# On One Evolution Equation of Parabolic Type with Fractional Differentiation Operator inS Spaces

Authors: V. V. Gorodetskiy; R. S. Kolisnyk; N. M. Shevchuk
Journal: International Journal of Differential Equations
 (2020)

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## Abstract

In the paper, we investigate a nonlocal multipoint by a time problem for the evolution equation with the operatorA=I−Δω/2, Δ=d2/dx2, and ω∈1;−2 is a fixed parameter. The operator A is treated as a pseudodifferential operator in a certain space of type S. The solvability of this problem is proved. The representation of the solution is given in the form of a convolution of the fundamental solution with the initial function which is an element of the space of generalized functions of ultradistribution type. The properties of the fundamental solution are investigated. The behavior of the solution at t⟶+∞ (solution stabilization) in the spaces of generalized functions of type S′ and the uniform stabilization of the solution to zero on ℝ are studied.


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## Body

## 1. Introduction

A rather wide class of differential equations with partial derivatives forms linear parabolic andB-parabolic equations, and the theory of which originates from the study of the heat conduction equation. The classical theory of the Cauchy problem and boundary-value problems for such equations and systems of equations is constructed in the works of I.G. Petrovskiy, S.D. Eidelman, S.D. Ivasyshen, M.I. Matiychuk, M.V. Zhitarashu, A. Friedman, S. Teklind, V.O. Solonnikov, and V.V. Krekhivskiy et al. The Cauchy problem with initial data from the spaces of generalized functions of the type of distributions and ultradistributions was studied by G.E. Shilov, B.L. Gurevich, M.L. Gorbachuk, V.I. Gorbachuk, O.I. Kashpirovskiy, Ya.I. Zhytomyrskiy, S.D. Ivasyshen, V.V. Gorodetskiy, and V.A. Litovchenko et al.A formal extension of the class of parabolic type equations is the set of evolution equations with the pseudodifferential operator (PDO), which can be represented asA=Jσ⟶x−1at,x;σJx⟶σ, x,σ⊂ℝn, t>0, where a is a function (symbol) that satisfies certain conditions and JJ−1 is the direct (inverse) Fourier or Bessel transform. The PDO includes differential operators, fractional differentiation and integration operators, convolution operators, and the Bessel operator Bν=d2/dx2+2ν+1x−1d/dx, ν>−1/2, which contains the expression 1/x in its structure and is formally represented as Bν=FBν−1−σ2FBν, where FBν is the Bessel integral transformation.IfA is a nonnegative self-adjoint operator in a Hilbert space H, then it is known [1] that a continuous on 0,T function ut is continuously differentiable solution of the operator differential equation u′t+Aut=0, t∈0,T, which refers to abstract equations of parabolic type, if and only if it is given as ut=e−tAf, f=u0∈H. It turns out [1] that all continuously differentiable functions within the interval 0,T solutions of this equation are described by the same formula where f is an element of the wider than H space Ha′ conjugate to the space Ha of analytic vectors of the operator A; the role of A is played by the extension A^ of the operator A to the space Ha′, and the boundary value of ut at the point 0 exists in the space Ha′.IfA=I−Δ1/2, Δ=d2/dx2, then A is a nonnegative self-adjoint operator in H=L2ℝ, since id/dx is a self-adjoint in L2ℝ operator with the domain Did/dx=φ∈L2ℝ:∃φ′∈L2ℝ. If Eλ, λ∈ℝ, is the spectral function of the operator id/dx, then, due to the basic spectral theorem for self-adjoint operators,(1)Aφ=I+iddx21/2φ=∫−∞+∞1+λ21/2dEλφ.It is known (see, for example, [2]) that(2)Eλφt=12π∫−∞λ∫−∞+∞φτeiστdτe−itσdσ.It follows from this thatdEλφt=1/2πFφλe−itλdλ. So,(3)Aφ=12π∫−∞+∞1+λ21/2Fφλe−itλdλ=F−11+λ21/2Fφ.Following [3], we call the operator A the Bessel operator of fractional differentiation. Therefore, A can be understood as a pseudodifferential operator constructed on the function symbol 1+λ21/2, λ∈ℝ. This allows us to interpret the function e−tAf that is a solution of the corresponding Cauchy problem as a convolution of the form Gt,⋅∗f [4, 5], where Gt,⋅=F−1e−1+λ21/2.In this paper, we give a similar depiction of the solution of a nonlocal multipoint time problem for the equation∂ut,x/∂t+Aut,x=0, t,x∈0,+∞×ℝ when the initial condition ut=0=f is replaced by the condition ∑k=0mαkBkut,⋅t=tk, where t0=0, t1,…,tm⊂0,+∞, α0,α1,…,αm⊂ℝ, and m∈ℕ are fixed and B1,…,Bm are pseudodifferential operators constructed of smooth symbols (if α0=1, α1=⋯=αm=0, B0=I, then obviously we have a Cauchy problem). This condition is interpreted in the classical or weak sense as if f is a generalized function (generalized element of the operator A) of ultradistribution type. Properties of the fundamental solution of the specified multipoint problem are investigated. The behavior of the solution at t⟶+∞ (solution stabilization) in the spaces of generalized functions of type S′ and the uniform stabilization of the solution to zero on ℝ are studied.Note that the nonlocal multipoint time problem relates to nonlocal problems for operator differential equations and partial differential equations. Such problems arise when modeling many processes and problems of practice with boundary-value problems for partial differential equations, when describing correct problems for a particular operator and constructing a general theory of boundary-value problems (see, for example, [6–11]).


