# Parabolic Equations of Infinite Order withL1 Data

Authors: Mohammed Housseine Abdou; Moussa Chrif; Said El Manouni
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We prove an existence result of a nonlinear parabolic equation under Dirichlet null boundary conditions in Sobolev spaces of infinite order, where the second member belongs toL1QT.


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## Body

## 1. Introduction

This paper is devoted to the study of the following strongly nonlinear parabolic problem of Dirichlet type in the cylinderQT:
(P)∂u∂t+Au+gt,x,u=ft,xinQT,u0,x=0,Dωu=0on⁡ST∀ω=0,1…,
where Ω is a bounded open subset of RN and QT=(0,T)×Ω is a cylinder with lateral surface ST=[0,T]×Γ, with Γ is the boundary of Ω. A is a nonlinear elliptic operator of infinite order defined by
(1)Au=∑α=0∞‍-1αDαAαt,x,Dγu,γ≤α.
Such operators include as a special case Leray-Lions types in the usual sense.The real functionsAα(t,x,ξ) are assumed to satisfy some growth and coerciveness conditions without supposing a monotonicity condition in ξ, for all multi-indices α.The nonlinear termg satisfies natural growth on |u| and has to fulfil a sign condition.The dataf is assumed to satisfy
(2)f∈L1QT.
In the case of infinite order, Dubinskiĭ [1] has proved, under some growth hypothesis and certain monotonicity conditions, the existence of solutions for the Dirichlet problem associated with the equation Au=f in some general functional Sobolev spaces of infinite order W0∞(aα,pα) of variables exponents pα, with α being a multi-indice. The same author has investigated the existence result for parabolic elliptic problems governed by operators of infinite orders. In fact, also in [1], Dubinskiĭ has proved by considering, further, the monotonicity of the operator A that the problem ∂u/∂t+Au=f has a solution in Lp(0,T,W0∞(aα,p)),  p>1, in the variational case (i.e., where f belongs to the dual space).Another work has been shown, in the variational case in [2], the existence of solutions for strongly parabolic nonlinear equations of infinite order related to the problem ∂u/∂t+Au+g(t,x,u)=f.Our purpose in this paper is to prove the existence of solutions for parabolic equations, in Sobolev spaces of infinite order withL1 data, associated with the problem ∂u/∂t+Au+g(t,x,u)=f.More precisely, we will assume more less restrictions on the operatorA (no monotonicity condition) and deal with a different approach by involving a truncation of the perturbations g. Next, we use the monotonicity of a part of approximate operator which contains a linear term of higher order of derivation that satisfies the monotonicity condition and prove the existence of solutions in the framework of function space Lp(0,T,W0∞(aα,p)),  p>1.Let us mention that an interesting result concerning the stationary counterpart of the problem(P) has been proved in [3, 4].


