) Suppose thatgn∈L2(Δ), n∈N. The series ∑n=1∞gnL2(Δ)2/λn2 converges if and only if the series ∑n=1∞gnvn converges in the space L2(Δ;V′).Proof.
(a) Let us defineg^=A-1g∈L2(Δ;V). Applying part (a) of Lemma 5 we have ∑n=1∞λn2g^nL2(Δ)2≤cag^L2(Δ;V)2, where(23)g^n=g^,vn=vn,g^=avn,g^λn2=ag^,vnλn2=Ag^,vnλn2=g,vnλn2=gnλn2,∀n∈N.Consequently, (24)∑n=1∞λn2g^nL2Δ2=∑n=1∞gnL2Δ2λn2≤caA-1gL2Δ;V2≤caA-12gL2Δ;V′2.
(b) The equivalence of the convergences of the series directly follows from the following inequalities(25)∑n=NMgnvnL2Δ;V′2=AA-1∑n=NMgnvnL2Δ;V′2≤A2∑n=NMgnA-1vnL2Δ;V2≤A2c^a∫Δa∑n=NMgnvnλn2,∑k=NMgkvkλk2dΔ=A2c^a∫Δ∑n=NMgn2λn2dΔ≤A2c^aca∫Δ∑n=NMgnvnλn2V2dΔ=A2c^aca∫Δ∑n=NMgnA-1vnV2dΔ=A2c^acaA-1∑n=NMgnvnL2Δ;V2≤A2caA-12c^a∑n=NMgnvnL2Δ;V′2,∀N,M∈N,N≤M.Lemma 7.
(a) Ifg∈L2(Δ;V1), then (26)∑n=1∞λn4gnL2Δ2≤gL2Δ;V12,gn=g,vn,n∈N.
(b) Suppose thatgn∈L2(Δ),  n∈N. The series ∑n=1∞λn4gnL2(Δ)2 converges if and only if the series ∑n=1∞gnvn converges in the space L2(Δ;V1).Proof.
(a) Since{vk}k∈N are eigenvectors of operator A and {vk}k∈N is orthonormal in H we have (Ag,vk)=a(g,vk)=a(vk,g)=λk2gk for all k∈N, and hence(27)∫ΔAg-∑n=1NgnAvn,Ag-∑k=1NgkAvkdΔ=∫ΔAg-∑n=1Ngnλn2vn,Ag-∑k=1Ngkλk2vkdΔ=∫ΔAg,AgdΔ-∫Δ∑n=1Nλn4gn2dΔ≥0for all N∈N, and tending N→∞ from the definition of the norm ·1 we obtain the required inequality.
(b) The equivalence of the convergences of the series directly follows from the following inequalities:(28)∑n=NMgnvnL2Δ;V12=∫Δ∑n=NMgnAvn,∑k=NMgkAvkdΔ=∫Δ∑n=NMgnλn2vn,∑k=NMgkλk2vkdΔ=∑n=NMλn4gnL2Δ2,∀N,M∈N,N≤M.Lemma 8.
(a) Ifg∈L2(Δ;V1′), then (29)∑n=1∞gnL2Δ2λn4≤A1A-12gL2Δ;V1′2,gn=g,vn1,n∈N.
(b) Suppose thatgn∈L2(Δ), n∈N. The series ∑n=1∞gnL2(Δ)2/λn4 converges if and only if the series ∑n=1∞gnvn converges in the space L2(Δ;V1′).Proof.
(a) Note that the operatorB=A1A:V1→V1′ is linear and continuous, and applying (14) we obtain that B satisfies the following coerciveness condition:(30)Bz,z1=A1Az,z1=Az,Az=z12∀z∈V1.Consequently, the operator B∈L(V1;V1′) is invertible and B-1∈L(V1′;V1) [47].
Let us defineg^=B-1g∈L2(Δ;V1). Applying part (a) of Lemma 7 we have ∑n=1∞λn4g^nL2(Δ)2≤g^L2(Δ;V1)2 and from equality (14) we obtain(31)g^n=g^,vn=vn,g^=avn,g^λn2=ag^,vnλn2=Ag^,vnλn2=Ag^,Avnλn4=A1Ag^,vn1λn4=g,vn1λn4=gnλn4for all n∈N. Consequently, (32)∑n=1∞λn4g^nL2Δ2=∑n=1∞gnL2Δ2λn4≤A1A-1gL2Δ;V12≤A1A-12gL2Δ;V1′2.
