A global in time parabolic equation for symmetric Lévy operators
Jean-Daniel Djida, Guy F. Foghem Gounoue, Yannick Kouakep Tchaptchie
aa r X i v : . [ m a t h . A P ] F e b A GLOBAL IN TIME PARABOLIC EQUATION FOR SYMMETRIC L´EVYOPERATORS
JEAN-DANIEL DJIDA, GUY F. FOGHEM GOUNOUE, AND YANNICK KOUAKEP TCHAPTCHIE
Abstract.
The overreaching goal of this paper is to investigate the existence and uniqueness ofweak solutions to a semilinear parabolic equation involving symmetric integrodifferential operatorsof L´evy type and a term called the interaction potential, that depends on the time-integral of thesolution over the entire interval of solving the problem. The existence and uniqueness of a weaksolution of the nonlocal complement value problem is proven under fair conditions on the interactionpotential. Introduction
Let Ω be an open bounded domain in R N ( N ≥ T >
0, we consider the following parabolicnonlocal problem with Dirichlet complement value: ∂ t u + L u + ϕ (cid:16) Z T u ( · , τ ) d τ (cid:17) u = 0 in Ω T := Ω × (0 , T ) ,u = 0 in Σ := ( R N \ Ω) × (0 , T ) ,u ( · ,
0) = u in Ω , (1.1)where u = u ( x, t ) is an unknown scalar function, and ϕ a scalar function that will be specified below.The initial state u : Ω → R is prescribed. Here, we restrict ourselves on a purely integrodifferentialoperator of L´evy type L , which is a particular type of nonlocal operators acting on a smoothmeasurable function u : R N → R as follows L u ( x ) := p . v . Z R N ( u ( x ) − u ( y )) ν ( x − y ) d y, ( x ∈ R N ) , (1.2)whenever the right hand side exists and makes sense. Here and henceforward, the function ν : R N \ { } → [0 , ∞ ] is the density of a symmetric L´evy measure. In other words, ν is positive andmeasurable such that ν ( − h ) = ν ( h ) for all h ∈ R N and Z R N (1 ∧ | h | ) ν ( h ) d h < ∞ . (1.3)Notationally, for a, b ∈ R , we write a ∧ b to denote min( a, b ). In addition, we will also assume that ν does not vanish on sets of positive measure. In view of the nonlocal character of the operator L , itappears natural tout prescribe the Dirichlet condition u = 0 the complement set ( R N \ Ω) × (0 , T ).A prototypical example of an operator L is the fractional Laplacian ( − ∆) s , which is obtained byletting ν ( h ) = C N,s | h | − s − N with h = 0 and fixed s ∈ (0 , C N,s (c.f [19, Chapter2] for an elementary computation) given by
Mathematics Subject Classification.
Key words and phrases.
Parabolic Integro-Differenial Equation (IDE), Symmetric L´evy operator, Weak solvability.The first author is supported by the Deutscher Akademischer Austausch Dienst/German Academic ExchangeService (DAAD).The second author is supported by the DFG via the Research Group 3013: “Vector-and Tensor-Valued SurfacePDEs”. C N,s := (cid:18)Z R N − cos( h ) | h | N +2 s d h (cid:19) − = 2 s Γ (cid:0) s + N (cid:1) π N (cid:12)(cid:12) Γ (cid:0) − s (cid:1)(cid:12)(cid:12) , is chosen so that the Fourier transform gives the relation \ ( − ∆) s u ( ξ ) = | ξ | s b u ( ξ ), ξ ∈ R N , holds forall u ∈ C ∞ c ( R N ). The fractional Laplacian is one of the most heavily studied integrodifferentialoperators. Some basics notions related to the fractional Laplacian can be found, for example, inthe references [4, 7, 17, 26] and many other references therein, for further related subjects, seealso [1, 5, 6, 9, 34]. The operator L in (1.2) arises naturally in the study of pure L´evy stochasticprocesses with the jump interaction measure ν ( h )d h . Analogous to the classical case, attemptshave been made recently (see, e.g., [19]), where elliptic and parabolic problems related to this typeof nonlocal operators are studied, see also [12, 13, 14, 15] where various other typical studies relatedto this of nonlocal operator are considered.Our main result (see Theorem 4.2 and Theorem 4.4) consist into proving the existence anduniqueness of a weak solution to the problem (1.1). It is noteworthy emphasizing that, we are ableto obtain the uniqueness for T sufficiently small by imposing some a boundedness condition on theinitial value u . As a side remark, we point out that under appropriate setting (see Remark 4.5),analogue achievements can be obtain replacing the homogeneous Dirichlet complement conditionin problem (1.1) with the homogeneous Neumann complement condition. We do this provided thatthe potential ϕ satisfies the following crucial assumption. Assumption 1.1.
The potential ϕ : R → [0 , ∞ ) is a continuous non-negative function such that ϕ (0) = 0 and τ ϕ ( τ ) τ is a non-decreasing differentiable function whose derivative is bounded onevery compact subset of R .This assumption admits functions ϕ that are not convex and not increasing as its argument tendsto + ∞ . Besides that, we do not impose any restrictions on the growth of ϕ at infinity.An interesting feature of the problem under consideration is that main equation in (1.1) containsa non-local in time term that depends on the integral over the whole interval (0 , T ) on which theproblem is being solved and a nonlocal operator of L´evy type in space. The mixture of nonlocalterms (spacial and time variables) appearing in (1.1) render the problem somehow fully nonlocal.Thus, the reason (1.1) is called nonlocal and global in time. It is noteworthy emphasizing that,similar analysis has been carry out in [30] where the classical Laplace operator − ∆ is used in place ofthe nonlocal operator L . There are several works that study problems with memory for parabolicequations which includes the integral of the solution from the initial to the current time and itis not difficult to find appropriate works on this subject. The problems with memory differ fromours. As pointed out in [30], we need to know the “future” in order to determine the coefficientin equation (1.1) and we need to deal with the two nonlocality terms in both spacial and timevariables. It is worth emphasizing that the problem in (1.1) cannot be reduced to known ones byany transformation. However, in the paper [29] in the classical scenario that study problems wherethe “future” stands in the boundary and initial data. We point that the classical problem appearswhile modelling a biological nano-sensor in the chaotic dynamics of a polymer chain or, as it is alsocalled, a polymer chain in an aqueous solution see for instance [30, 31, 32] and references therein.In [31], the weak solvability of the problem is proven for the case where u is a positive boundedfunction and ϕ is the so called Flory–Huggins potential. The positiveness is a natural requirementsince u is a density of probability. The Flory–Huggins potential is a convex increasing functionthat tends to infinity as its argument approaches a certain positive value. GLOBAL IN TIME PARABOLIC EQUATION FOR SYMMETRIC L´EVY OPERATORS 3
The original work of [30, 31] demonstrate that, in the case where only the Laplace operator isinvolved, the problem (1.1) makes sense. In parallel with the article [30], we take this work tothe next stage by using the generator of a pure jump stochastic process of L´evy type, which is asymmetric nonlocal type operator of the form L , to prove further results on weak solvability forthis type of problem.The rest of the paper is structured as follows. In Section 2, we provide some preliminaries well-known results and functions spaces which are useful in this paper. In Section 3, we prove auxiliaryresults which are the milestones to prove our core result. Finally, Section 4 is devoted to the proofof the existence and uniqueness of a weak solution to the problem (1.1) thereby constituting themain goal of this article. We prove the existence with the aid of the Tychonoff fixed-point theoremand prove the uniqueness for sufficiently small T . Acknowledgement:
The authors thank Victor St21 for helpful and productive discussions onTheorem 3.2 and Lemma 4.3. 2.
