Asymptotic properties of PDEs in compact spaces

In this article we combine the study of solutions of PDEs with the study of asymptotic properties of the solutions via compactification of the domain. We define new spaces of functions on which study the equations, prove a version of Ascoli–Arzelà Theorem, develop the fixed point index results necessary to prove existence and multiplicity of solutions in these spaces and also illustrate the applicability of the theory with an example.


Introduction
The use of topological methods in the study of PDE is a classical field of research-see [15,16,23,29]. Unfortunately, the most modern and sophisticated methods that have been recently developed for ODEs-see, for instance, [1,3,27,38]-have been difficult to apply to PDEs. The reasons for this are, to cite some, the greater effort needed to check the, if rather weak, cumbersome hypotheses, the lower availability of explicit expressions of Green's functions for PDEs, the higher complexity of the domain of definition and the higher regularity that is necessary in order to obtain existence and uniqueness results.
Even then, there has been a recent effort to overcome this difficulties, mainly by imposing some kind of symmetry on the operator that defines the equation and, in particular, searching for radial solutions [7,8,[17][18][19][20][21]28]. More general approaches also appear for elliptic PDEs and systems of PDEs [24][25][26].
On the other hand, the study of ODEs on unbounded domains has progressed steadily [10,13,[33][34][35]. The key to deal with unbounded domains is to use some kind of relatively compactness criterion such as [37, Theorem 1]see for instance [9,37]. These kind of criteria are reworkings of the classical Ascoli-Arzelà Theorem and have been used in a different way in [4,5]. In these works the authors are able to apply Ascoli-Arzelà Theorem by compactifying the domain of the functions involved in the ODE, thus allowing for a study of the asymptotic properties of the solutions.
In this article we combine both the study of solutions of PDEs with the study of asymptotic properties of the solutions via compactification of the domain. Furthermore, we take the opportunity to fix some of the shortcomings in [4,5] and provide an example of application. It is worth noticing that our results are not constrained to partial differential equations of a particular type (parabolic, elliptic, hyperbolic) since the results obtained are presented for the integral form of the equations; but also that, in general, the conditions to be checked for a particular problem can become quite unwieldy, which can be a limiting factor when it comes to apply the results to more convoluted problems.
The structure of this article is as follows: On Section 2 we deal with the basic topological notions necessary for understanding compactifications and provide some examples thereof. In Section 3 we provide the definition of the family of Banach spaces we will be dealing with. We also prove basic results regarding its structure as Banach space as well as a version of Ascoli-Arzelà Theorem (Theorem 3.9) necessary for the results to come. It is in Section 4 that we apply the usual topological methods regarding the fixed point index in order to obtain existence results of an integral problem in several variables which, in general, can be seen as a transformation on a PDE problem. Finally, in Section 5 we provide an example of a hyperbolic equation of which a solution with a predetermined asymptotic behavior can be found.

Preliminaries: compactifications and extensions
In order to understand asymptotic behavior on a metric space (X, d) we first need to formalize the notion of point of infinity. In an intuitive way, we can picture a point of infinity as a point far away from any point in X. For instance, the usual order relation on the real numbers makes us think of a number bigger than any other. Those points of infinity must live some place, and that place is what we call a compactification. A compactification X has a topological structure that allows us to formalize the notion of asymptotic behavior of those functions defined on X in a precise way. This is because, once we have a topology in X, we can take limits. Furthermore, the topology of the compactification, when metrizable, allows us to study the relations and relative positions of the different points of infinity.
We proceed now to formally define the concept of compactification through an adequate map.