(b) SinceBvn=A1Avn=λn2A1vn=λn2Avn=λn4vn, we have vn/λn4=B-1vn and the equivalence of the convergences of the series directly follows from the following inequalities:(33)∑n=NMgnvnL2Δ;V1′2=BB-1∑n=NMgnvnL2Δ;V1′2≤B2∑n=NMgnB-1vnL2Δ;V12=B2∑n=NMgnvnλn4L2Δ;V12=B2A∑n=NMgnvnλn4L2Δ;H2=B2∑n=NMgnvnλn2L2Δ;H2=B2∫Δ∑n=NMgn2λn4dΔ≤B2B-12∑n=NMgnvnL2Δ;V1′2for all N,M∈N,  N≤M.


## 3. Main Result

In the present paper we investigate nonlocal problem (1), (2) applying the following variational formulation: find a vector-function u∈W,  d(trt2=0u)/dt1∈L2(0,T1;V1′),  d(trt1=0u)/dt2∈L2(0,T2;V1′), which satisfies equation(34)∂∂t1u,v+∂∂t2u,v+au,v=f,v,∀v∈V,in the sense of distributions on ω, and initial conditions (35)tr⁡t1=0ut2=φt2,0≤t2≤T2,tr⁡t2=0ut1=αtr⁡t2=ξut1+ψt1,0≤t1≤T1,in the spaces L2(0,T2;H) and L2(0,T1;H), respectively. Applying Theorem 3 and by taking account of the density of linear combinations of products ϕv,  ϕ∈D(ω),  v∈V, in L2(ω;V) we obtain that (35) is equivalent to (1), which is considered as equation in the space L2(ω;V′).Let us prove the following lemmas, which will be used to obtain the existence of solution of problem (34), (35). We denote by ω1={(t1,t2)∈ω;t2>t1} and ω2={(t1,t2)∈ω;t1>t2} the parts of the rectangle ω above and below the bisector {(t1,t2);t1=t2}, respectively; ω-1 and ω-2 are closures of ω1 and ω2.Lemma 9.
Ifg∈L2(0,T2) and g~∈L2(0,T1), then the following estimates are valid(36)∫ω1e-2λn2t1g2t2-t1dt1dt2≤12λn2∫0T2g2sds,∫ω2e-2λn2t2g2t1-t2dt1dt2≤12λn2∫0T1g2sds.Proof.
Taking into account the definition of the setsω1 and ω2 we obtain(37)∫ω1e-2λn2t1g2t2-t1dt1dt2≤∫0T2∫0t2e-2λn2t2-sg2sdsdt2=∫0T2∫sT2e-2λn2t2-sg2sdsdt2=∫0T21-e-2λn2T2-s2λn2g2sds,which implies the first estimate. Likewise we obtain the second estimate.Lemma 10.
Ifg∈H1(ω), then the following estimates are valid (38)∫ω1∫0t1e-λn2t1-τgτ,τ+t2-t1dτ2dt1dt2≤cλn4gH1ω2,∫ω2∫0t2e-λn2t2-τgτ+t1-t2,τdτ2dt1dt2≤cλn4gH1ω2.Proof.
Assuming thatg∈C1(ω-),  ω-=[0,T1]×[0,T2], we have(39)∫ω1∫0t1e-λn2t1-τgτ,τ+t2-t1dτ2dt1dt2=∫ω1gt1,t2λn2-e-λn2t1g0,t2-t1λn2-∫0t1e-λn2t1-τλn2∂g∂τ1τ,τ+t2-t1+∂g∂τ2τ,τ+t2-t1dτ2dt1dt2≤4λn4∫ω1g2t1,t2dt1dt2+4λn4∫0T2∫0t2g20,sdsdt2+4T1T2λn4∫ω1∂g∂τ1t1,t22dt1dt2+∫ω1∂g∂τ2t1,t22dt1dt2.From continuity of the trace operator H1(ω)→L2(∂ω) we obtain(40)∫0T2∫0t2g20,sdsdt2≤T2∫0T2g20,sds≤T2cgH1ω2.The latter inequality implies the first estimate of the lemma for functions from C1(ω-) and since C1(ω-) is dense in H1(ω), by passing to the limit we obtain the estimate for functions from H1(ω). Likewise we obtain the second estimate.Lemma 11.
Ifg∈H1(ω), then the following estimates are valid (41)∫0ξT1∫0t1e-λn2t1-τgτ,τ+ξ-t1dτ2dt1≤cλn4gH1ω2,∫ξT1T1∫0ξe-λn2ξ-τgτ+t1-ξ,τdτ2dt1≤cλn4gH1ω2,where ξT1=min⁡{ξ,T1}.Proof.