## 2. Spaces of Test and Generalized Functions

Gelfand and Shilov introduced in [12] a series of spaces, which they called the spaces S. They consist of infinitely differentiable on ℝ functions, which satisfy certain conditions on the decrease at infinity and the growth of derivatives. These conditions are given by the inequalities xkφmx≤ckm, x∈ℝ, k,m⊂ℤ+, where ckm is some double sequence of positive numbers. If there are no restrictions on elements of the sequence ckm, then obviously we have L. Schwartz’s space S≡Sℝ of quickly descending at infinity functions. However, if the numbers ckm satisfy certain conditions, then the corresponding specific spaces are contained in S and they are called the spaces of S type. Let us define some of them.For anyα, β>0, let us put(4)Sαβℝ≡Sαβ≔φ∈S∃c>0∃A>0∃B>0∀m,n⊂ℤ+∀x∈ℝ:xmφnx≤cAmBnmmαnnβ.The introducedS spaces can also be described as in [12].Sαβ consists of those infinitely differentiable on ℝ functions φx that satisfy the following inequalities:(5)φnx≤cBnnnβexp−ax1/α,n∈ℤ+,x∈ℝ,with some positive constants c, B, and a dependent only on the function φ.If0<β<1 and α≥1−β, then Sαβ consists of only those φ∈C∞ℝ, which admit an analytic extension into the complex plane ℂ such that(6)φx+iy≤cexp−ax1/α+by1/1−β,c>0,a>0,b>0.The spaceSα1 consists of functions φ∈C∞ℝ, which can be analytically extended into some band Imz<δ (dependent on φ) of the complex plane, so that the estimate(7)φx+iy≤cexp−ax1/α,c,a>0,∀y∈ℝ,is carried out.The spacesSαβ, α,β>0, are nontrivial if α+β≥1, and they form dense sets in L2ℝ.The topological structure inSαβ is defined as follows. The symbol Sα,Aβ,B, A,B>0, denotes the set of functions φ∈Sαβ that satisfy the condition(8)∀A¯>A∀B¯>B:xkφmx≤cA¯kB¯mkkαmmβ,k,m⊂ℤ+,x∈ℝ.This set is transformed into a complete, countably normed space, if the norms in it are defined by means of relations:(9)φδρ=supx,k,mxkφmxA+δkB+ρmkkαmmβ,δ,ρ⊂1,12,13,….The specified norm system is sometimes replaced by an equivalent norm system:(10)φδρ′=supx,mexpa1−δx1/α⋅φmxB+ρmmmβ,δ,ρ⊂12,13,…,a=αeA1/α.IfA1<A2, B1<B2, then Sα,A1β,B1 is continuously embedded into Sα,A2β,B2 and Sαβ=∪A,B>0Sα,Aβ,B, that is, Sαβ is endowed by the inductive limit topology of the spaces Sα,Aβ,B [12]. Therefore, the convergence of a sequence φν,ν≥1⊂Sαβ to zero in the space Sαβ is the convergence in the topology of some space Sα,Aβ,B, to which all the functions φν belong. In other words (see [12]), φν⟶0 in Sαβ as ν⟶∞, if and only if for every n∈ℤ+, the sequence φνn,ν≥1, converges to zero uniformly on an arbitrary segment a,b⊂ℝ and for some c, a,B>0 independent of ν, and the inequality(11)φνnx≤cBnnnβexp−ax1/α,x∈ℝ,n∈ℤ+.holds.InSαβ, the continuous translation operation Tx: φξ⟶φξ+x is defined. This operation is also differentiable (even infinitely differentiable [12]) in the sense that the limit relation φx+h−φxh−1⟶φ′x, h⟶0, is true for every function φ∈Sαβ in the sense of convergence in the Sαβ-topology. In Sαβ, the continuous differentiation operator is also defined. The spaces of type S are perfect [12] (that is, the spaces and all bounded sets of which are compact) and closely related to the Fourier transform, namely, the formula FSαβ=Sβα, α,β>0, is correct, where(12)FSαβ≔ψ:ψσ=∫ℝφxeiσxdx,φ∈Sαβ.Moreover, the operatorF : Sαβ⟶Sβα is continuous.LetSαβ′ denote the space of all linear continuous functionals on Sαβ with weak convergence. Since the translation operator Tx is defined in the space Sαβ of test functions, the convolution of a generalized function f∈Sαβ′ with the test function φ∈Sαβ is given by the following formula:(13)f∗φ=f,T−xφˇ⋅≡f,φx−⋅,φˇξ=φ−ξ.It follows from the infinite differentiability property of the argument translation operation inSαβ that the convolution f∗φ is a usual infinitely differentiable function on ℝ.We define the Fourier transform of a generalized functionf∈Sαβ′ by the relation(14)Ff,φ=f,Fφ,∀φ∈Sβα,where the operator F: Sαβ′⟶Sβα′ is continuous.Letf∈Sαβ′. If f∗φ∈Sαβ, ∀φ∈Sαβ, and the convergence φν⟶0 in the Sαβ-topology as ν⟶∞ implies that f∗φν⟶0 in the Sαβ-topology as ν⟶∞, then the function f is called a convolutor in Sαβ. If f∈Sαβ′ is a convolutor in Sαβ, then for an arbitrary function φ∈Sαβ, the formula Ff∗φ=FfFφ is valid, where Ff is a multiplier in Sαβ [12].Recall that the functiong∈C∞ℝ is called a multiplier in Sαβ if gφ∈Sαβ for an arbitrary function φ∈Sαβ, and the mapping φ⟶gφ is continuous in the space Sαβ.