Preliminaries Notions
The purpose of this section is to introduce notations and some preliminary results. Let us collectsome basics on nonlocal Sobolev-like spaces in the L setting that are generalizations of Sobolev–Slobodeckij spaces and which will very helpful in the sequel. Let us emphasize that, those functionspaces are tailor made elliptic complement value problem involving symmetric L´evy operators oftype L . We refer the reader to [19] more extensive discussions on this topic.From now on, unless otherwise stated, Ω ⊂ R N is an open bounded set. We also assumethat ν : R N \ { } → [0 , ∞ ] has full support, satisfies the L´evy integrability condition, i.e., ν ∈ L ( R N , ∧ | h | d h ) and is symmetric, i.e., ν ( h ) = ν ( − h ) for all h ∈ R N . We define the space V ν (Ω | R N ) := n u : R N → R meas : E ( u, u ) < ∞ o , (2.1)where E ( · , · ) is the bilinear form defined by E ( u, v ) := 12 Z Z Q (Ω) ( u ( x ) − u ( y ))( v ( x ) − v ( y )) ν ( x − y ) d y d x (2.2)and Q (Ω) is the cross-shaped set on Ω given by Q (Ω) := (Ω × Ω) ∪ (Ω × ( R N \ Ω)) ∪ (( R N \ Ω) × Ω) . We endow the space V ν (Ω | R N ) with the norm k u k V ν (Ω | R N ) := (cid:16) Z Ω | u ( x ) | d x + E ( u, u ) (cid:17) . We point out that the notation V ν (Ω | R N ) is to emphasize that the integral of the measurable map( x, y ) ( u ( x ) − u ( y )) ν ( x − y ) performed over Ω × R N is finite. From the local scenario pointof view, it is fair to see the space V ν (Ω | R N ) as the nonlocal replacement of the classical Sobolevspace H (Ω). In order to study the Dirichlet problem (1.1) we also need to define the subspace offunctions in V ν (Ω | R N ) that vanishes on the complement of Ω, i.e., X ν (Ω | R N ) = n u ∈ V ν (Ω | R N ) : u = 0 a.e. on R N \ Ω o , JEAN-DANIEL DJIDA, GUY F. FOGHEM GOUNOUE, AND YANNICK KOUAKEP TCHAPTCHIE where V ν (Ω | R N ) is defined as in (2.1). The space X ν (Ω | R N ) is clearly a closed subspace of V ν (Ω | R N ).Furthermore, we have that k u k X ν (Ω | R N ) = (cid:16) Z Z R N R N ( u ( x ) − u ( y )) ν ( x − y ) d y d x (cid:17) defines an equivalent norm on X ν (Ω | R N ). Indeed, in virtue of the Poincar´e-Friedrichs inequalityon X ν (Ω | R N ), there exists a constant C = C ( N, Ω , ν ) > N, Ω and ν such that k u k L (Ω) ≤ C k u k X ν (Ω | R N ) for every u ∈ X ν (Ω | R N ) . (2.3)This can be verified by observing that R N \ B R ( x ) ⊂ R N \ Ω for all x ∈ Ω, where R = diam(Ω) > u ∈ X ν (Ω | R N ), we recall that u = 0 a.e on R N \ Ω. Hence we have k u k X ν (Ω | R N ) ≥ Z Ω | u ( x ) | d x Z R N \ Ω ν ( x − y )d y ≥ Z Ω | u ( x ) | d x Z R N \ B R ( x ) ν ( x − y )d y = 2 k ν R k L ( R N ) k u k L (Ω) . It suffices to take C= (2 k ν R k L ( R N ) ) − / with ν R = ν R N \ B R (0) . According to [22], as in the classicalcase, the Poincar´e-Friedrichs inequality (2.3) remains true if Ω is only bounded in one direction.Now, we define T ν ( R N \ Ω) the trace space of V ν (Ω | R N ), i.e., the space of restrictions to R N \ Ωof functions of V ν (Ω | R N ). To be more precise, we have T ν ( R N \ Ω) = n v : R N \ Ω → R meas. such that v = u | R N \ Ω with u ∈ V ν (Ω | R N ) o . We equip T ν ( R N \ Ω) with its natural norm k v k T ν (Ω | R N ) = inf n k u k V ν (Ω | R N ) : u ∈ V ν (Ω | R N ) with v = u | R N \ Ω o . Next, we consider the weighted L -spaces on R N \ Ω denoted by L ( R N \ Ω , ν K ) where for a givenmeasurable set K ⊂ Ω with 0 < | K | < ∞ , we define ν K ( x ) := ess inf y ∈ K ν ( x − y ) and ˚ ν K ( x ) = Z K ∧ ν ( x − y )d y. The aforementioned spaces are Hilbert spaces and are somewhat linked. Let ( V ν (Ω | R N )) ∗ and( X ν (Ω | R N )) ∗ be the dual spaces of V ν (Ω | R N ) and X ν (Ω | R N ) respectively. Therefore, we have thefollowing Gelfand evolution triple embeddings X ν (Ω | R N ) ֒ → L (Ω) ֒ → ( X ν (Ω | R N )) ∗ and V ν (Ω | R N ) ֒ → L (Ω) ֒ → ( V ν (Ω | R N )) ∗ . In addition we have the continuous embeddings X ν (Ω | R N ) ֒ → V ν (Ω | R N ) ֒ → T ν (Ω | R N ) ֒ → L ( R N \ Ω , ν K ) . (2.4)It is worth of noticing that these interactions in (2.4) between the spaces V ν (Ω | R N ), X ν (Ω | R N ), T ν (Ω | R N ), and L ( R N \ Ω , ν K ) respectively are analogous to the ones between the classical Sobolevspaces H (Ω), H (Ω), H ( ∂ Ω), and L ( ∂ Ω). Alternatively, the embeddings in (2.4) remain true ifthe weight ν K is replaced with ˚ ν K . The next result borrowed from [19, 22, 10], provides sufficientconditions under which the spaces X ν (Ω | R N ) and V ν (Ω | R N ) are compactly embedded in L (Ω). GLOBAL IN TIME PARABOLIC EQUATION FOR SYMMETRIC L´EVY OPERATORS 5
Theorem 2.1.