Assuming thatg∈C1(ω-) we get(42)∫0ξT1∫0t1e-λn2t1-τgτ,τ+ξ-t1dτ2dt1=∫0ξT1gt1,ξλn2-e-λn2t1g0,ξ-t1λn2-∫0t1e-λn2t1-τλn2∂g∂τ1τ,τ+ξ-t1+∂g∂τ2τ,τ+ξ-t1dτ2dt1≤4λn4∫0ξT1g2t1,ξdt1+4λn4∫0ξg20,sds+4T1λn4∫ω1∂g∂τ12+∂g∂τ22dt1dt2.The continuity of the trace operator H1(ωξ)→L2(∂ωξ), ωξ=(0,T1)×(0,ξ), implies that (43)max⁡∫0ξT1g2t1,ξdt1,∫0ξg20,sds≤cgH1ωξ2≤cgH1ω2.From the latter inequality we obtain the first estimate of the lemma for functions from C1(ω-) and since C1(ω-) is dense in H1(ω), by passing to the limit we obtain the estimate for functions from H1(ω). Likewise we obtain the second estimate.The following existence and uniqueness theorem is valid for nonlocal problem (34), (35).Theorem 12.
Ifφ∈L2(0,T2;H),  φ′∈L2(0,T2;V1′),  ψ∈L2(0,T1;H),  ψ′∈L2(0,T1;V1′), φ and ψ satisfy compatibility condition φ(0)=αφ(ξ)+ψ(0) and f,∂f/∂t1,∂f/∂t2∈L2(ω;V′), then problem (34), (35) has a unique solution and the following estimate is valid:(44)uW≤cφL20,T2;H+φ′L20,T2;V1′+ψL20,T1;H+ψ′L20,T1;V1′+fL2ω;V′+∂f∂t1L2ω;V′+∂f∂t2L2ω;V′.Proof.
Applying embedding theorem [46] we have that φ∈C([0,T2];V′),  ψ∈C([0,T1];V′) and the compatibility condition is the equality in V′. First, we prove the existence of a solution. Let us consider the sequence of approximate solutions wN(t1,t2)=∑n=1Nun(t1,t2)vn, where un is a solution of the following nonlocal problem for the first-order partial differential equation:(45)∂un∂t1+∂un∂t2+λn2un=fn,t1,t2∈ω,(46)un0,t2=φnt2,0≤t2≤T2,unt1,0=αunt1,ξ+ψnt1,0≤t1≤T1,where φn=(φ,vn),ψn=(ψ,vn),  fn=f,vn, n∈N. Note that φn∈H1(0,T2),  ψn∈H1(0,T1),  fn∈H1(ω) and hence φn∈C[0,T2],  ψn∈C[0,T1].
Let us denote byg1n and g2n the following functions:(47)g1nt1,t2=∫0t1e-λn2t1-τfnτ,τ+t2-t1dτ,t1,t2∈ω-1,g2nt1,t2=∫0t2e-λn2t2-τfnτ+t1-t2,τdτ,t1,t2∈ω-2.Functions g1n and g2n are defined for all (t1,t2)∈ω-1 and (t1,t2)∈ω-2, respectively, because the trace of fn on each line (48)lt1t2=τ,τ+t2-t1;0<τ<t1fort2≥t1,τ+t1-t2,τ;0<τ<t2fort1≥t2,which is parallel to the bisector {(t1,t2);t1=t2}, which belongs to L2(lt1t2).