## 3. Nonlocal Multipoint by Time Problem

Consider the functionaσ=1+σ2ω/2, σ∈ℝ, where ω∈1,2 is a fixed parameter. Obviously, the function aσ satisfies the inequality(15)aσ≤cεexpεσω,σ∈ℝ,where cε=2ω/2max1,1/ε for an arbitrary ε>0. Using direct calculations and the Stirling formula, we find that(16)Dσmaσ≤c0B0mm!≤c1B1mmm,m∈ℕ,σ∈ℝ,where B0=B0ω>0, B1=B1ω>0, and c0,c1>0. It follows from (15) and (16) that aσ is a multiplier in S1/ω1. Indeed, let φ∈S1/ω1, that is, the function φ and its derivatives satisfy the inequality(17)Dσkφσ≤cAkkkexp−a|σ|ω,k∈ℤ+,σ∈ℝ,with some positive constants c, A, and a. Then, using the Leibniz formula for differentiating the product of two functions, as well as inequalities (15)–(17), we find that(18)aσφσk≤∑l=0kCklalσφk−lσ=aσφkσ++∑l=1kCklalσ⋅φk−lσ≤cεcAkkkexp−aσω+c1c∑l=1kCklB1lllAk−lk−lk−lexp−aσω.Sinceε>0 is arbitrary, we can put ε=a/2. Then,(19)aσφσk≤cεcAkkkexp−a2σω+c1c∑l=1kCklB1lllAk−lk−lk−l×exp−a2σω≤c2B2kkkexp−a2σω,σ∈ℝ,k∈ℤ+,where c2=ccε+c1 and B2=2maxA,B1. It follows from (19) that a⋅φ is an element of S1/ω1.The operation of multiplying byaσ is continuous in the space S1/ω1. In fact, let φn,n≥1 be a sequence of functions from S1/ω1 convergent to 0 in this space. This means that φn,n≥1⊂S1/ω,A01,B0 with some constants A0, B0>0 and(20)φnδρ=supk,σexpa1−δσω⋅φnkσB0+ρkkk⟶0,n⟶∞,δ,ρ⊂12,13,…,a=1ωeA0ω.In other words, for an arbitraryε˜>0, there exists a number n0=n0ε˜ such that, for n≥n0,(21)φnkσ≤ε˜B0+ρkkkexp−a1−δσω,σ∈ℝ,k∈ℤ+.Using inequalities (15) and (16) (when putting in (15), ε=a/21−δ), we get(22)aσφnσk≤c3ε˜B˜+ρkexp−a21−δσω,k∈ℤ+,n≥n0,where c3=cε+c1 and B˜=2maxB0,B1. It follows from the last inequality that aφnδρ≤c3ε˜, ∀n≥n0, that is, the sequence aφn,n≥1 converges to zero in the space S1/ω,A˜1,B˜ where A˜=21/ωA0 and B˜=2maxB0,B1. This means that the sequence aφn,n≥1 converges to zero in the space S1/ω1, which is what we needed to prove.Remark 1.
It follows from the proven property that the functionaσ=1+σ2ω/2, σ∈ℝ, is also a multiplier in every space S1/ωβ, where β>1. Therefore, in the space Sβ1/ω, β≥1, defined as a continuous linear pseudodifferential operator A, constructed by the function aσ:(23)Aφ=F−1aσFφ,∀φ∈Sβ1/ω,where Aφ=I−Δω/2φ, Δ=d2/dx2.
Let us consider the evolution equation with the operatorA (the Bessel fractional differentiation operator):(24)∂u∂t+Au=0,t,x∈0,+∞×ℝ≡Ω.
By a solution of equation (24), we mean the function ut,x, t,x∈Ω, such that (1) ut,⋅∈C10,+∞ for every x∈ℝ ; (2) u⋅,x∈Sβ1/ω for every t∈0,+∞; (3) ut,x is continuous at every point 0,x of the boundary ΓΩ=0×ℝ of the region Ω; (4) ∃M: ℝ⟶0,∞,∀t∈0,+∞ : utt,x≤Mx, intℝMxdx<∞; and (5) ut,x, t,x∈Ω satisfies equation (24).
For equation (24), we consider the nonlocal multipoint by time problem of finding a solution of equation (24) that satisfies the condition(25)μu0,x−μ1B1ut1,x−⋯−μmBmutm,x=fx,x∈ℝ,f∈Sβ1/ω,where u0,x≔limt⟶+0ut,x, x∈ℝ, m∈ℕ, μ,μ1,…,μm⊂0,∞, and t1,…,tm⊂0,∞ are fixed numbers, 0<t1<t2<⋯<tm<+∞, μ>∑k=1mμk, and B1,…,Bm are pseudodifferential operators in Sβ1/ω constructed by functions (characters) gk : ℝ⟶0,+∞, respectively, Bk=F−1gkσF, k∈1,…,m. The functions gk∈C∞ℝ and k∈1,…,m satisfy the conditions(26)∀ε>0,∀σ∈ℝ:gkσ≤expεσω,∃Lk>0,∀s∈ℕ:Dσsgkσ≤Lksss.