1. If y n ∈ κ(X), since κ −1 | κ(X) is continuous, we have that x n,j → κ −1 (y n ) and, therefore, since f is continuous, there exists b n ∈ N such that, for j b n , 2. On the other hand, if y n ∈ Y \κ(X), then f (y n ) := lim κ x→yn f (x) and so there exists We have that (κ(x n,j )) j∈N converges to y n , so there exists b n ∈ N such that, for j b n , d(κ(x n,j ), y n ) < δ and, therefore, Hence, for j j n := max{a n , b n }, Let us define z n := x n, jn for every n ∈ N. By the triangle inequality, we have that Since y n → y, we have that d(y n , y) → 0. Thus, d(κ(z n ), y) → 0. Then, we are in one of the following cases: 1. If y ∈ Y \κ(X), f (y) = lim κ x→y f (x). Then, since d(κ(z n ), y) → 0, d(f (z n ), f (y)) → 0 and, as a consequence, lim n→∞ f (z n ) = f (y). 2. If y ∈ κ(X) then, by the continuity of κ −1 | κ(X) , we have that z n → κ −1 (y) and hence, by the continuity of f , lim In any case, Consequently, The uniqueness of the extension is due to Proposition 2.10.

The space of continuously n-differentiable (κ, ϕ)-extensions
It is now our objective to study the analytic properties of functions in metric compactifications of closures of open sets in R n . The key to achieve this is to construct an adequate Banach space that we will call C m κ,ϕ . First, let us introduce some notation. If α = (α 1 , . . . , α n ) ∈ ({0} ∪ N) n , we define |α| := n j=1 α j ; and In particular, for k ∈ {1, . . . , n} and e k = (δ 1,k , . . . , δ n,k ), we simply denote Let n, m ∈ N, A ⊂ R n open and connected and unbounded, X a compact (and thus complete and totally bounded-see [36,Theorem 45.1]) metric space, κ : A → X a compactification and ϕ ∈ C m (A, R + ). We denote by C(X, R) the space of continuous functions from X to R. C(X, R) is a Banach space with the usual supremum norm: Furthermore, the f p are unique.
Proof. Let f ∈ C m κ,ϕ (A), p ∈ P m . Since for every x ∈ X\κ(A) and every p ∈ P m there exists lim κ y→x ∂ p (f/ϕ)(x), by Theorem 2.11, there exists a continuous map On the other hand, if f ∈ C m (A, R) is such that for every p ∈ P m there exists f p ∈ C(X, R) satisfying ∂ p (f/ϕ) = f p •κ, by Theorem 2.11, there exists lim κ y→x ∂ p (f/ϕ)(x) for every x ∈ X\κ(A) and p ∈ P m . Finally, if g, h ∈ C(X, R) are such that g • κ = h • κ, since g and h are continuous and κ(A) is dense in X, g = h. Therefore, the f p are unique. This is not true in general, as the following example shows. Nonetheless, it is enough to define

Remark 3.2. In [4] the authors identify the spaces
to obtain the identity.
for every x ∈ R, k 2. Observe that supp g k = [k − 1 k , k + 1 k ], so supp g k ∩ supp g j = ∅ for every k, j 2, k = j. Therefore, the function f (x) := ∞ k=2 g k (x), x ∈ R, is well defined. The function f and its derivative are illustrated in Fig. 1.
Observe that |g k (x)| 1 k , so lim x→∞ f (x) = 0. Furthermore, since g k ∈ C 1 (R, R) for every k 2, f ∈ C 1 (R, R), but it is not true that there exists lim x→∞ f (x). Indeed: so lim x→∞ f (x) does not exist.
In the next result we will prove that C m κ,ϕ (A) is a Banach space. To do that we first consider the Banach space BC m (A) of m-times continuously differentiable bounded real functions f with the norm Ξ is clearly linear and injective and we can induce the norm of for k N . Thus, for k, j N , since f k,p and f j,p are continuous, This means ( f k,p ) k∈N is a Cauchy sequence and, since Since ε was fixed arbitrarily, h p • κ = ∂ p g. Thus, f ∈ C m κ,ϕ (A). Lemma 3.1 allows us to define, for every p ∈ P m , a function where Γ p f is the unique function satisfying the equality Γ p f • κ = ∂ p (f/ϕ).