Note thatg1n∈L2(ω1) and g2n∈L2(ω2). Their generalized derivative is given by(49)∂g1n∂t1=fnt1,t2-λn2∫0t1e-λn2t1-τfnτ,τ+t2-t1dτ-∫0t1e-λn2t1-τ∂fn∂τ2τ,τ+t2-t1dτ,∂g1n∂t2=∫0t1e-λn2t1-τ∂fn∂τ2τ,τ+t2-t1dτfor almost all (t1,t2)∈ω1, (50)∂g2n∂t1=∫0t2e-λn2t2-τ∂fn∂τ1τ+t1-t2,τdτ,∂g2n∂t2=fnt1,t2-λn2∫0t2e-λn2t2-τfnτ+t1-t2,τdτ+∫0t2e-λn2t2-τ∂fn∂τ1τ+t1-t2,τdτfor almost all (t1,t2)∈ω2. Applying Lemma 10 we obtain(51)∂g1n∂t1L2ω12≤3fnL2ω12+3cfnH1ω2+T1∫0T2∫0min⁡t2,T1∫0t1∂fn∂τ22⁡τ,τ+t2-t1dτdt1dt2≤3c+1fnH1ω2+T1T2∫ω1∂fn∂τ22dt1dt2≤cfnH1ω2,(52)∂g1n∂t2L2ω12≤T1∫0T2∫0min⁡t2,T1∫0t1∂fn∂τ22τ,τ+t2-t1dτdt1dt2≤T1T2∫ω1∂fn∂τ22dt1dt2≤T1T2fnH1ω2,(53)∂g2n∂t1L2ω22≤T2∫0T1∫0min⁡t1,T2∫0t2∂fn∂τ12τ+t1-t2,τdτdt2dt1≤T1T2∫ω2∂fn∂τ12dt1dt2≤T1T2fnH1ω2,(54)∂g2n∂t2L2ω22≤3fnL2ω22+3cfnH1ω2+T2∫0T1∫0min⁡t1,T2∫0t2∂fn∂τ12τ+t1-t2,τdτdt2dt1≤3c+1fnH1ω2+T2T1∫ω2∂fn∂τ12dt1dt2≤cfnH1ω2.Therefore, g1n∈H1(ω1) and g2n∈H1(ω2).
Applying Lemma11 and continuity of the trace operator H1(ωξ)→L2(∂ωξ) we obtain(55)g1nt1,ξL20,ξT12≤cλn4fnH1ω2,g2nt1,ξL2ξ,T12≤cλn4fnH1ω2forT1>ξ,dg1nt1,ξdt1L20,ξT12≤3fnt1,ξL20,ξT12+3λn4∫0ξT1∫0t1e-λn2t1-τfnτ,τ+ξ-t1dτ2dt1+3T1∫0ξT1∫0t1∂fn∂τ22τ,τ+ξ-t1dτdt1≤cfnH1ω2,dg2nt1,ξdt1L2ξ,T12≤ξ∫ξT1∫0ξ∂fn∂τ12τ+t1-ξ,τdτdt1≤ξfnH1ω2forT1>ξ.Hence, g1n(t1,ξ)∈H1(0,ξT1) and g2n(t1,ξ)∈H1(ξ,T1) for T1>ξ, and from embedding theorem we obtain that g1n(t1,ξ)∈C[0,ξT1],  g2n(t1,ξ)∈C[ξ,T1] for T1>ξ.
Note that the classical problem for (45) with initial conditions un(0,t2)=φn(t2) and un(t1,0)=dn(t1),  dn∈H1(0,T1),  φn(0)=dn(0), has a unique solution un∈H1(ω),  dun(t1,0)/dt1∈L2(0,T1),  dun(0,t2)/dt2∈L2(0,T2), which is given by(56)unt1,t2=e-λn2t1φnt2-t1+g1nt1,t2,t1,t2∈ω1,e-λn2t2dnt1-t2+g2nt1,t2,t1,t2∈ω2.From nonlocal conditions (46) for function dn we have(57)dnt1=αe-λn2t1φnξ-t1+g1nt1,ξ+ψnt1,t1∈0,ξT1,αe-λn2ξdnt1-ξ+g2nt1,ξ+ψnt1,t1∈ξT1,T1,from which we obtain expression of dn for t1>ξ,  T1>ξ,(58)dnt1=αe-λn2ξkdnt1-kξ+∑i=0k-1αe-λn2ξiαg2nt1-iξ,ξ+ψnt1-iξfor kξ<t1≤min⁡{(k+1)ξ,T1},  k=1,…,K, where K=[T1/ξ], if T1/ξ∉N, and K=T1/ξ-1, if T1/ξ∈N;  [x] denotes the integer part of x∈R.