Note that the above properties of the functionsgk imply that gk, k∈1,…,m, is a multiplier in Sβ1/ω.
We are looking for the solution of problems (24) and (25) via the Fourier transform. Due to condition (19),(27)Fut=∫ℝutt,xeiσxdx=∂∂t∫ℝut,xeiσxdx=∂∂tFu.
We introduce the notationFut,x=vt,σ. Given the form of the operators A, B1, …, Bm, we get(28)FAut,x=FF−1aσFu=aσFut,x=aσvt,σ,FBkut,x=gkσFut,x=gkσvt,σ,k∈1,…,m.
So, for the functionv:Ω⟶ℝ, we arrive at a problem with parameter σ:(29)dvt,σdt+aσvt,σ=0,t,σ∈Ω,(30)μv0,σ−∑k=1mμkgkσvtk,σ=f˜σ,σ∈ℝ,where f˜σ=Ffσ. The general solution of equation (29) has the form(31)vt,σ=cexp−taσ,t,σ∈Ω,where c=cσ is determined by condition (30). Substituting (30) into (31), we find that(32)c=f˜σμ−∑k=1mμkgkσexp−tkaσ−1,σ∈ℝ.
Now, putGt,x≔F−1Qt,σ and Qt,σ=Q1t,σQ2σ, and thus,(33)Q1t,σ=exp−taσ,Q2σ=μ−∑k=1mμkgkσQ1tk,σ−1.
Then, thinking formally, we come to the relation(34)ut,x=∫ℝGt,x−ξfξdξ=Gt,x∗fx.
Indeed,(35)ut,x=2π−1∫ℝQt,σ∫ℝfξeiσξdξe−iσxdσ=∫ℝ2π−1∫ℝQt,σe−iσx−ξdσfξdξ=∫ℝGt,x−ξfξdξ=Gt,x∗fx,t,x∈Ω.
The correctness of the transformations performed, the convergence of the corresponding integrals and, consequently, the correctness of formula (35) follow from the properties of the function G, which are given below. The properties of G are determined by the properties of Q because G=F−1Q. So, let us first examine the properties of the function Qt,σ as a function of the variable σ.Lemma 1.
For derivatives ofQ1t,σ, t,σ∈Ω, the estimates(36)DσsQ1t,σ≤cAstγsssexp−tσω,are valid, where γ=0 if 0<t≤1 and γ=1 if t>1, and the constants c>1 and A>0 do not depend on t.Proof.
To prove the statement, we use the Faa di Bruno formula for differentiation of a complex function:(37)DσsFgσ=∑p=1sdpdgpFg∑s!p1!…pl!ddσgσp1…1l!dldσlgσpl,s∈ℕ,where the sum sign is applied to all the solutions in positive integers of the equation p1+2p2+⋯+lpl=s, p1+⋯+pl=m. Put F=eg and g=−taσ. Then,(38)Dσse−taσ=e−taσ∑p=1s∑s!p1!…pl!Λ,where the symbol Λ denotes the expression, and(39)Λ≔ddσ−taσp1⋅12!d2dσ2−taσp2…1l!dldσl−taσpl.
Given estimate (16) is fulfilled, we find that(40)Λ≤c0p1+⋯+plB0p1+2p2+⋯+lpltp1+⋯+pl≤c˜0stpB0s,c˜0=max1,c0.
Using (40) and the Stirling formula, we arrive at the inequalities(41)DσsQ1t,σ≤c˜0sB0stγss!exp−taσ≤cAstγsssexp−tσω,σ∈ℝ,where γ=0 if 0<t≤1 and γ=1 if t>1, and the values c>1 and A>0 do not depend on t.
Lemma is proved.Remark 2.
It follows from estimate (41) that Q1t,⋅∈S1/ω1 for every t>0.Lemma 2.
The functionQ2 is a multiplier in S1/ω2.Proof.
To prove the assertion, let us estimate the derivatives ofQ2. For this we use formula (37) in which we put F=φ−1 and φ=R, where(42)Rσ=μ−∑k=1mμkgkσexp−tkaσ.
Then,Q2=Fφ=R−1 and(43)DσsQ2σ=∑p=1sdpdRpR−1∑s!p1!…pl!ddσRσp1…1l!dldσlRσpl,s∈ℕ.
Given the properties ofg1,…,gm and inequality (41), we find that(44)1l!djdσjRσ≤1j!∑k=1mμk∑i=0jCjiDσigkσ⋅Dσj−ie−tkaσ≤1j!∑k=1mμk∑j=0iCjiLkii!c˜0j−itkγj−ij−i!e−tkaσ≤∑k=1mμk∑i=0jCjiLkic˜0j−itγj−i,where we took into account that i!j−i!≤j!).
Let(45)L˜=maxL1,…,Lm,L0=2maxL˜,c˜0T,then(46)1j!djdσjRσ≤α0L0j,α0=∑k=1mμk,j∈1,…,l,ddσRσp1…1l!dldσlRσpl≤α0L0pα0L02p2…α0L0lpl=α0p1+⋯+plL0p1+2p2+⋯+lpl=α0pL0s.
In addition,dp/dRpR−1=−1pp!R−p+1 and(47)R−1σ=Q2σ=μ−∑k=1mμkgkσexp−tkaσ−1≤μ−∑k=1mμkexp−t1σω+εσω≤μ−∑k=1mμk−1≡β0>0,ε=t1.