## 2. Preliminaries

LetΩ be a bounded domain in RN,  aα≥0,  p>1 real numbers for all multi-index α, and ∥·∥p the usual Lebesgue norm in the space Lp(Ω). The Sobolev space of infinite order is the functional space defined by
(3)W∞aα,pΩ=u∈C∞Ω:u∞p=∑α=0∞‍aαDαupp<∞.
Here Dα=∂|α|/(∂x1)α1⋯(∂xN)αN.We denote byC0∞(Ω) the space of all functions with compact support in Ω with continuous derivatives of arbitrary order.Since we will deal with the Dirichlet problem, we will use the functional spaceW0∞(aα,p)(Ω) defined by
(4)W0∞aα,pΩ=u∈C0∞Ω:u∞p=∑α=0∞‍aαDαupp<∞.
In contrast with the finite order Sobolev space, the very first question, which arises in the study of the spaces W∞(aα,p)(Ω), is the question of their nontriviality (or nonemptiness), that is, the question of the existence of a function u such that u∞<∞.Definition 1 (see [1]).
The spaceW0∞(aα,p)(Ω) is called nontrivial space if it contains at least one function which is not identically equal to zero; that is, there is a function u∈C0∞(Ω) such that u∞<∞.It turns out that the answer of this question depends not only on the given parametersaα and p of the spaces W∞(aα,p)(Ω), but also on the domain Ω.The dual space ofW0∞(aα,p)(Ω) is defined as follows:
(5)W-∞aα,p′Ω=f:f=∑α=0∞‍-1αDαfα,f-∞p′=∑α=0∞‍aαfαp′p′<∞,
where fα∈Lp′(Ω) for all multi-indices α and p′ is the conjugate of p;  that is,  p′=p/p-1 (for more details about these spaces, see [1, 5]).By the definition, the duality of the spaceW-∞(aα,p′)(Ω) and W0∞(aα,p)(Ω) is given by relation
(6)f,v=∑|α|=0∞‍aα∫Ω‍fα(x)Dαv(x)dx,
which, as it is not difficult to verify, is correct.Let us denote byLp(0,T,W0∞(aα,p)) the space of functions u(t,x) which has finite norm
(7)up,∞p=∫0T‍u∞pdt
and is equal to zero together with all derivatives Dωu on the lateral surface S. In other words one has
(8)Lp(0,T,W0∞(aα,p))=umeasurable:up,∞p=∑|α|=0∞‍aα∫0T‍Dαuppdt<∞,DωuS=0,ω=0,1,…umeasurable:∥u∥p,∞p=∑|α|=0∞‍aα∫0T‍∥Dαu∥ppdt.
Further, let Lp′(0,T,W-∞(aα,p′)) be the dual space of the space Lp(0,T,W0∞(aα,p)), that is, the space of generalized functions f(t,x) having a form
(9)f(t,x)=∑|α|=0∞‍-1|α|aαDαfα(t,x),
where fαt,x∈Lp′(QT) and
(10)ρ′f=∑α=0∞‍aα∫0T‍fαt,xp′p′dt<∞.
The value of f(t,x)∈Lp′(0,T,W-∞(aα,p′)) on an element v(t,x)∈Lp(0,T,W0∞(aα,p)) is defined by the formula
(11)f,v=∑α=0∞‍aα∫0T‍∫Ω‍fαt,xDαvt,xdxdt,
which, as easy to see, is correct.Sobolev spaces of infinite order have extensive applications to the theory of partial differential equations and, among their number, in mathematical physics. The basis of these applications is the nonformal algebra of differential operators of infinite orders as the operators, acting in the corresponding Sobolev spaces of infinite order. This makes it possible, by considering∂/∂x as a parameter, to solve a partial equation as ordinary differential equation, to which are adjoined the initial or boundary conditions.More explicitly, we cite the following examples of operators of infinite order which are closely inspired from the ones used in Dubinskiĭ [1].Example 2.
Consider the following operator:(12)Au=-I+▵u.
Our technique here consists in exploiting certain results in the setting of functional spaces of infinite order. Thus as in [1], we can write the operator A as follows:
(13)Au=(-I+▵)u=∑k=0∞‍ak-▵ku,
where ak>0, k=0,1,…, are real numbers which guarantee the nontriviality of the corresponding functional space defined by
(14)W0∞(ak,2)(Ω)=u∈C0∞Ω:ρu=∑k=0∞‍ak∇u22<∞.
Moreover for any f∈L2(0,T,W-∞(ak,2)(Ω)), the parabolic problem
(15)∂u∂t+∑k=0∞‍ak-▵ku=f,x∈Ω
has a solution u∈L2(0;T;W0∞(ak;2)(Ω)), in the variational sense.
By using the recent work of authors (see Theorem 3.1. in [2]), the strongly nonlinear parabolic problem
(16)∂u∂t+∑k=0∞‍ak-▵ku+g(t,x,u)=fx∈Ω,
has also a solution u∈L2(0;T;W0∞(ak;2)(Ω)), in the variational sense, for any f∈L2(0,T,W-∞(ak,2)(Ω)). Here g is a nonlinear term which has to fulfil a sign condition (see Section 3).Remark 3.
For examples of the nontriviality of Sobolev spaces of infinite order, we refer the reader to [1, 5–7] for details.