Since functionsφn,  ψn,  g1n(t1,ξ), and g2n(t1,ξ) are continuous, we have dn∈C[0,ξT1]. Let us assume that T1>ξ and then ξT1=ξ. Note that g1n(0,ξ)=0,  g1n(ξ,ξ)=g2n(ξ,ξ) and from compatibility condition we have φn(0)=αφn(ξ)+ψn(0). Hence, we get(59)dnξ=αe-λn2ξφn0+g1nξ,ξ+ψnξ=αe-λn2ξαφnξ+ψn0+g2nξ,ξ+ψnξ=αe-λn2ξdn0+g2nξ,ξ+ψnξ,which imply continuity of dn at point ξ. Moreover, we have(60)dnk+1ξ=αe-λn2ξkdnξ+∑i=0k-1αe-λn2ξiαg2nk-i+1ξ,ξ+ψnk-i+1ξ=αe-λn2ξk+1dn0+αe-λn2ξkαg2nξ,ξ+ψnξ+∑i=0k-1αe-λn2ξiαg2nk-i+1ξ,ξ+ψnk-i+1ξ=αe-λn2ξk+1dn0+∑i=0kαe-λn2ξiαg2nk-i+1ξ,ξ+ψnk-i+1ξ.Consequently, dn∈C[0,T1] and its generalized derivative dn′ belongs to L2(0,T1), since(61)dn′t1=α-λn2e-λn2t1φnξ-t1-e-λn2t1φn′ξ-t1+dg1nt1,ξdt1+ψn′t1for almost all t1∈(0,ξT1) and (62)dn′t1=αe-λn2ξkdn′t1-kξ+∑i=0k-1αe-λn2ξiαdg2nt1-iξ,ξdt1+ψn′t1-iξfor T1>ξ and for almost all t1∈(kξ,min⁡{(k+1)ξ,T1}),  k=1,…,K.
Sinceg1n(0,t2)=0 and g1n(0,t2)=0, from construction of dn it follows that un satisfies conditions (46).
Note that the trace on the bisectorlTT,  T=min{T1,T2}, of restriction of function un on ω1 equals the trace on the bisector lTT of restriction of function un on ω2, because g1n(t,t)=g2n(t,t),  0<t≤T,  φn(0)=αφn(ξ)+ψn(0), and(63)unt,t=e-λn2tφn0+g1nt,t=e-λn2tαφnξ+ψn0+g2nt,t=e-λn2tdn0+g2nt,t.
The generalized derivatives of functionun are given by(64)∂un∂t1=-λn2e-λn2t1φnt2-t1-e-λn2t1φn′t2-t1+∂g1n∂t1t1,t2,t1,t2∈ω1,e-λn2t2dn′t1-t2+∂g2n∂t1t1,t2,t1,t2∈ω2,∂un∂t2=e-λn2t1φn′t2-t1+∂g1n∂t2t1,t2,t1,t2∈ω1,-λn2e-λn2t2dnt1-t2-e-λn2t2dn′t1-t2+∂g2n∂t2t1,t2,t1,t2∈ω2.Therefore, we have ∂un/∂t1,∂un/∂t2∈L2(ω) and un satisfies (45) in L2(ω).
Now we obtain estimates forun, which permit one to prove the convergence of the sequence of approximate solutions (wN)N≥1.
Applying Lemmas9 and 10 we obtain(65)unL2ω2≤2∫ω1e-2λn2t1φn2t2-t1dt1dt2+2∫ω1g1n2t1,t2dt1dt2+2∫ω2e-2λn2t2dn2t1-t2dt1dt2+2∫ω2g2n2t1,t2dt1dt2≤1λn2φnL20,T22+cλn4fnH1ω2+1λn2dnL20,T12.To obtain estimate for dn without loss of generality we assume that T1>ξ. Applying Lemmas 9–11 we get(66)dnL20,T12=dnL20,ξ2+∑k=1K∫kξmin⁡k+1ξ,T1αe-λn2ξkdnt1-kξ+∑i=0k-1αe-λn2ξiαg2nt1-iξ,ξ+ψnt1-iξ2dt1≤∫0ξdn2t1dt1+K2K+1max⁡α2K,α2∫0ξdn2sds+K2K+1max⁡α2K,α2∫ξT1g2n2t1,ξdt1+K2K+1max⁡1,α2K-1∫ξT1ψn2sds≤1+K2K+1max⁡α2K,α23α2∫0ξφn2ξ-t1dt1+3α2∫0ξg1n2t1,ξdt1+3∫0ξψn2t1dt1+K2K+1max⁡α2K,α2cλn4fnH1ω2+K2K+1max⁡1,α2K-1∫ξT1ψn2sds≤cφnL20,T22+ψnL20,T12+1λn4fnH1ω2.From the latter estimate we have (67)unL2ω2≤cφnL20,T22λn2+ψnL20,T12λn2+fnH1ω2λn4.