Proof. Γ p is linear by the linearity of ∂ p . To see that it is continuous observe that, due to the density of κ(A) in X and the continuity of the functions Γ p f , Remark 3.6. There is an interesting relation between the norm in C m κ,ϕ (A) and that of the Γ p . Remember that, due to the density of κ(A) in X and the continuity of the functions Γ p f , In order to successfully develop the next section we need a precompactness criterion for subsets in C m κ,ϕ (A). Unfortunately, Ascoli-Arzelà Theorem, as normally stated, cannot be applied to C m κ,ϕ (A) nor BC m (A) directly as A is not compact. Nevertheless, the theorem does apply to C(X, R) since X is a Hausdorff compact topological space and R is a complete metric space.
The question is, what is the relation between compactness in C m κ,ϕ (A) and in C(X, R)? The answer to this question comes from Lemma 3.1, as the following result shows: On the other hand, assume is an open cover of V pr and, for every s = 1, . . . , h r , Consider the family U := {U fr,s } hr j=1 . U is a finite subset of U. Let us check that it is a subcover. Take g ∈ F . Then Γ pr g ∈ B C(X,R) (Γ pr f r,s , δ r,s /2) for some s = 1, . . . , h r , so Γ pr g − Γ pr f r,s ∞ < δ r,s . We now want to prove that Γ q g − Γ q f r,s ∞ < δ r,s for every q ∈ {1, . . . , r − 1} because then, by expression (3.1), g − f r,s κ,ϕ < δ r,s < and, thus, By expression (3.2), g − f l,j κ,ϕ < δ l,j /2 and f r,s − f l,j κ,ϕ < δ l,j /2 for every j ∈ F l,fr,s and l ∈ {1, . . . , r − 1}, which, by expression (3.2), implies for every q ∈ P m , j ∈ F l,fr,s and l ∈ {1, . . . , r − 1}. Hence, as we wanted to show.
For every x ∈ A, ε ∈ R + and p ∈ P m there exists some δ x,p ∈ R + such that Proof. Assume 1, 2 and 3 hold. Fix p ∈ P m and x ∈ X.
Step 1: We will show that Γ p (F ) is equicontinuous at x. We study the following two cases: (a) If x ∈ κ(A), then x = κ(z) for some z ∈ A and for every ε ∈ R + there exists δ x,p such that for every f ∈ F , y ∈ A, y − z < δ x,p , Since κ : Since κ(A) is dense in X and Γ p is continuous, for every f ∈ F , which implies that Γ p (F ) is equicontinuous at x in C(X, R). (b) If x ∈ X\κ(A), we resort to an analogous argument using 3 instead of 2. There exists δ x,p such that for every f ∈ F , y ∈ X, d(x, κ(y)) < δ x,p , Since κ(A) is dense in X and Γ p is continuous, for every f ∈ F , which implies that Γ p (F ) is equicontinuous at x in C(X, R).
Step 2: We will show that Γ p (F ) is uniformly bounded at x. We again have two cases as follows: (a) If x = κ(z) for some z ∈ A, by 1, we have that Hence, Γ p F is uniformly bounded at x. We conclude that Γ p F is compact for every p ∈ P m and, thus, F is compact.
Finally, assume F is compact. Then so is Γ p F for every p ∈ P m . Fix p ∈ P m and x ∈ X. Γ p F is uniformly bounded at x, so there exist M x,p ∈ R + such that, for every f ∈ F , Thus, given x ∈ A, for every f ∈ F , Γ p F is also equicontinuous at x. So for every ε ∈ R + there exists δ x,p ∈ R + such that, for every f ∈ F , and 2 holds. If x ∈ X\κ(A), then, by the continuity of Γ p , for every f ∈ F , y ∈ X, d(x, κ(y)) < δ x,p ,  Remark 3.10. Conditions 2 and 3 in Theorem 3.9 can be synthesised as a single one as follows: 4. For every x ∈ X, ε ∈ R + and p ∈ P m there exist some δ x,p ∈ R + such that if f ∈ F and y ∈ A is such that d(x, κ(y)) < δ x,p . In practice it is convenient to keep them separated as 2 avoids the direct use of the compactification.