## 3. Main Result

In this section we formulate and prove the main result. We denote byλα the number of multi-indices γ such that |γ|≤|α|. Let A be the nonlinear operator of infinite order defined as in (1), with Aα:(0,T)×Ω×Rλα→R being a real function.Let us now formulate the following assumptions.(A1)
Aα(t,x,ξγ) is a Carathéodory function for all α,|γ|≤|α|.(A2)
For a.e.(t,x)∈QT, all m∈N*, all ξγ,ηα,|γ|≤|α|, and some constant c0>0, we assume that
(17)∑α=0m‍Aαt,x,ξγηα≤c0∑α=0m‍aαξαp-1ηα,wherep>1,  aα≥0 are real numbers for all multi-indices α.(A3)
There exist constantsc1>0,  c2≥0 such that
(18)∑|α|=0m‍Aα(t,x,ξγ)ξα≥c1∑|α|=0m‍aαξαp-c2for allm∈N*, for all ξγ,ξα;|γ|≤|α| and for a.e. (t,x)∈QT.(A4)
The spaceW0∞(aα,p)(Ω) is nontrivial.As regards the nonlinear termg, we assume that g satisfies the following natural growth on |u| and the classical sign condition.(G)
g:QT×R→R is a Carathéodory function satisfying
(19)gt,x,s≤b1sp-1+b2,g(t,x,s)s≥0,for a.e.(t,x)∈QT,  s∈R, and some constants b1 and b2.Concerning the second memberf, we assume that
(20)f∈L1(QT).We will prove the following existence theorem.Theorem 4.
Under assumptions(A1)–(A4) and (G), for any right side f∈L1(QT) there exists at least a function u such that(1)
u(t,x)∈Lp(0,T,W0∞(aα,p)), ∂u/∂t∈Lp′(0,T,W-∞(aα,p′));(2)
u(0,x)=0;(3)
for any functionv(t,x)∈L∞(0,T,W0∞(aα,p)), the following identity
(21)∫0T‍∂u∂t,vdt+∑α=0∞‍aα∫0T‍∫Ω‍Aαt,x,DγuDαvdxdt+∫0T‍∫Ω‍gt,x,uvdxdt=∫0T‍〈f,v〉dt
is valid.Proof.
we proceed by steps in order to prove our result.
Step  1. The approximate problem.
Set for a.e.(t,x)∈QT(22)fkt,x=Tkft,x,gk(t,x,u)=Tkg(t,x,u),
where Tk is the usual truncation given by
(23)Tkη=ηifη<kkηηifη≥k.
It is clear that fk≤k for a.e. (t,x)∈QT. Thus, it follows that fk∈L∞(QT).
Further, we have(24)fk⟶ffora.e.t,x∈QT,fk≤f∈L1(QT),
and from Lebesgue’s dominated convergence theorem, see [8], we conclude that
(25)fk⟶finL1(QT).
Let k∈N* sufficiently large. Define the operator A2k+2 of order 2k+2 by
(26)A2k+2u=∑α=k+1‍-1k+1cαD2αu+∑|α|=0k‍-1|α|Dα(Aα(t,x,Dγu)).
Note that cα are constants small enough such that they fulfil the conditions of the following lemma introduced in [1].
In fact, such a condition imposed on eachcα is required to ensure the nontriviality of the space W0∞(cα,2).
Lemma  5 (cf. [1]).For any nontrivial spaceW0∞(aα,pα)there exists a nontrivial spaceW0∞(cα,2)such thatW0∞(aα,pα)⊂W0∞(cα,2).
The operatorA2k+2 is clearly monotone since the term of higher order of derivation is linear and satisfies the monotonicity condition (see [1, 3]). Moreover, thanks to the truncation Tk as in [9] and from assumptions (A1), (A2), and (A3), we deduce that the operator A2k+2+gk is bounded, coercive, and pseudo-monotone. Then, it is well known (see Lions [10]) that there exists uk∈Lp(0,T,W0k+1,p(Ω)) such that