Assume ν ∈ L ( R N , ∧ | h | ) and Ω ⊂ R N is open bounded. If ν L ( R N ) thenthe embedding X ν (Ω | R N ) ֒ → L (Ω) is compact. Furthermore, the embedding V ν (Ω | R N ) ֒ → L (Ω) isalso compact if Ω has a Lipschitz boundary, ν L ( R N ) and lim δ → δ Z B δ (0) | h | ν ( h )d h = ∞ . (2.5)It is worthwhile noticing that we have the natural continuous and dense embeddings L (0 , T ; X ν (Ω | R N )) ֒ → L (0 , T ; L (Ω)) ֒ → L (0 , T ; ( X ν (Ω | R N )) ∗ ) . We recall from [25], since X ν (Ω | R N ) is a real Hilbert , if we set H ν (0 , T ) = n ζ ∈ L (0 , T ; X ν (Ω | R N )) : ∂ t ζ ∈ L (0 , T ; (( X ν (Ω | R N )) ∗ o , then, H ν (0 , T ) is a Hilbert space endowed with the norm given by k ζ k H ν (0 ,T ) = k ζ k L (0 ,T ; X ν (Ω | R N )) + k ∂ t ζ k L (0 ,T ;( X ν (Ω | R N )) ∗ ) . Proposition 2.2.
With the assumptions of Theorem 2.1 in force, the following assertions are true. ( i ) Lions-Magenes Lemma [2, Theorem II.5.12]: the following embedding is continuous H ν (0 , T ) ֒ → C ([0 , T ]; L (Ω)) . (2.6)( ii ) Lions-Aubin Lemma [2, Theorem II.5.16]: the following embedding is compact H ν (0 , T ) ֒ → L (0 , T ; L (Ω)) . (2.7)Now we state the integration by parts formula contained in [19, 8] for smooth functions. Preciselyfor every φ, ψ ∈ C ∞ c ( R N ) following nonlocal Gauss-Green formula holds true E ( φ, ψ ) = Z Ω ψ L φ ( x ) d x + Z R N \ Ω ψ ( y ) N φ ( y ) d y (2.8)where, the bilinear form E ( · , · ) is defined in (2.2) and N φ denotes the nonlocal normal derivative of φ across the boundary of Ω with respect to ν and is defined by N φ ( x ) := Z Ω (cid:0) φ ( x ) − φ ( y ) (cid:1) ν ( x − y ) d y, x ∈ R N \ Ω . (2.9)With the aforementioned function spaces at hand, and motivated by the Gauss-Green formula (2.8),we are now in position to define the notion solution of a weak to the problem (1.1). Definition 2.3.
A function u : R N × (0 , T ) → R will be a weak solution of problem (1.1) if( i ) u ∈ L (0 , T ; X ν (Ω | R N )) and ∂ t u ∈ L (0 , T ; (cid:0) X ν (Ω | R N ) (cid:1) ∗ );( ii ) For every ψ ∈ L (cid:0) , T ; X ν (Ω | R N ) (cid:1) , u satisfies u ( · ,
0) = u and Z Ω ∂ t u ψ d x d t + E ( u, ψ ) d t + Z Ω ϕ ( v ) u ψ d x d t = 0 for all 0 ≤ t ≤ T . (2.10)In particular, we have Z T Z Ω ∂ t u ψ d x d t + Z T E ( u, ψ ) d t + Z T Z Ω ϕ ( v ) u ψ d x d t = 0 . Our proof of the existence of a weak solution to the problem (1.1) relies upon the following Tychonofffixed-point Theorem 2.4 which is a generalization of the Brouwer and Schauder fixed-point theorems.
JEAN-DANIEL DJIDA, GUY F. FOGHEM GOUNOUE, AND YANNICK KOUAKEP TCHAPTCHIE
Theorem 2.4 (Tychonoff [33]) . Let X be a reflexive separable Banach space and G ⊂ X be aclosed convex bounded set. If a mapping π : G → G is weakly sequentially continuous, then π hasat least one fixed point in G . It is noteworthy emphasizing that for the particular case where G is compact and convex, Theorem2.4 is known as the Schauder fixed-point theorem while the finite dimension case dim X < ∞ isknown as the Brouwer fixed-point theorem.3. Nonlocal elliptic and parabolic problem
The overreaching goal of this section is to investigate weak solutions to two specific nonlocalproblems which is of interest in the proof of our main result. The first problem is an ellipticnonlocal problem and the second one is a parabolic nonlocal problem.3.1.
Nonlocal elliptic problem.
Given a measurable function f : Ω → R , we consider the ellipticproblem consisting into finding a function v : R N → R satisfying of the following problem: ( L v + ϕ ( v ) v = f in Ω ,v = 0 on R N \ Ω . (3.1)Heuristically, the problem (3.1) results from the evolution problem (1.1) by integrating with respectto t from 0 to T . In a sense, the functions v and f correspond to R T u ( · , t ) d t and u − u ( · , T ),respectively. Problems of type (3.1) are considered in the classical scenario in [3, 18, 20, 30]) withthe operator L replaced with − ∆. There, the difficulties with the integrability of the term ϕ ( v ) v were handled. In our case, we consider the function f ∈ L (Ω), so that we expect more from thesolution of the problem such as ϕ ( v ) ∈ L (Ω). We follow the strategy of the proof presented in[30]. We use the following notation χ ( τ ) = ϕ ( τ ) τ. A function v ∈ X ν (Ω | R N ) is said to be a weak solution of problem (3.1) if χ ( v ) ∈ L (Ω) and E ( v, ψ ) + ( χ ( v ) , ψ ) = ( f, ψ ) for all ψ ∈ X ν (Ω | R N ) . (3.2)Next, we want to show that the above variational problem (3.2) is well-posed in the sense ofHadamard. In other words, it possesses a unique solution which continuously depends upon theinitial data. Let us start with the following stability lemma. Lemma 3.1.