Since, by assumption,μ>∑k=1mμk (assuming the properties of g1,…,gm, and 0<t1<t2<⋯<tm). So,(48)dpdRpR−1≤β0p+1p!,DσsQ2σ≤s!∑p=1sβ0p+1p!α0pL0s≤β0L0ss!2∑p=1sβ0p≤β0L0ss⋅s!2β1s≤β2β3ss2s,s∈ℕ.
It follows from the last inequality and boundedness of the functionQ2σ on ℝ that Q2 is a multiplier in S1/ω2.Corollary 1.
For everyt>0, the function Qt,σ=Q1t,σQ2σ, σ∈ℝ, is an element of the space S1/ω2, and the estimates(49)DσsQt,σ≤c˜A˜stγss2sexp−tσω,s∈ℤ+,t,σ∈Ω,are valid, where the constants c˜ and A˜>0 do not depend on t.Taking into account the properties of the Fourier transform (direct and inverse) and the formulaF−1S1/ω2=S21/ω, we get that Gt,⋅∈S21/ω for every t>0. We remove in the estimates of derivatives of the function G (in the variable x), the dependence on t, assuming t>1. To do this, we use the relations(50)xkDxsFφx=ik+sFσsφσk=ik+s∫ℝσsφσkeiσxdσ,k,s⊂ℤ+,φ∈S1/ω2.So,(51)xkDxsGt,x=2π−1ik+s−1s∫ℝσsQt,−σkeiσxdσ.Applying the Leibniz formula for differentiating the product of two functions and estimating the derivatives of the functionQt,σ, we find that(52)σsQt,−σk=∑p=0kCkpσspQk−pt,−σ≤σsQkt,−σ+ksσs−1Qk−1t,−σ+kk−12!ss−1σs−2Qk−2t,−σ+⋯≤c˜A˜ktγkB˜st−s/ωmks+ksA˜k−1tγk−1B˜s−1t−s−1/ωmk−1,s−1+12!kk−1ss−1A˜k−2tγk−2B˜s−2t−s−2/ωmk−2,s−2+…e−t/2σω,where mks=k2kss/ω. Taking into account the results in (see [12], p. 236–243), we find that this double sequence satisfies the inequality(53)ksmk−1,s−1mks≤γ˜k+s,γ˜>0.Bearing in mind the last inequality and also thatt>1, we obtain(54)σsQt,−σk≤c˜A˜ktkB˜smks1+ksA˜B˜mk−1,s−1mks+12!kk−1ss−11A˜2B˜2mk−2,s−2mks+⋯e−t/2σω≤c˜A˜ktkB˜smks×1+ksA˜B˜mk−1,s−1mks+12!1A˜2B˜2ksmk−1,s−1mksk−1s−1mk−2,s−2mk−1,s−1+⋯e−t/2σω≤c˜A˜ktkB˜smks1+γ˜A˜B˜k+s+12!γ˜2A˜B˜2k+s2+⋯e−t/2σω≤c1A¯ktkB¯smkse−t/2σω,A¯=A˜expγ˜A˜B˜,B¯=B˜expγ˜A˜B˜.So,(55)xkDxsGt,x≤2π−1c1A¯ktkB¯smks∫ℝe−t/2σωdσ=c2A¯ktkB¯st−1/ωk2kss/ω,k,s⊂ℤ+.Then,(56)DxsGt,x≤c2B¯st−1/ωss/ωinfkA¯kk2kt−1xk≤c3B¯st−1/ωss/ωexp−a0t−1/2x1/2,where the constants c3, B¯, and a0>0 do not depend on t; here, we used the well-known inequality (see [12], p. 204)(57)e−α/eξ1/α≤infkLkkkαξk≤ce−α/eξ1/α,c=eαe/2,in which α=2 and L=A˜. Thus, such a statement is correct.Lemma 3.
The derivatives (by variablex) of the function Gt,x for t>1 satisfy the inequality(58)DxsGt,x≤c3B¯sss/ωt−1/ωexp−a0t−1/2x1/2,s∈ℤ+,where the constants c3, B¯, and a0>0 do not depend on t.Remark 3.
In what follows, we assume that in condition (25), β=2.
Here are a few more properties of the functionGt,x.Lemma 4.
The functionGt,⋅, t∈0,+∞, as an abstract function of t with values in the space S21/ω, is differentiable in t.Proof.
It follows from the continuity of the Fourier transform (direct and inverse) that, to prove the statement, it suffices to establish that the functionFGt,⋅=Qt,⋅, as a function of a parameter t with values in the space S1/ω2, is differentiable in t. In other words, it is necessary to prove that the boundary value relation(59)ΦΔtσ≔1ΔtQt+Δt,⋅−Qt,⋅⟶∂∂tQt,⋅,Δt⟶0,is performed in the sense as follows:(1)
DσsΦΔtσ⟶Δt⟶0Dσs−aσQt,σ, s∈ℤ+, uniformly on each segment a,b⊂ℝ(2)
DσsΦΔtσ≤c¯B¯ss2sexp−a¯σω, s∈ℤ+, where the constants c¯,B¯,anda¯>0 do not depend on Δt if Δt is small enough
The functionQt,σ, t,σ∈Ω, is differentiable in t in the usual sense. Due to the Lagrange theorem on finite increments,(60)ΦΔtσ=−aσQt+θΔt,σ.