(R, d) is a compact metric space. Consider the compactification κ : R → R defined as κ(x) = x for every x ∈ R. Take ϕ(x) = x for every for every x ∈ R and consider the family F := {f n } n∈N where f n (x) = e −(x−n) 2 , x ∈ R. F ⊂ C 1 κ,ϕ (R) is equicontinuous and uniformly bounded (in R), so it satisfies conditions 1 and 2 in Theorem 3.9, but not 3, as Γ 0 (F ) ⊂ C(R, R) is not compact. Indeed, f n −f m 1−e −1 for every m, n ∈ N, m = n. This means that F admits no Cauchy sequence and, thus, no convergent subsequence, so F cannot be compact.
Remark 3. 13. We observe that in [4, Theorem 3.2] condition 3 of Theorem 3.9 is missing. Also, [37, Theorem 1] can be considered as a particular instance of Theorem 3.9 for the case of the compactification in Example 3.12. Condition 3 of Theorem 3.9 is called regularity in [37]. A similar condition appears in [9, Section 2.12, p. 62] under the name equiconvergence in a setting that would correspond to the compactification κ = κ| [0,∞) where κ is taken as in Example 3.12.

Fixed points of integral equations
In this section we will prove the existence of fixed points of integral equations. To that end, we will develop a method based on the fixed point index theory on abstract cones.
With the notation introduced in the previous section, let n, m ∈ N, A ⊂ R n be open, connected and unbounded, X a compact metric space, κ : A → X for t ∈ A (it is defined as the limit of such expressions on X\κ(A)), where G : A × A → R. Since the kernel G is defined on A × A, we need to introduce the following notation: given p ∈ ({0} ∪ N) n , and using the natural injection . . . , p n , 0, . . . , 0), G(t, ·). This notation is not to be confused with ∂ p (G(t, ·)), where we fix the value of t and differentiate with respect to the rest of the derivatives. The same applies to ∂ p (G(·, s)) for a fixed s. The next result will provide sufficient conditions for the operator T : C m κ,ϕ (A) → C m κ,ϕ (A) to be well defined, continuous and compact. We will make use of the following hypotheses: In particular, from the definition of C m κ,ϕ (A) and Theorem 3.4, this implies that there exist Taking into account the definition of function Γ p f as the unique function satisfying the equality Γ p f •κ = ∂ p (f/ϕ), and the proof of Theorem 2.11, it occurs that z x p (s) = Γ p (G(·, s)) (x). (C 2 ) For every ε ∈ R + and p ∈ P m , there exist δ ∈ R + and a measurable function w p such that, if x, y ∈ X satisfy d(x, y) < δ, then (C 3 ) The nonlinearity f : R n × R → [0, ∞) satisfies the following conditions: for all y ∈ R with |y| < r and a. e. t ∈ R n .
(C 4 ) For every r > 0, x ∈ X\κ(A) and p ∈ P m it holds that M p Φ r , |z x p | Φ r , w p Φ r ∈ L 1 (A). Proof. We shall divide the proof into several steps as follows: Step 1: Let us prove first that operator T is well defined, that is, that it maps C m κ,ϕ (A) to C m κ,ϕ (A). From the general rules of differentiability of integrals (see [2,Corollary 2.8.7]) it holds that or, which is the same, Hence, Now, from (C 4 ), it is ensured the existence of some positive constant c such that the previous expression is upperly bounded by ε c. Consequently, ∂ p T u ϕ is continuous on A for every p ∈ P m , that is, T u ϕ ∈ C m (A, R) and, since ϕ ∈ C m (A, R + ), we conclude that T u ∈ C m (A, R).

Let us show now that for every x ∈ X\κ(A) there exists lim
Therefore, we conclude that T u ∈ C m κ,ϕ (A).