Let f i ∈ L (Ω) , i = 1 , . Assume that v i ∈ X ν (Ω | R N ) satisfies E ( v i , ψ ) + ( χ ( v i ) , ψ ) = ( f i , ψ ) for all ψ ∈ X ν (Ω | R N ) . Then for some constant C = C ( N, Ω , ν ) > only depending only N, Ω and ν such that k v − v k X ν (Ω | R N ) ≤ C k f − f k ( X ν (Ω | R N )) ∗ . Proof.
Combining both equation and testing with ψ = v − v yields E ( v − v , v − v ) + (cid:0) χ ( v ) − χ ( v ) , v − v (cid:1) L (Ω) = ( f − f , v − v ) L (Ω) . Observing that τ χ ( τ ) = ϕ ( τ ) τ is non-decreasing, is equivalent to saying that( χ ( τ ) − χ ( τ ))( τ − τ ) ≥ τ , τ ∈ R , (3.3)the above relation implies k v − v k X ν (Ω | R N ) ≤ k f − f k ( X ν (Ω | R N )) ∗ k v − v k X ν (Ω | R N ) . The desired estimate follows from the Poincar´e-Friedrichs inequality (2.3). (cid:3)
GLOBAL IN TIME PARABOLIC EQUATION FOR SYMMETRIC L´EVY OPERATORS 7
Theorem 3.2.
Let Assumption 1.1 be be in force and let f ∈ L (Ω) . Then the problem (3.1) hasa unique weak solution v ∈ X ν (Ω | R N ) . Moreover the fiollowing estimates hold true: ( i ) E ( v, v ) ≤ C k f k L (Ω) ; ( ii ) k ϕ ( v ) v k L (Ω) ≤ k f k L (Ω) , with C > only depending on N , Ω , and ν ; ( iii ) k ϕ ( v ) k L (Ω) ≤ δ k f k L (Ω) + | Ω | , with δ > only depending on ϕ .Proof. Note that the uniqueness immediately follows from Lemma 3.1. We prove the remainingresults of Theorem 3.2 in several steps by adapting the strategy of the proof of [30, Lemma 3.1].
Step 1:
We are interested in establishing the well-posedness of problem (3.1) using the Galerkinmethod which consists into projecting the latter on suitable finite dimensional space. First of all,we mention that bounded functions are dense in V ν (Ω | R N ) and hence in X ν (Ω | R N ). Thus, there isan orthonormal basis { φ k } of X ν (Ω | R N ) whose elements are bounded, i.e., φ k ∈ L ∞ (Ω).We emphasize that the inner product in X ν (Ω | R N ) is defined as ( ψ , ψ ) X ν (Ω | R N ) = E ( ψ , ψ ) for ψ , ψ ∈ X ν (Ω | R N ). Let V k be the subspace of X ν (Ω | R N ) spanned by the basis functions { φ , . . . , φ k } .For each k ∈ N , we claim the existence of a function v k ∈ V k such that E ( v k , ψ ) + ( χ ( v k ) , ψ ) = ( f, ψ ) for all ψ ∈ V k . (3.4)We prove this in two different ways. First, note that (3.4) is equivalent to the minimization problem J ( v k ) = min w ∈ V k J ( w ) with J ( w ) := 12 E ( w, w ) + Z Ω G ( w )d x + Z Ω f w d x where we define the function G ( w ) = R w χ ( τ )d τ = R w ϕ ( τ ) τ d τ . Note that G is non-negative since ϕ ( τ ) ≥ w J ( w ) is continuous on V k . Furthermore, with the aid of thePoincar´e-Friedrichs inequality (2.3) we find that J ( w ) → ∞ , as k w k X ν (Ω | R N ) → ∞ and w ∈ V k .Alternatively, as highlighted in [30], we obtain the existence of v k using the Brouwer fixed-pointtheorem as follows. Let w ∈ V k , necessarily ϕ ( w ) is a bounded function since φ k ’s are also bounded.The Lax-Milgram lemma implies there is a unique function b w ∈ V k such that E ( b w, ψ ) + ( ϕ ( w ) b w, ψ ) = ( f, ψ ) for all ψ ∈ V k . In particular, the Poincar´e–Friedrichs inequality (2.3) yields E ( b w, b w ) + Z Ω ϕ ( w ) b w d x ≤k f k L (Ω) k b w k L (Ω) ≤ C k f k L (Ω) k b w k X ν (Ω | R N ) Thus, letting R = C k f k L (Ω) , since ϕ ≥ k b w k X ν (Ω | R N ) ≤ R and Z Ω ϕ ( w ) b w d x ≤ R . (3.5)We let B R = (cid:8) w ∈ V k : k w k X ν (Ω | R N ) ≤ R (cid:9) , be the closed ball in V k of radius R centered at theorigin. Clearly, (3.5) implies that the mapping T : V k → B R with T w = b w is well defined. Itremains to prove that T is a continuous mapping. Indeed, let { w n } be a sequence in V k with w n = λ ,n φ + · · · + λ k,n φ k converging in V k to a function w = λ φ + · · · + λ k φ k , i.e., λ ℓ,n n →∞ −−−→ λ ℓ , ℓ = 1 , , · · · , k . By continuity we have ϕ ( w n ) n →∞ −−−→ ϕ ( w ) almost everywhere. In addition, theconvergence in L (Ω) also holds, i.e., k ϕ ( w n ) − ϕ ( w ) k L (Ω) n →∞ −−−→ n ≥ k ϕ ( w n ) k L ∞ (Ω) < ∞ because sup n ≥ k w n k L ∞ (Ω) < ∞ . On the other side, in virtue of the first JEAN-DANIEL DJIDA, GUY F. FOGHEM GOUNOUE, AND YANNICK KOUAKEP TCHAPTCHIE estimate in (3.5), the sequence { T w n } is bounded in finite dimensional space V k and thus convergesin V k up to a subsequence to some w ∗ ∈ V k . Altogether, it follows that, for all ψ ∈ V k ⊂ L ∞ (Ω)( f, ψ ) = lim n →∞ E ( b w n , ψ ) + ( ϕ ( w n ) b w n , ψ ) = E ( w ∗ , ψ ) + ( ϕ ( w ) w ∗ , ψ ) . The uniqueness of b w entails w ∗ = b w = T w and hence the whole sequence { T w n } converges in T w in V k , which gives the continuity of T . Therefore, by the Brouwer fixed-point theorem, T has afixed point v k ∈ V k , i.e., v k = T v k which clearly satisfies (3.4) as announced.Furthermore, recalling R = C k f k L (Ω) , from (3.5) we get the following estimates for all k ∈ N , k v k k X ν (Ω | R N ) ≤ R and Z Ω χ ( v k ) v k d x ≤ R . (3.6)Therefore, the sequence { v k } is clearly bounded in X ν (Ω | R N ). The compactness Theorem 2.1 yieldsthe existence of a subsequence, still denoted by { v k } , weakly converging in X ν (Ω | R N ) and stronglyconverging in L (Ω) to a function v . Wherefore, due to the continuity of the function χ , we get χ ( v k ) → χ ( v ) almost everywhere in Ω . (3.7) Step 2:
Next, we prove that the functions { χ ( v k ) } are uniformly integrable. In view of the estimate(3.6), for each measurable set Γ ⊂ Ω and each Λ >
0, we let Γ k Λ = { x ∈ Γ : | v k ( x ) | ≥ Λ } so that Z Γ k Λ | χ ( v k ) | d x ≤ Z Ω χ ( v k ) v k d x ≤ R Λ . Since χ is non-decreasing, putting γ (Λ) = Λ max { ϕ ( − Λ) , ϕ (Λ) } , we get | χ ( τ ) | ≤ γ (Λ) for all τ ∈ [ − Λ , Λ] . Therefore, the following relation holds Z Γ \ Γ k Λ | χ ( v k ) | d x ≤ γ (Λ) | Γ | , where | Γ | is the Lebesgue measure of the set Γ. These inequalities imply that Z Γ | χ ( v k ) | d x ≤ R Λ + γ (Λ) | Γ | . Patently, for an arbitrary ε >
0, we take Λ = 2 R /ε and δ = ε/ (2 γ (Λ)). Therefore, we find thatsup k ≥ Z Γ | χ ( v k ) | d x < ε for an arbitrary measurable set Γ ⊂ Ω such that | Γ | < δ . This, is precisely the uniform integrabilityof χ ( v k ). This fact together with (3.7) and the Vitali convergence theorem (see, e.g., [19, TheoremA.19]) enable us to conclude that χ ( v ) ∈ L (Ω) and χ ( v k ) → χ ( v ) in L (Ω) as k → ∞ . Now passingto the limit in (3.4) as k → ∞ we find that v satisfies (3.2), which along with Lemma 3.1, meansthat v is a unique weak solution of problem (3.1). Step 3:
Next, we prove the estimates appearing in ( i ) − ( iii ). The first estimate follows from (3.6).In order to prove the second one, we introduce the truncated function for every ℓ ∈ N as follows: v ℓ ( x ) = ℓ, v ( x ) ≥ ℓ,v ( x ) , − ℓ < v ( x ) < ℓ, − ℓ, v ( x ) ≤ − ℓ. Since χ ( v ) χ ( v ℓ ) ≥ χ ( v ℓ ) and E ( v ℓ , χ ( v ℓ )) ≥ ψ = χ ( v ℓ ) in (3.2) yields GLOBAL IN TIME PARABOLIC EQUATION FOR SYMMETRIC L´EVY OPERATORS 9 k χ ( v ℓ ) k L (Ω) ≤ E ( v ℓ , χ ( v ℓ )) + k χ ( v ℓ ) k L (Ω) ≤ k f k L (Ω) k χ ( v ℓ ) k L (Ω) . Which, as { χ ( v ℓ ) } → χ ( v ) a.e. in Ω, in virtue of Fatou lemma implies the second estimate k χ ( v ) k L (Ω) ≤ lim inf ℓ →∞ k χ ( v ℓ ) k L (Ω) ≤ k f k L (Ω) . Finally, by continuity of ϕ , there exists δ > ϕ ( τ ) ≤ τ ∈ [ − δ, δ ]. Hence, lettingΓ δ = { x ∈ Ω : | v ( x ) | ≥ δ } , the second estimate implies the third one as follows Z Ω ϕ ( v ) d x = Z Γ δ ϕ ( v ) d x + Z Ω \ Γ δ ϕ ( v ) d x ≤ δ Z Ω ϕ ( v ) v d x + Z Ω \ Γ δ ϕ ( v ) d x ≤ δ k f k L (Ω) + | Ω | . (cid:3) Next, define the mapping V : L (Ω) → X ν (Ω | R N ) such that, for f ∈ L (Ω), v = V ( f ) is the unique weak solution of problem (3.1). (3.8)We derive in the lemma below, some convergence results for the sequence { V ( f k ) } which are decisivefor the application the Tychonoff fixed-point Theorem 2.4. Lemma 3.3.
Let { f k } be a sequence in L (Ω) that weakly converges to f in L (Ω) . If v k = V ( f k ) and v = V ( f ) , then as k → ∞ we have ( i ) v k → v strongly in X ν (Ω | R N ) ; ( ii ) ϕ ( v k ) ⇀ ϕ ( v ) weakly in L (Ω) .Proof. Let us identify f k − f in ( X ν (Ω | R N )) ∗ with the linear form w Z Ω ( f k ( x ) − f ( x )) w ( x )d x. Since, the space X ν (Ω | R N ) is reflexive, for each k ≥ w k ∈ X ν (Ω | R N ) such that (c.f.[21, Theorem 2] or [24, Chapter 6]), k w k k X ν (Ω | R N ) ≤ k f k − f k X ν ((Ω | R N )) ∗ = Z Ω ( f k ( x ) − f ( x )) w k ( x )d x. According to the compactness Theorem 2.1 we may assume that { w k } strongly converges to some w in L (Ω). Therefore, the weakly convergence of { f k } implies that k f k − f k X ν ((Ω | R N )) ∗ = Z Ω ( f k ( x ) − f ( x )) w k ( x )d x k →∞ −−−→ . The convergence in X ν (Ω | R N ) follows immediately from Lemma 3.1 since k v k − v k X ν (Ω | R N ) ≤ C k f k − f k ( X ν (Ω | R N )) ∗ k →∞ −−−→ . On the other hand, we also have the strong convergence of { v k } in L (Ω) and the continuity of ϕ imply that { ϕ ( v k ) } converges almost everywhere to ϕ ( v ) up a subsequence. Furthermore, since { f k } is bounded, as in the proof of Lemma 3.1, one easily gets that k ϕ ( v k ) k L (Ω) ≤ C for all k ≥ C > k . Thus, { ϕ ( v k ) } has a further subsequence weakly convergingin L (Ω). The Banach-Saks theorem [27, Appendix A] or [28, Proposition 10.8] infers the existenceof a further subsequence whose C´esaro mean converges strongly in L (Ω) and almost everywherein Ω to the same limit. Necessarily, since { ϕ ( v k ) } converges almost everywhere to ϕ ( v ), the entiresequence { ϕ ( v k ) } weakly converges in L (Ω) to ϕ ( v ). (cid:3) Nonlocal parabolic problem.