Applying expressions for∂un/∂t1 and ∂un/∂t2, Lemma 9, and estimates (51)–(54) we obtain (68)∂un∂t1L2ω2≤3λn4e-λn2t1φnt2-t1L2ω12+3e-λn2t1φn′t2-t1L2ω12+3∂g1n∂t1L2ω12+2e-λn2t2dn′t1-t2L2ω22+2∂g2n∂t1L2ω22≤3λn22φnL20,T22+32λn2φn′L20,T22+1λn2dn′L20,T12+5cfnH1ω2.In order to estimate dn′ without loss of generality we assume that T1>ξ. Applying Lemmas 9–11 we have(69)dn′L20,T12=dn′L20,ξ2+∑k=1K∫kξmin⁡k+1ξ,T1αe-λn2ξkdn′t1-kξ+∑i=0k-1αe-λn2ξiα∂g2n∂t1t1-iξ,ξ+ψn′t1-iξ2dt1≤dn′L20,ξ2+K2K+1max⁡α2K,α2dn′L20,ξ2+K2K+1max⁡α2K,α2∫ξT1dg2nt1,ξdt12dt1+K2K+1max⁡1,α2K-1ψn′L2ξ,T12≤1+K2K+1max⁡α2K,α24α2λn4∫0ξφn2ξ-t1dt1+4α2∫0ξφn′ξ-t12dt1+4α2∫0ξdg1nt1,ξdt12dt1+4∫0ξψn′2t1dt1+K2K+1max⁡α2K,α2ξfnH1ω2+K2K+1max⁡1,α2K-1ψn′L2ξ,T12≤cλn4φnL20,T22+φn′L20,T22+ψn′L20,T12+fnH1ω2.
From the estimate for∂un/∂t1L2(ω)2 and (69) we get(70)∂un∂t1L2ω2≤cλn2φnL20,T22+1λn2φn′L20,T22+1λn2ψn′L20,T12+fnH1ω2.By using Lemma 9 and estimates (52), (54), (66), and (69) we obtain(71)∂un∂t2L2ω2≤2e-λn2t1φn′t2-t1L2ω12+2∂g1n∂t2L2ω12+3λn4e-λn2t2dnt1-t2L2ω22+3e-λn2t2dn′t1-t2L2ω22+3∂g2n∂t2L2ω22≤1λn2φn′L20,T22+cfnH1ω2+3λn22dnL20,T12+3dn′L20,T122λn2≤cλn2φnL20,T22+φn′L20,T22λn2+λn2ψnL20,T12+1λn2ψn′L20,T12+fnH1ω2.
Applying parts (a) of Lemmas4, 6, and 8 and inequalities (67), (70), and (71) we obtain(72)∑n=1∞λn2unL2ω2≤cφL20,T2;H2+ψL20,T1:H2+fL2ω:V′2,∑n=1∞1λn2∂un∂tiL2ω2≤cφL20,T2;H2+φ′L20,T2;V1′2+ψL20,T1:H2+ψ′L20,T1:V1′2+fL2ω;V′+∂f∂t1L2ω;V′+∂f∂t2L2ω;V′,i=1,2.Consequently, from parts (b) of Lemmas 5 and 6 we get the sequence (wN)N≥1 and hence the series (73)ut1,t2=∑n=1∞unt1,t2vnconverges in the space W. Since the trace operators trt1=0,trt2=0, and trt2=ξ are continuous, from (46) we obtain that u satisfies conditions (35), and, by (72), the estimate (44) is valid. Moreover, (74)tr⁡t1=0u=limN→∞⁡∑n=1Nun0,t2vn∈L20,T2;H⊂D′0,T2;H,tr⁡t2=0u=limN→∞⁡∑n=1Nunt1,0vn∈L20,T1;H⊂D′0,T1;H,and since the operator of differentiation is continuous on the space of distributions we have(75)dtr⁡t1=0udt2=limN→∞⁡∑n=1Ndun0,t2dt2vn∈D′0,T2;H,dtr⁡t2=0udt1=limN→∞⁡∑n=1Ndunt1,0dt1vn∈D′0,T1;H.Applying part (a) of Lemma 8 we obtain (76)∑n=1N1λn4dun0,t2dt2L20,T22=∑n=1Nφn′L20,T22λn4≤A1A-12φ′L20,T2;V1′2,and by using parts (a) of Lemmas 4–6 and (69) we get(77)∑n=1N1λn4dunt1,0dt1L20,T22=∑n=1Ndn′L20,T12λn4≤cφnL20,T22+φn′L20,T22λn4+ψn′L20,T12λn4+fnH1ω2λn4≤cφL20,T2;H2+φ′L20,T2;V1′2+ψ′L20,T1:V1′2+fL2ω;V′+∂f∂t1L2ω;V′+∂f∂t2L2ω;V′.Consequently, from part (b) of Lemma 8 we obtain(78)dtr⁡t1=0udt2=limN→∞⁡∑n=1Ndun0,t2dt2vn∈L20,T2;V1′,dtr⁡t2=0udt1=limN→∞⁡∑n=1Ndunt1,0dt1vn∈L20,T1;V1′.