So,(61)DσsΦΔtσ=−∑l=0sCslDσlaσDσs−lQt+θΔt,σ,(62)DσsΦΔtσ−∂∂tQt,σ=−∑l=0sCslDσlaσDσs−lQt+θΔt,σ−Dσs−lQt,σ.
Since(63)Dσs−lQt+θΔt,σ−Dσs−lQt,σ=Dσs−l+1Qt+θ1Δt,σ,0<θ1<1,it follows from this and estimate (49) that(64)Dσs−l+1Qt+θ1Δt,σθΔt⟶0,Δt⟶0,uniformly on an arbitrary segment a,b⊂ℝ. Then,(65)DσsΦΔtσ⟶Dσs∂∂tQt,σ,Δt⟶0,uniformly on each segment a,b⊂ℝ, too. Therefore, condition 1 is satisfied.
Taking into account (61), estimates (15), (16), and (49) for aσ, Qt,σ, and their derivatives, we find that(66)DσsΦΔtσ≤c1cε∑l=0sCslB1lllA˜s−ltγs−ls−l2s−lexp−t+θΔtσω+εσω,where ε>0 is arbitrary fixed. Take ε=t/2, and then(67)DσsΦΔtσ≤c¯B¯ss2sexp−a¯σω,s∈ℤ+,where B¯=2maxB1,A˜tγ, a¯=t/2, and all the constants do not depend on Δt.Corollary 2.
The formula(68)∂∂tf∗Gt,⋅=f∗∂Gt,⋅∂t,∀f∈S21/ω′,t>0,is correct.Proof.
According to the definition of a convolution of a generalized function with a test one, we have(69)f∗Gt,⋅=fξ,T−xGˇt,ξ,Gˇt,ξ=Gt,−ξ.
Then,(70)∂∂tf∗Gt,⋅=limΔt⟶01Δtf∗Gt+Δt,⋅−f∗Gt,⋅=limΔt⟶0fξ,1ΔtT−xGˇt+Δt,ξ−T−xGˇt,ξ.
Due to Lemma4, the relation(71)1ΔtT−xGˇt+Δt,ξ−T−xGˇt,ξ⟶∂∂tT−xGˇt,ξ,Δt⟶0,is performed in the sense of S21/ω topology. So,(72)∂∂tf∗Gt,⋅=fξ,limΔt⟶01ΔtT−xGˇt+Δt,ξ−T−xGˇt,ξ=fξ,∂∂tT−xGˇt,ξ=fξ,T−x∂∂tGˇt,ξ=f∗∂∂tGt,⋅.
The statement is proved.
Since(73)FGt,⋅=Qt,σ=Q1t,σQ2σ=∫ℝGt,xeiσxdx,t>0,we get the formula(74)∫ℝGt,xdx=Q1t,0Q20=λ0e−t,λ0=μ−∑k=1mμkgk0e−tk−1,where a0=1 is taken into account.Lemma 5.
InS21/ω′, the following relations are correct:(75)1Gt,⋅⟶F−1Q2,t⟶+0;2μGt,⋅−∑k=1mμkBkGtk,⋅⟶δ,t⟶+0,where δ is the Dirac delta function.Proof.
(1) Taking into account the continuity property of the Fourier transform (direct and inverse) in spaces ofS′ type, it is sufficient to establish that(76)FGt,⋅=Q1t,⋅Q2⋅⟶Q2⋅,t⟶+0,in the space S1/ω2′. To do this, we take an arbitrary function φ∈S1/ω2, and using the fact that Q2 is a multiplier in the space S1/ω2 and the Lebesgue theorem on the limit passage under the Lebesgue integral sign, we find that(77)Q1t,⋅Q2⋅,φ=Q1t,⋅,Q2⋅φ⋅=∫ℝQ1t,σQ2σφσdσ⟶t⟶+0∫ℝQ2σφσdσ=1,Q2⋅φ⋅=Q2,φ.
This leads us to statement (1) of Lemma5.
(2) Given statement (1) and the form of the operatorsB1,…,Bm, we find that(78)μGt,⋅−∑l=1mμlBlGtl,⋅⟶t⟶+0μF−1Q2−∑l=1mμlBlGtl,⋅=μF−1Q2−∑l=1mμlF−1gl⋅Q1tl,⋅Q2⋅=F−1μQ2⋅−∑l=1mμlgl⋅Q1tl,⋅Q2⋅=F−1μ−∑l=1mμlgl⋅Q1tl,⋅Q2⋅=F−1μ−∑l=1mμlgl⋅Q1tl,⋅⋅μ−∑l=1mμlgl⋅Q1tl,⋅−1=F−11=δ.
Therefore, relation (75) is fulfilled in the space S21/ω′.
Lemma5 is proved.Remark 4.
Ifμ=1, μ1=⋯=μm=0, then problems (24) and (25) degenerate into the Cauchy problem for equation (24). In this case, Q2σ=1, ∀σ∈ℝ, Gt,x=F−1e−taσ, and Gt,⋅⟶F−11=δ for t⟶+0 in the space S11/ω′.Corollary 3.
Let(79)ωt,x=f∗Gt,x,f∈S2,∗1/ω′,t,x∈Ω,where S2,∗1/ω′ is a class of convolutors in S21/ω. Then, in S21/ω′, the following boundary relation holds:(80)μωt,⋅−∑k=1mμkBkωtk,⋅⟶f,t⟶+0.Proof.