We consider the following parabolic problem: ∂ t u + L u + ζu = 0 in Ω T ,u = 0 in Σ ,u ( · ,
0) = u , in Ω , (3.9)where u , ζ ∈ L (Ω) with ζ ≥
0. We also assume ν L ( R N ) so that by Theorem 2.1, theembedding X ν (Ω | R N ) ֒ → L (Ω) is compact. Therefore, by the standard Galerkin superpositionmethod (see for instance [19, Section 4.6]), a weak solution u of the problem (3.9) can be easilyobtained in L (cid:0) , T ; X ν (Ω | R N ) ∩ L (Ω , ζ ) (cid:1) . Here L (Ω , ζ ) is the Hilbert space with the norm k u k L (Ω ,ζ ) = Z Ω | u ( x ) | ζ ( x )d x. We omit the proof as well as various justifications (see also [11, 16]). Another possibility, is toobserve that [23] there exists a unique semigroup with generator A on L (Ω) associated to theclosed bilinear form a ( u, v ) = ( u, v ) X ν (Ω | R N ) + ( u, v ) L (Ω ,ζ ) , with u, v ∈ X ν (Ω | R N ) ∩ L (Ω , ζ ), suchthat a ( u, v ) = h Au, v i . Thus u ( x, t ) = e − tA u ( x ) , ≤ t ≤ T , is the unique weak solution to (3.9).The weak solution of problem (3.9) satisfies the energy estimate:12 k u ( · , t ) k L (Ω) + Z t E ( u, u ) d τ + Z t Z Ω ζ u d x d τ ≤ k u k L (Ω) (3.10)for all t ∈ [0 , T ]. Besides that, ∂ t u belongs to the space L (cid:0) , T ; ( X ν (Ω | R N ) ∩ L (Ω , ζ )) ∗ (cid:1) , where( X ν (Ω | R N ) ∩ L (Ω , ζ )) ∗ is the conjugate space to X ν (Ω | R N ) ∩ L (Ω , ζ ). As a consequence of thisfact, from (2.6), we find that u ∈ C (0 , T ; L (Ω)). Thus, the function u T = u ( · , T ) is well defined asan element of L (Ω) and (3.10) holds for all t ∈ [0 , T ].For each ζ ∈ L (Ω), ζ ≥
0, define the mapping U : ζ U ( ζ ) where U ( ζ ) ∈ L (cid:0) , T ; X ν (Ω | R N ) ∩ L (Ω , ζ ) (cid:1) is the unique weak solution of problem (3.9).Define also U T ( ζ )( · ) = U ( ζ )( · , T ). We now investigate the dependence of U and U T on ζ . Lemma 3.4.
Let u ∈ L (Ω) and { ζ k } be a sequence of non-negative functions converging weaklyin L (Ω) to a function ζ . Then U T ( ζ k ) ⇀ U T ( ζ ) weakly in L (Ω) as k → ∞ .Proof. For brevity, we denote u k = U ( ζ k ) and u = U ( ζ ). Let ψ : R N × (0 , T ) → R be an arbitrarysmooth function such that ψ = 0 in R N \ Ω × (0 , T ). As it follows from (3.10), for all k ∈ N ,12 k u k ( · , T ) k L (Ω) + Z T E ( u k , u k ) d t + Z T Z Ω ζ k u k d x d t ≤ k u k L (Ω) . On the other hand, using the weak the formulation of u k we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T h ∂ t u k , ψ i d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Z T E ( u k , ψ ) d t − Z T Z Ω ζ k u k ψ d x d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T E ( u k , u k ) d t ! Z T E ( ψ, ψ ) d t ! + Z T Z Ω ζ k u k d x d t ! Z T Z Ω ζ k ψ d x d t ! ≤ k u k L (Ω) Z T E ( ψ, ψ ) d t + Z T Z Ω ζ k ψ d x d t ! . GLOBAL IN TIME PARABOLIC EQUATION FOR SYMMETRIC L´EVY OPERATORS 11
The boundedness of { ζ k } implies that for a constant C ψ > ψ and u such thatsup k ≥ (cid:12)(cid:12)(cid:12) Z T h ∂ t u k , ψ i d t (cid:12)(cid:12)(cid:12) ≤ C ψ . The uniform boundedness principle implies that { ∂ t u k } is bounded in L (cid:0) , T ; ( X ν (Ω | R N )) ∗ ) andthus { u k } is bounded in H ν (0 , T ). Therefore, taking into account the compactness result fromProposition 2.2, the sequence { u k } has a subsequence still denoted by { u k } such that, as k → ∞ , u k ( · , T ) ⇀ h weakly in L (Ω) ,u k ⇀ w weakly in L (cid:0) , T ; X ν (Ω | R N )) ,u k → w strongly in L (cid:0) , T ; L (Ω)) . It turns out that h = w ( · , T ) and the strong convergence of { u k } in L (cid:0) , T ; L (Ω)) implies that Z T Z Ω ζ k u k ψ d x d t → Z T Z Ω ζuψ d x d t as k → ∞ . For each k ≥
1, by definition of u k , we get Z T Z Ω u k ∂ t ψ d x d t + Z T E ( u k , ψ ) d t + Z T Z Ω ζ k u k ψ d x d t d t = − Z Ω ( u k ( · , T ) ψ ( · , T ) − u ψ ( · , x. Finally, letting k → ∞ , we get w ( · ,
0) = u and Z T Z Ω ∂ t wψ d x + Z T E ( w, ψ ) d t + Z T Z Ω ζwψ d x d t = 0 . Thus w is a weak solution to (3.9) and by uniqueness we have w = u . The desired result follows. (cid:3) Weak solvability and uniqueness of the solution
Armed with the above auxiliaries results, let us turn our attention to the proof of the weaksolvability of problem (1.1). In order to apply the Tychonoff fixed-point Theorem 2.4, we take X = L (Ω), G = { w ∈ L (Ω) : k w k L (Ω) ≤ k u k L (Ω) } which is clearly closed, convex andbounded. The next result provides the existence of a weak solution to the problem (1.1). Theorem 4.1.