(Pk)∂uk∂t+A2k+2uk+gk(t,x,uk)=fk(t,x),uk(0,x)=0.
In the variational formulation, we get
(27)∫0T‍∂uk∂t,vdt+∫0T‍A2k+2uk,vdt+∫0T‍∫Ω‍gkt,x,ukvdxdt=∫0T‍〈fk,v〉dt,
for any v∈L∞(0,T,W0k+1(Ω)).
Step  2 (a priori estimates). Let us choose v=uk as a test function in (Pk). Then using the sign condition in (G), one has the estimates
(28)∑|α|=k+1‍cα∫0T‍Dαuk22dt+∑|α|=0k‍aα∫0T‍Dαukppdt≤c2,(29)∫Q‍gk(t,x,uk)ukdxdt≤c2.
In the sequel c2,c3,c4,…  designate arbitrary constants not depending on k.
From the first equality in(Pk) and estimates (28) and (29), we remark that ∂uk/∂t∈Lp′(0,T,W0-k-1,p′(Ω)). In addition, for any v∈L∞(0,T,W0∞(aα,p)) the following equality is valid:
(30)∫0T‍∂uk∂t,vdt≤Q1+Q2+Q3,
where
(31)Q1=∫0T‍∫Ω‍fkt,xvdxdt,Q2=∫0T‍∫Ω‍gkt,x,ukvdxdt,Q3=∑|α|=k+1‍cα∫0T‍∫Ω‍DαukDαvdxdt+∑α=0k‍aα∫0T‍∫Ω‍Dαukp-1Dαvdxdt.
Regarding the quantityQ1, one has
(32)Q1=∫0T‍∫Ω‍fkt,xvdxdt≤∫0T‍∫Ω‍ft,xvdxdt≤∫0T‍∫Ω‍f·vdxdt≤fL1(QT)vL∞(QT),
and so
(33)Q1≤c3vL∞(QT).
We also have
(34)Q2≤∫0T‍∫Ω‍ukp-1v+vdxdt≤∫0T‍ukpp-1vpdt+c4∫0T‍vppdt1/p≤∫0T‍ukppdt1/p′∫0T‍vppdt1/p+c4∫0T‍vppdt1/p≤(c2+c4)vp,∞,
where c2 is the constant of the estimate (28). Then one gets
(35)Q2≤c5vp,∞.
Moreover, for the last term Q3, one has
(36)Q3=J1+J2,
where
(37)J1=∑α=k+1‍cα∫0T‍∫Ω‍DαukDαvdxdt≤∑α=k+1‍cα∫0T‍Dαuk22dt1/2×∑α=k+1‍cα∫0T‍Dαv22dt1/2≤c21/2vp,∞,J2=∑α=0k‍aα∫0T‍∫Ω‍Dαukp-1Dαvdxdt≤∑α=0k‍aα∫0T‍Dαukppdt1/p′×∑α=0k‍aα∫0T‍Dαvppdt1/p≤c21/p′vp,∞.
Then, one deduces that
(38)Q3≤c6vp,∞.
Combining (30), (33), (35), and (38), it follows that
(39)∫0T‍∂uk∂t,vdt≤c7vp,∞+c3vL∞(QT).
This implies that
(40)∂uk∂tp′,-∞≤c8;
that is, the derivatives ∂uk/∂t form a bounded set in the space Lp′(0,T,W-∞(aα,p′)).
Now, estimates (28) and (40) permit us to apply the well known lemma of compactness (see Lions [11]).
LetB0, B, and B1 be Banach spaces. Let us set
(41)Y=u:u∈Lp00,T,B0,u′∈Lp10,T,B1,
where p0>1, p1>1 are real numbers.
Lemma  6 (cf. [1]).Let the imbeddings(42)B0⊂B⊂B1hold; moreover, let the imbeddingB0⊂Bbe compact. Then(43)Y⊂Lp0(0,T,B)and this imbedding is compact.
In order to apply this lemma, define(44)B0=WS+1(aα,p)=ux:∑α=0S‍aαDαupp<∞,B=WS(aα,p),B1=W-∞(aα,p′);p0=p,p1=p′,
where S≥0 is arbitrary and p′=p/p-1.
Step  3 (convergence of the approximate problem (Pk)). In view of (28) and (40), we deduce that the family uk of solutions of problems (Pk) is compact in the space Lp(0,T,WS(aα,p)), where S is arbitrary. Consequently, by similar argument as in the elliptic case (using the diagonal process), see [3] or [1], one gets that the sequence uk converges strongly together with all derivatives Dωuk to a function u∈Lp(0,T,W0∞(aα,p)).
Letting nowm>0 be fixed, E a measurable subset of QT, and ɛ>0, we have