Let us prove that the relation(81)Fμωt,⋅−∑k=1mμkBkωtk,⋅⟶Ff,t⟶+0takes place in the space S1/ω2′. Since f∈S2,∗1/ω′ and Gt,⋅∈S21/ω for every t>0, we have(82)Fωt,⋅=Ff∗Gt,⋅=FfFGt,⋅=FfQt,⋅.
So, we need to prove that(83)FfμQt,⋅−∑k=1mμkgk⋅Qtk,⋅⟶Ff,in S1/ω2′, when t⟶+0. Since Qt,⋅=Q1t,⋅Q2⋅⟶Q2⋅ for t⟶+0 in the space S1/ω2′ (see the proof of assertion 1 and Lemma 5), the correlation(84)μQt,⋅−∑k=1mμkgk⋅Qtk,⋅⟶t⟶+0μQ2⋅−∑k=1mμkgk⋅Q1tk,⋅Q2⋅=μ−∑k=1mμkgk⋅Q1tk,⋅Q2⋅=1,is realized in the space S1/ω2′. Thus, relation (81) and (80) are fulfilled in the corresponding spaces. The statement is proved.
The functionGt,⋅ satisfies equation (24) as t>0. Indeed,(85)∂∂tGt,x=∂∂tF−1Qt,σ=F−1∂∂tQt,σ=F−1aσQt,σ,AGt,x=F−1aσF−1Gt,⋅=F−1aσQt,σ.
So,(86)∂Gt,x∂t+AGt,x=0,t,x∈Ω,which is what had to be proved.
It follows from Corollary3 that the nonlocal multipoint by the time problem for equation (24) can be formulated in the following way: find a function ut,x, t,x∈Ω that satisfies equation (24) and the condition(87)μlimt⟶+0ut,⋅−∑k=1mμkBkutk,⋅=f,f∈S2,∗1/ω′,where boundary relation (87) is considered in the space S21/ω′, and the constraints on the parameters μ, μ1, …, μm, t1, …, tm are the same as in case of problems (24) and (25)).Theorem 1.
By nonlocal multipoint by time problem (24), (87) is solvable and its solution is given by the formula(88)ut,x=f∗Gt,x,t,x∈Ω,where ut,⋅∈S21/ω for every t>0.Proof.
Make sure that the functionut,x, t,x∈Ω, satisfies equation (24). Indeed (see Corollary 2),(89)∂ut,x∂t=∂∂tf∗Gt,⋅=f∗∂Gt,⋅∂t,Aut,x=F−1aσFf∗Gt,⋅.
Sincef is a convolutor in S21/ω,(90)Ff∗Gt,⋅=FfFGt,⋅=FfQt,⋅.
So,(91)Aut,x=F−1aσQt,σFfσ=−F∂∂tQt,⋅Ff=−FF∂∂tGt,⋅⋅Ff=−F−1Ff∗∂Gt,⋅∂t=−f∗∂Gt,⋅∂t.
It follows from this that the functionut,x, t,x∈Ω, satisfies equation (24). From Corollary 3, we get that ut,x satisfies condition (87) in the defined sense. The theorem is proved.Remark 5.
If, in condition (87), B1=⋯=Bm=I (I is the identity operator), then we can prove that problems (24) and (87) are well posed and its solution is given by the formula(92)ut,x=f∗Gt,x,f∈S1,∗1/ω′,t,x∈Ω,where Gt,⋅=F−1Qt,⋅∈S11/ω for every t>0, and(93)Qt,σ=e−taσμ−∑k=1mμke−tkaσ−1.Theorem 2.
Supposeut,x, t,x∈Ω, is the solution of problems (24) and (87). Then, ut,⋅⟶0 in the space S21/ω′ as t⟶+∞.Proof.
Recall that the solution of problems (24) and (87) is given by the formula(94)ut,x=f∗Gt,x=fξ,Gt,x−ξ,f∈S2,∗1/ω′,t,x∈Ω.
Let us prove thatut,⋅,φ⟶0 as t⟶+∞ for an arbitrary function φ∈S21/ω. Put(95)Ψtξ=∫−∞+∞Gt,x−ξφxdx,Ψt,Rξ=∫−RRGt,x−ξφxdx,R>0,t>1.
In these notations, we prove that (a) for everyt>1 and R>0, the function Ψt,Rξ belongs to the space S21/ω and Ψt,Rξ⟶Ψtξ in the space S21/ω as R⟶∞; (b) Ψtξ∈S21/ω for every t>1. From this, we get(96)ut,⋅,φ=∫−∞+∞fξ,Gt,x−ξφxdx=fξ,∫−∞+∞Gt,xφx+ξdx=fξ,∫−∞+∞Gt,−yφ−y−ξdy=fξ,∫−∞+∞Gt,−yφˇy−ξdy,φˇz=φ−z,where ut,⋅ is interpreted as a regular generalized function from S21/ω′ for every t>0.
So, let us set property (a). For fixedk,m⊂ℤ+, we have(97)ξkDξmΨt,Rξ≤∫−R+RξkφxDξmGt,x−ξdx≤∫−∞+∞ξkφξ+ηDηmGt,ηdη.