Let u ∈ L (Ω) and T > . Let the mapping π : G → G be defined for w ∈ G , by π ( w ) = U T ( ϕ ( v )) , where v = V ( u − w ) (defined as in (3.8) ) is the unique weak solution to (3.1) with f = u − w . Then π has a fixed point u T that is u T = π ( u T ) = U T (cid:0) ϕ ( v ) (cid:1) with v = V ( u − u T ) .Moreover, v = Z T u d t and u = U (cid:0) ϕ ( v ) (cid:1) is a weak solution of the problem (1.1) .Proof. Let w ∈ G then from Lemma 3.1 we know that ϕ ( v ) ∈ L (Ω) with v = V ( u − w ). Forthe non-negative function ζ = ϕ ( v ) ∈ L (Ω), the function U ( ζ ) satisfies (3.10) which impliesthat k U T ( ζ ) k L (Ω) ≤ k u k L (Ω) . In particular, k π ( w ) k L (Ω) ≤ k u k L (Ω) for all w ∈ G and thus, π ( G ) ⊂ G . It remains to prove the weak sequential continuity of π . Let { w k } be an arbitrarysequence in G that converges to w ∈ G weakly in L (Ω). We need to prove that π ( w k ) ⇀ π ( w )weakly in L (Ω) as k → ∞ . In virtue of Lemma 3.3, v k = V ( u − w k ) → v = V ( u − w )strongly in X ν (Ω | R N ) and ϕ ( v k ) ⇀ ϕ ( v ) weakly in L (Ω) as k → ∞ , where v k = V ( u − w k ) and v = V ( u − w ). In turn, Lemma 3.4 implies that π ( w k ) = U T ( ϕ ( v k )) ⇀ U T ( ϕ ( v )) = π ( w ) weakly in L (Ω) as k → ∞ . Thus, according to Theorem 2.4, π has a fixed point u T = π ( u T ). Next, knowingthat u T is a fixed point of the mapping π , we show that u = U ( ϕ ( v )), with v = V ( u − u T ), is aweak solution to the problem (1.1). Indeed, recall that u = U ( ϕ ( v )) is the unique weak solutionto the problem (3.9) with ζ = ϕ ( v ), i.e., u ( · ,
0) = u and for all ψ ∈ L (0 , T ; X ν (Ω | R N )), Z T Z Ω ∂ t u ψ d x d t + Z T E ( u, ψ ) d t + Z T Z Ω ϕ ( v ) u ψ d x d t = 0 (4.1)in particular for ψ ∈ X ν (Ω | R N ) (time independent) we get E (cid:16) Z T u d t, ψ (cid:17) + Z Ω ϕ ( v ) ψ Z T u d t d x = Z Ω [ u − u T ] ψ d x. Thus, according to Lemma 3.1, v = R T u ( x, t )d t is the unique weak solution to the elliptic problem L v + ϕ ( v ) v = u − u T in Ω and v = 0 on R N \ Ω . (4.2)We have shown that v = V ( u − u T ) = R T u d t. Therefore, we obtain u = U (cid:16) ϕ (cid:16) Z T u d t (cid:17)(cid:17) (4.3)which, according to the relation (4.1), implies that u is a weak solution to the problem (1.1). (cid:3) The main result of the paper is the following theorem.
Theorem 4.2.
Let u ∈ L (Ω) , T > and ϕ satisfies Assumption 1.1. The problem (1.1) has aweak solution u ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; X ν (Ω | R N )) such that ϕ ( v ) ∈ L (Ω) , ϕ ( v ) v ∈ L (Ω) , ϕ ( v ) u ∈ L (Ω T ) , and u ∈ C (0 , T ; L (Ω)) , where v = Z T u d t . Moreover, the following estimates hold true k u k L ∞ (0 ,T ; L (Ω)) + k u k L (0 ,T ; X ν (Ω | R N )) + Z T Z Ω ϕ ( v ) u d x d t ≤ k u k L (Ω) k ∂ t u k L (0 ,T ;( X ν (Ω | R N ) ∩ L (Ω ,ϕ ( v )) ∗ ) ≤ k u k L (Ω) . Proof.
The existence of a weak solution to the problem (1.1) follows immediately from Theorem4.1. Furthermore, mimicking the estimate (3.10) yields12 k u k L ∞ (0 ,T ; L (Ω)) + k u k L (0 ,T ; X ν (Ω | R N )) + Z T Z Ω ϕ ( v ) u d x d t ≤ k u k L (Ω) . (4.4)Now, each for ψ ∈ L (0 , T ; X ν (Ω | R N ) ∩ L (Ω , ϕ ( v )), by definition of u , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T h ∂ t u, ψ i d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Z T E ( u, ψ ) d t − Z T Z Ω ϕ ( v ) uψ d x d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T E ( u, u )d t ! Z T E ( ψ, ψ )d t ! + Z T Z Ω ϕ ( v ) u d x d t ! Z T Z Ω ϕ ( v ) ψ d x d t ! ≤ k u k L (Ω) Z T E ( ψ, ψ ) d t + Z T Z Ω ϕ ( v ) ψ d x d t ! . This implies that
GLOBAL IN TIME PARABOLIC EQUATION FOR SYMMETRIC L´EVY OPERATORS 13 k ∂ t u k L (0 ,T ;( X ν (Ω | R N ) ∩ L (Ω ,ϕ ( v )) ∗ ) ≤ k u k L (Ω) . Therefore, we have u ∈ L (cid:0) , T ; X ν (Ω | R N ) ∩ L (Ω , ζ ) (cid:1) and ∂ t u ∈ L (cid:0) , T ; ( X ν (Ω | R N ) ∩ L (Ω , ζ )) ∗ (cid:1) with ζ = ϕ (cid:16) R T u d t (cid:17) which implies that ϕ ( v ) u ∈ L (Ω T ). By Proposition 2.2 we get u ∈ C (cid:0) , T ; L (Ω) (cid:1) . On other hand, we know that v = R T u d t is the unique weak solution to theproblem (4.2) and hence from Theorem 3.2 we have ϕ ( v ) , ϕ ( v ) v ∈ L (Ω). This ends the proof. (cid:3) Next, we prove that problem (1.1) has a unique solution, provided that the initial condition u isbounded. Before, we need to establish the following maximum principle result. Lemma 4.3.