(45)∫E‍gkt,x,ukdxdt≤∫E∩uk≤m‍gkt,x,ukdxdt+1m∫E∩uk>m‍gkt,x,ukukdxdt≤∫E∩uk≤m‍(b1ukp-1+b2)dxdt+1m∫QT‍gk(t,x,uk)ukdxdt≤mp-1+1E+c2m,
where c2 is the constant of (29) which is independent of k.
For|E| sufficiently small and c2/m<ɛ/2, we obtain
(46)∫E‍gk(t,x,uk)≤ɛ.
Using Vitali’s theorem, we get
(47)gk(x,t,uk)⟶gx,t,uinL1(QT).
On the other hand, in view of Fatou’s lemma and (29), we obtain
(48)∫QT‍gx,t,uuds≤lim⁡k→+∞∫QT‍gkx,t,ukukds≤c2;
this implies that
(49)g(x,t,u)u∈L1(QT).
Now, we will prove that
(50)lim⁡k→+∞∫0T‍A2k+2uk,vdt=∫0T‍Au,vdt
for all v∈L∞(0,T,W0∞(aα,p)).
In fact, letk0 be a fixed number sufficiently large (k>k0) and let v∈L∞(0,T,W0∞(aα,p)). Set
(51)∫0T‍Au-A2k+2uk,vdt=I1+I2+I3,
where
(52)I1=∑|α|=0k0‍∫0T‍Aαt,x,Dγu-Aαt,x,Dγuk,Dαvdt,I2=∑|α|=k0+1∞‍∫0T‍Aαt,x,Dγu,Dαvdt,I3=-∑α=k0+1k‍∫0T‍Aαt,x,Dγuk,Dαv-∑α=k+1‍cαDαu,Dαvdt,
or, in another form,
(53)I3=-∑α=k0+1k+1‍∫0T‍Aαt,x,Dγuk,Dαvdt,
with Aα(t,x,ξγ)=cαξα and cα≥0 for |α|=k+1 (cα are constants given in Lemma 5).
We will go to limit ask→+∞ to prove that I1, I2, and I3 tend to 0. Starting by I1, we have I1→0 since A(t,x,ξγ) is of Carathéodory type.
The termI2 is the remainder of a convergence series; hence I2→0.
For what concernsI3, for all ɛ>0, there holds k(ɛ)>0 (see [8, page 56]) such that
(54)∑|α|=k0+1k+1‍∫0T‍Aαt,x,Dγuk,Dαvdt≤∑α=k0+1k+1‍∫0T‍Aαt,x,Dγuk,Dαvdt≤c0∑α=k0+1k+1‍aα∫0T‍∫Ω‍Dαukp-1Dαvdxdt≤c0∑|α|=k0+1k+1‍aα∫0T‍Dαukpp-1Dαvpdt≤ɛc0∑α=k0+1k+1‍aα∫0T‍Dαukppdt+c0kɛ∑α=k0+1k+1‍aα∫0T‍Dαvppdt≤ɛc0c2+c0k(ɛ)∑|α|=k0+1∞‍aα∫0T‍Dαvppdt,
where c2 is the constant given in the estimate (28). Moreover the term
(55)∑|α|=k0+1∞‍aα∫0T‍Dαvppdt
is the remainder of a convergent series; therefore I3→0 holds.
Finally, we conclude that(56)∫0T‍A2k+2uk,vdt⟶∫0T‍Au,vdtask⟶+∞
for all v∈L∞(0,T,W0∞(aα,p)).
Moreover, it is clear that(57)∫0T‍fk,vdt⟶∫0T‍f,vdt
as k→+∞ since fk→finL1(QT).
Consequently, by passing to the limit in(Pk), we obtain
(58)∫0T‍∂u∂t,vdt+∫0T‍Au,vdt+∫QT‍g(t,x,u)vdxdt=∫0T‍f,vdt,
for all v∈L∞(0,T,W0∞(aα,p)).
That is,(59)∫0T‍∂u∂t,vdt+∑α=0∞‍aα∫0T‍∫Ω‍Aαt,x,DγuDαvdxdt+∫0T‍∫Ω‍g(t,x,u)vdxdt=∫0T‍〈f,v〉dt,
for all v∈L∞(0,T,W0∞(aα,p)).
This completes the proof.


## 4. Example

The following example of an operator of infinite order is closely related to the one used in [12].Let us consider the operator:(60)Au=∑α=0∞‍-1αDαaαDαup-2Dαu,
where aα≥0 is a sequence of numbers, p>1 is a number such that the space W∞(aα,p)(Ω) is not trivial (e.g., if aα=[(2α)!]-p,p>1 and dim⁡Ω=1); then the conditions A1,  A2, and A3 are satisfied.As regards a functiong that satisfies the condition (G), let us consider
(61)g(t,x,s)=ssrh(x),withr>0,
where h∈L1(Ω), h(x)≥0, a.e.Consequently, for the described above operatorA and the nonlinear term g  the existence result follows immediately from Theorem 4.



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