Sinceφ∈S21/ω, the inequality(98)ξkDξmφξ≤cLkMmk2kmm/ω,k,m⊂ℤ+,with some c, L, M>0, is valid. It follows from this that for every η∈ℝ,(99)supξ∈ℝξkφξ=supy∈ℝy−ηkφη=supy∈ℝ∑l=0kCklyl−ηk−lφy≤∑l=0kCklηk−lsupy∈ℝykφy≤c∑l=0kCklLll2lηk−l.
Next, we will use estimate (58). Then,(100)ξkDξmΨt,Rξ≤c∑l=0kCklLll2l∫−∞+∞ηk−lDηmGt,ηdη≤cc3B¯mmm/ωt−1/ω∑l=0kCklLll2l∫−∞+∞ηk−lexp−a0t−1/2η1/2dη.
By the direct calculations, we find that(101)∫−∞+∞ηk−lexp−a0t−1/2η1/2dη≤c4L˜kk2k−ltk−l+1,c4,L˜>0.
So,(102)ξkDξmΨt,Rξ≤cc3c4B¯mt1−1/ωmm/ω∑l=0kCklLlL˜k−ltk−ll2lk2k−l≤c¯B¯mL¯kk2kmm/ω,where c¯=cc3c4t1−1/ω and L¯=2maxL,L˜t. Therefore, Ψt,Rξ∈S21/ω for every t>1 and an arbitrary R>0. Furthermore, we make sure that Ψt,Rξ⟶Ψtξ as R⟶∞, with all its derivatives uniformly in ξ on each segment a,b⊂ℝ. In addition, the set of functions ξkDξmΨt,Rξ, k,m⊂ℤ+, is uniformly bounded (by R) in the space S21/ω (this property follows from estimate (102) in which c¯, B¯, and L¯>0 do not depend on R). This means that condition (a) is fulfilled.
Condition (b) follows from (a) because, in a perfect space, every bounded set is compact.
Using properties (a) and (b), we obtain the equality(103)ut,⋅,φ=∫−∞+∞Gt,−yf∗φˇydy.
Sincef∈S21/ω′ is a convolutor in S21/ω, we have f∗φˇ∈S21/ω. Then, given estimate (58) (at s=0), we find that(104)ut,⋅,φ≤∫−∞+∞Gt,−yf∗φˇydy≤ct1/ω∫−∞+∞f∗φˇydy≤c0t1/ω⟶0,t⟶+∞,for an arbitrary function φ∈S21/ω, i.e., ut,⋅⟶0 in the space S21/ω′ as t⟶+∞. The theorem is proved.
If the generalized functionf in (87) is finite (i.e., its support suppf is a finite set in ℝ), then we can say on uniform tending a solution ut,x of problems (24) and (87) to zero on ℝ as t⟶+∞. Note that every finite generalized function is a convolutor in S spaces. This property follows from the general result concerning the theory of perfect spaces (see [12], p. 137): if Φ is a perfect space with differential translation operation, then every finite functional is a convolutor in Φ. Finite functionals form a fairly wide class. In particular, every bounded set F⊂ℝ is a support of some generalized function [12].Theorem 3.
Letut,x be a solution of problems (24) and (87) with the boundary function f in condition (87), which is an element of S2β′⊂S21/ω′, β>1, with finite support in ℝ. Then, ut,x⟶0 uniformly on ℝ as t⟶+∞.Here is a scheme for proving this statement. Letsuppf⊂a1,b1⊂a2,b2⊂ℝ. Consider the function φ∈S2β, β>1, such that φx=1, x∈a1,b1, and suppφ⊂a2,b2. This function exists when β>1 because S2β contains finite functions [12]. Consider the function(105)ut,x=fξ,φξGt,x−ξ+fξ,γξGt,x−ξ,where γ=1−φ. Since suppγξGt,x−ξ∩suppφ=∅, we have(106)ut,x=t−1/ωfξ,t1/ωφξGt,x−ξ.To prove the formulated above statement, it remains to establish that the set of functionsΦt,xξ=t1/ωφξGt,x−ξ is bounded in the space S2β, β>1, for large t and x∈ℝ.For example, if, in condition (87), f=δ, then δ is a convolutor in the space S, suppδ=0, and ut,x=δ∗Gt,x=Gt,x. It follows directly from estimate (58) that Gt,x⟶0 uniformly on ℝ as t⟶+∞.


## 4. Conclusion

In this paper, the solvability of a nonlocal multipoint by the time problem for the evolutionary equation∂u/∂t+I−d2/dx2ω/2u=0, t,x∈0,+∞×ℝ is proved, herewith the operator I−d2/dx2ω/2 is interpreted as a pseudodifferential operator in the space S21/ω constructed by function 1+σ2ω/2, σ∈ℝ, while the nonlocal multipoint by time condition also contains pseudodifferential operators constructed on smooth symbols. The representation of the solution is given in the form of a convolution of the fundamental solution with the initial function which is an element of the space of generalized functions of the ultradistribution type (the coagulator in space S21/ω). The behavior of the solution ut,x, t⟶+∞ in the space of generalized functions S21/ω′, is investigated. The conditions for the initial generalized function are found under which the solution is uniformly stabilized to zero on ℝ. The method of research of a nonlocal multipoint by time problem offered in this paper allows to interpret differential-operator equations of a form ∂u/∂t+φi∂/∂xu=0 as an evolutionary equation with a pseudodifferential operator φi∂/∂x=F−1φ⋅F constructed by the function φ acting in certain countable-normalized space of infinite-differential functions (the choice of space depends on the properties of the function which is the symbol of the operator φi∂/∂x).



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