Let u = u ( x, t ) be a weak solution of the problem (1.1) , i.e., satisfies (2.10) with u ∈ L (Ω) ∩ L ∞ (Ω) then k u k L ∞ (0 ,T ; L ∞ (Ω)) ≤ k u k L ∞ (Ω) on Ω T , i.e., | u | ≤ k u k L ∞ (Ω) on a.e. Ω T . Proof.
Set Λ = k u k L ∞ (Ω) and consider the convex function F : R → [0 , ∞ ) defined by F ( τ ) = (cid:0) τ + Λ (cid:1) if τ < − Λ0 if | τ | ≤ Λ (cid:0) τ − Λ (cid:1) if τ > Λ . So that, F ( τ ) = 0 if and only if | τ | ≤ Λ, in particular F ( u ) = 0 a.e. on Ω. By convexity, F ′ isnon-decreasing, i.e., ( F ′ ( τ ) − F ′ ( τ ))( τ − τ ) ≥ τ , τ ∈ R in particular, since F ′ (0) = 0, wehave F ′ ( τ ) τ ≥ τ ∈ R . Furthermore, F ′ ( u ( · , t )) ∈ X ν (Ω | R N ) because u ( · , t ) ∈ X ν (Ω | R N )and one can check that F ′ is Lipschitz since F ′′ is bounded. Therefore, testing the equation (2.10)against ζ = F ′ ( u ) gives d d t Z Ω F ( u ( x, t ))d x = − E ( u, F ′ ( u )) − Z Ω ϕ ( v )( x ) F ′ ( u ( x, t )) u ( x, t )d x ≤ . Since F ( u ) = 0 almost everywhere on Ω, integrating the inequality gives Z Ω F ( u ( x, t ))d x ≤ ≤ t ≤ T. Thus, F ( u ( x, t )) = 0 a.e. on Ω T , and hence | u ( x, t ) | ≤ Λ a.e. on Ω T . (cid:3) Theorem 4.4.
Assume that ϕ satisfies Assumption 1.1., u ∈ L (Ω) ∩ L ∞ (Ω) and that for Λ = k u k L ∞ (Ω) we have | ϕ ′ ( τ ) | ≤ κ for τ ∈ [ − Λ T, Λ T ] for some constant κ > . Then the weak solutionof problem (1.1) is unique provided that κ Λ T < .Proof. Suppose that problem (1.1) has two weak solutions u and u , and put v i ( x ) = Z T u i ( x, t ) d t , i = 1 ,
2. Then u = u − u is a weak solution to ∂ t u + L u + ϕ ( v ) u − ϕ ( v ) u = 0 in Ω T ,u = 0 in Σ ,u ( · ,
0) = 0 in Ω . The maximum principle in Lemma 4.3, implies that | u i | ≤ Λ a.e. in Ω T and hence | v i | ≤ Λ T , i = 1 ,
2, a.e. in Ω. Testing the above equation with u leads to the following equality:12 dd t k u ( · , t ) k L (Ω) + E ( u, u ) + Z Ω ϕ ( v ) u d x + Z Ω (cid:0) ϕ ( v ) − ϕ ( v ) (cid:1) u u d x = 0 which implies that12 k u ( · , t ) k L (Ω) + Z t E ( u, u )d τ ≤ κ Λ Z t Z Ω | v ( x ) | | u ( x, τ ) | d x d τ for all t ∈ [0 , T ], where v = v − v = R T u ( · , τ )d τ. Noticing that, k v k L (Ω) = Z Ω (cid:12)(cid:12)(cid:12) Z T u ( x, τ )d τ (cid:12)(cid:12)(cid:12) d x ≤ T Z T k u ( · , τ ) k L (Ω) d τ, we get Z t Z Ω | v ( x ) | | u ( x, τ ) | d x d τ ≤ T (cid:16) Z T k u ( · , τ ) k L (Ω) d τ (cid:17) / (cid:16) Z t k u ( · , τ ) k L (Ω) d τ (cid:17) / , Therefore, we obtain the following inequality for all t ∈ [0 , T ] k u ( · , t ) k L (Ω) ≤ κ Λ T (cid:16) Z T k u ( · , τ ) k L (Ω) d τ (cid:17) / (cid:16) Z t k u ( · , τ ) k L (Ω) d τ (cid:17) / . (4.5)In short we rewrite the above inequality as follows ̺ ′ ( t ) ≤ κ Λ T ̺ / ( T ) ̺ / ( t ) with ̺ ( t ) = Z t k u ( · , τ ) k L (Ω) d τ. A routine integration yields that ̺ / ( t ) ≤ κ Λ T ̺ / ( T ) and, in particular, ̺ / ( T ) ≤ κ Λ T ̺ / ( T ).The latest inequality holds true only if ̺ ( T ) = 0 since κ Λ T <
1, which implies that u = 0. (cid:3) We now point out the following the closing remark which shows how the function spaces consid-ered in this note extends our studies to a sightly different type of problems.
Remark 4.5.
Analogous results to those obtained in this notes can be established replacing theDirichlet complement condition u = 0 in ( R N \ Ω) × (0 , T ) , the problem 1.1 with the Neumanncomplement condition N u = 0 in ( R N \ Ω) × (0 , T ) , where N u represents the nonlocal normalderivative of u across as defined in (2.9). To this end, it is decisive to taking into account thesetting of Theorem 2.1, namely that Ω is bounded and Lipschitz and that ν satisfies the asymptoticcondition (2.5), in such a way that the compactness of the embedding V ν (Ω | R N ) ֒ → L (Ω) holdstrue. Wherefrom, one readily obtains (see [19]) the Poincar´e type inequality k u k L (Ω) ≤ C E ( u, u ) for all u ∈ V ν (Ω | R N ) ⊥ , for some constant C > V ν (Ω | R N ) ⊥ = (cid:8) V ν (Ω | R N ) : R Ω u d x = 0 (cid:9) . These observations,alongside of our procedure, allow to replace the space X ν (Ω | R N ) with the space V ν (Ω | R N ) ⊥ . References [1] H. Antil and M. Warma. Optimal control of fractional semilinear PDEs*.
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