Let us show thatu∈W,  d(tr⁡t2=0u)/dt1∈L2(0,T1;V1′),  d(tr⁡t1=0u)/dt2∈L2(0,T2;V1′) is a solution of (34). Indeed, since the system {vk}k∈N is complete in V, it follows that for each v∈V there exists a sequence (zk)k≥1, where zk is a linear combination of v1,…,vk, such that zk→v in V as k→∞. By taking account of the properties of eigenvectors {vk}k∈N, from (45) for each ψ∈D(ω) and v∈V we have(79)-∫ωwN,zN∂ψ∂t1dt1dt2-∫ωwN,zN∂ψ∂t2dt1dt2+∫ωawN,zNψdt1dt2=0,∀N∈N.By passing to the limit in the last relation, we find that u satisfies (34). So, the existence of solution is proved.
To complete the proof, we prove the uniqueness of solution. Note that the classical problem for (34) with initial conditions (80)tr⁡t1=0ut2=φ~t2,0≤t2≤T2,tr⁡t2=0ut1=ψ~t1,0≤t1≤T1,has at most one solution. Indeed, if u~ is a solution of the homogeneous classical problem (34), (80), that is, φ~≡0,  ψ~≡0 and f≡0, then, by multiplying (1), which is equivalent to (34), by u, integrating on (0,T1)×(0,T2), and applying Theorem 2 we get(81)12∫0T2tr⁡t1=T1u~t2H2dt2+12∫0T1tr⁡t2=T2u~t1H2dt1+∫0T1∫0T2au~,u~dt2dt1=0,and, consequently, u~≡0 in (0,T1)×(0,T2).
Since problem (34), (35) for α=0 is a classical problem, from the proof of the existence it follows that the classical problem for (34) with initial conditions (80), φ~∈L2(0,T2;H),  φ~′∈L2(0,T2;V1′),  ψ~∈L2(0,T1;H), ψ~′∈L2(0,T1;V1′),  φ~(0)=ψ~(0), has a unique solution u∈W,  trt1=0u∈L2(0,T2;H),  d(trt1=0u)/dt2∈L2(0,T2;V1′),  trt2=0u∈L2(0,T1;H),  d(trt2=0u)/dt1∈L2(0,T1;V1′), which can be represented in the form of convergent in W series (73), where un is given by (56). Therefore, the uniqueness of the solution of nonlocal problem (34), (35) follows from the uniqueness of the solution of problem (45), (46), which completes the proof.


## 4. Application

In this section, we consider an application of Theorem12 obtained for abstract nonlocal problem to nonclassical problem for ultraparabolic partial differential equation with nonlocal initial condition. Let Ω⊂Rn,  n∈N, be a bounded domain with Lipschitz boundary Γ [47]. By Hk(Ω) we denote the Sobolev space of functions in L2(Ω) whose generalized partial derivatives up to the order k∈N belong to L2(Ω). We denote the closure in Hk(Ω) of the set D(Ω) by H0k(Ω) and its dual space by H-kΩ. We consider the second-order elliptic operator: (82)A≡-∑q,j=1n∂∂xqaqjx∂∂xj+a0x,where aqj(x),a0(x)∈L∞(Ω),  q,j=1,…,n, and the following inequalities are valid(83)a0x≥0,∑q,j=1naqjxηjηq≥cA∑j=1nηj2,cA=const>0,for almost all x∈Ω, ηj  (j=1,…,n) are arbitrary real numbers. Partial derivatives in the definition of A are treated as generalized derivatives with respect to the corresponding variables. Note that the first-order partial derivatives of a function in L2(Ω) belongs to the space H-1(Ω). Since the multiplication of a function in the space L2(Ω) by a function in L∞(Ω) leaves it in L2(Ω), for each function v∈H01(Ω) we have Av∈H-1(Ω). The elliptic operator A is a continuous operator from H01(Ω) to the space H-1(Ω), which, by virtue of (83) and Poincare inequality, satisfies the following self-adjointness and coerciveness conditions: (84)Av,w∗=Aw,v∗,Av,v∗≥αvH1Ω2,∀v,w∈H01Ω,where α=const>0 and  〈·,·〉∗ is the duality relation between the spaces H-1(Ω) and H01(Ω). We denote by D(A)={v∈H01(Ω);Av∈L2(Ω)} the domain of operator A:D(A)→L2(Ω). The bilinear form a(v,w)=〈Av,w〉∗ corresponding to the operator A is of the form (85)av,w=∫Ω∑q,j=1naqjx∂v∂xj∂w∂xq+a0xvwdxand satisfies conditions (11). Since the continuous embedding of H01(Ω) in L2(Ω) is compact, it follows from general theorem for self-adjoint coercive operators mapping a Hilbert space into its dual [47] that there exists a system of eigenfunctions {vk}k=1∞ of the operator A:H01(Ω)→H-1(Ω) corresponding to the eigenvalues {λk2}k=1∞, such that {vk}k=1∞ is orthonormal in L2(Ω) and complete in H01(Ω), and 0<λ1≤λ2≤⋯,  λk→∞ as k→∞.Let us consider nonclassical problem for ultraparabolic partial differential equation(86)∂u∂t1+∂u∂t2+Au=finΩ×0,T1×0,T2,with nonlocal initial and homogeneous boundary conditions (87)ux,0,t2=φx,t2,x,t2∈Ω×0,T2,ux,t1,0=αux,t1,ξ+ψx,t1,x,t1∈Ω×0,T1,(88)ux,t1,t2=0,x,t1,t2∈Γ×0,T1×0,T2,where α∈R. We identify functions defined on Ω×ω with vector-functions defined on ω and ranging in the corresponding function spaces defined on Ω. Hence, the nonlocal in time problem (86)–(88) can be stated as the problem of finding a vector-function u∈L2(ω;H01(Ω)),  ∂u/∂t1,∂u/∂t2∈L2(ω;H-1(Ω)),  d(trt2=0u)/dt1∈L2(0,T1;DA′),  d(trt1=0u)/dt2∈L2(0,T2;DA′), which satisfies (86) in the space D′(ω;H-1(Ω)) of distributions on ω ranging in H-1(Ω), where the partial derivatives ∂u/∂t1 and ∂u/∂t2 are treated as the first-order generalized derivatives ∂u/∂t1,∂u/∂t2∈D′(ω;H01(Ω)). The unknown function u satisfies conditions (87) in the spaces L2(0,T2;L2(Ω)) and L2(0,T1;L2(Ω)), respectively. The boundary conditions (88) are valid, because the trace of function from H01(Ω) vanishes on the boundary of the domain Ω. Thus, nonlocal in time problem (86)–(88) is a particular case of the nonclassical problem (34), (35) for V=H01(Ω),H=L2(Ω), and V1=D(A). The following existence and uniqueness theorem is valid for nonlocal in time problem (86)–(88).Theorem 13.
Ifφ∈L2(0,T2;L2(Ω)),  φ′∈L2(0,T2;DA′),  ψ∈L2(0,T1;L2(Ω)), ψ′∈L2(0,T1;DA′), φ and ψ satisfy compatibility condition φ0,v∗=α〈φ(ξ),v〉∗+〈ψ(0),v〉∗ for all v∈H01(Ω) and f,∂f/∂t1,∂f/∂t2∈L2(ω;H-1(Ω)), then nonlocal problem (86)–(88) has a unique solution u∈L2(ω;H01(Ω)), ∂u/∂t1,∂u/∂t2∈L2(ω;H-1(Ω)), d(trt2=0u)/dt1∈L2(0,T1;DA′), d(trt1=0u)/dt2∈L2(0,T2;DA′), and the following estimate is valid:(89)uL2ω;H01Ω2+∑i=12∂u∂tiL2ω;H-1Ω2≤cφL20,T2;L2Ω2+φ′L20,T2;DA′2+ψL20,T1;L2Ω2+ψ′L20,T1;DA′2+fL2ω;H-1Ω2+∂f∂t1L2ω;H-1Ω2+∂f∂t2L2ω;H-1Ω2.


## 5. Conclusions

In this paper, we investigated the existence and uniqueness of solution of nonclassical problem for ultraparabolic equation with two time variables and nonlocal initial condition, which connect values of the unknown vector-function at the initial and some later moment of time. The considered problem is a generalization of the classical problem for ultraparabolic equation and the obtained results give new well-posedness result for the classical problem, as well as for nonlocal one, in suitable function spaces with minimal regularity that is necessary to define the traces on the boundary of the time domain. Note that applying the results obtained in this paper and the presented method of investigation one can study various initial-boundary value problems for ultraparabolic equations with several time variables and more general nonlocal initial conditions. The proof of the existence of solution of the nonclassical problem is constructive and can be used to obtain approximate solutions of the nonlocal problem.



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