Step 2: Continuity: Let (u n ) n∈N ⊂ C m κ,ϕ (A) be a sequence which converges to u in C m κ,ϕ (A) and let us show that (T u n ) n∈N converges to T u in C m κ,ϕ (A). The convergence of (u n ) n∈N to u in C m κ,ϕ (A) implies that, in particular, u n (s) → u(s) for a.e. s ∈ A and so, from (C 3 ), f (s, u n (s)) → f (s, u(s)) for a.e. s ∈ A.
Following similar arguments to the ones above, we have that, for every p ∈ P m , Remark 4.2. We must note that in the proof of the previous theorem it is necessary to show that the operator T that we are considering has enough regularity. In particular, we have proved that under hypotheses (C 1 )-(C 4 ), operator T maps the space C m κ,ϕ (A) to itself. This means that, given u ∈ C m κ,ϕ (A), we have proved two different things: first, the regularity of T u, and, second, the asymptotic properties of T u.
We observe that the general hypotheses that we have asked the kernel to satisfy in order to prove the regularity of T u might be too restrictive or too difficult to check in practice. In this sense, we must take into account the fact that in many examples it will be possible to prove directly the regularity of T u, even if hypotheses (C 1 )-(C 4 ) are not satisfied. This will be the case, for example, of integral equations whose origin is a differential equation, as in this case it is clear that the inverse operator of a differential one will always have enough regularity. In fact, this will be the case that we will consider in our example in the last section of this paper. Now, following the line of [14], we will consider an abstract cone in the space C m κ,ϕ (A) defined by In order to choose a cone K α such that T maps the cone into itself, we will require functional α to satisfy the following condition: (C 5 ) For all u ∈ K α , T u ∈ K α . Now we will use the well-known fixed point index theory to prove the existence of a fixed point of operator T . In order to do so, we will define some suitable subsets of the cone K α and give some conditions to ensure that the index of these subsets is either 1 or 0.
Let us consider the following subsets: is well defined, that is, the set on which the supremum is taken is nonempty for every ρ and the supremum is finite.
The next lemma compiles some classical results regarding the fixed point index formulated in [22,Theorems 6.2,7.3 and 7.11] in a more general framework.
In particular, given X a Banach space, K ⊂ X a cone and Ω ⊂ K an arbitrary open subset, ∂ Ω will denote the boundary of Ω in the relative topology in K, induced by the topology of X. Moreover, let us denote by Fix(T ) the set of fixed points of T .
Proof. Let us prove that T u = μ u for all u ∈ ∂K β,ρ α and every μ 1. Suppose, on the contrary, that there exist some u ∈ ∂K β,ρ α and some μ 1 69 Page 20 of 27 Lucía López-Somoza and A. F. Tojo JFPTA such that μ u(t) = T u(t) for all t ∈ A. In such a case, taking β on both sides of the equality and using the fact that we obtain that which is a contradiction. Thus, i Kα (T, K β,ρ α ) = 1.
Proof. Let us prove that u = T u + λ e for all u ∈ ∂K γ,ρ α and every λ > 0, where e is given im (C 10 ). Suppose, on the contrary, that there exist some u ∈ ∂K γ,ρ α and some λ > 0 such that u(t) = T u(t) + λ e(t) for all t ∈ A. In such a case, taking γ on both sides of the equality, we obtain that which is a contradiction. Thus, i Kα (T, K γ,ρ α ) = 0.
Finally, we will give an existence result. We note that, although we will formulate sufficient conditions to ensure the existence of one or two fixed points of operator T , it is possible to give similar results to show the existence of three or more fixed points. Theorem 4.6. Let (C 1 )-(C 10 ) hold. Then, we observe the following: 1. If the function b given in (C 10 ) is well defined and two constants ρ 1 , ρ 2 ∈ (0, ∞) with ρ 2 > b(ρ 1 ) such that (I 0 ρ1 ) and (I 1 ρ2 ) hold, then T has at least a fixed point.
Proof. The proof follows as a direct consequence of previous lemmas, taking into account that K β,ρ , in case functions b and/or c are well defined.
Let us check that the operator T : C 0 κ,ϕ (A) → C 0 κ,ϕ (A) (defined as the continuous extension of T ) is well defined, continuous and compact.