Linear non-local diffusion problems in metric measure spaces

Proceedings of the Royal Society of Edinburgh,146A, 833–863, 2016 DOI:10.1017/S0308210515000724 Linear non-local diffusion problems in metric measure spaces An´ıbal Rodr´ıguez-Bernal Departamento de Matem´atica Aplicada, Universidad Complutense de Madrid, 28040, Madrid, Spain ([email protected]) and Instituto de Ciencias Matem´aticas, Campus Cantoblanco UAM, C/ Nicol´as Cabrera, 13–15, 28049, Madrid, Spain Silvia Sastre-G´omez School of Mathematical Sciences, Western Gateway Building, Western Road, University College Cork, Ireland ([email protected]) (MS received 17 December 2014; accepted 22 May 2015) The aim of this paper is to provide a comprehensive study of some linear non-local diffusion problems in metric measure spaces. These include, for example, open subsets in RN, graphs, manifolds, multi-structures and some fractal sets. For this, we study regularity, compactness, positivity and the spectrum of the stationary non-local operator. We then study the solutions of linear evolution non-local diffusion problems, with emphasis on similarities and differences with the standard heat equation in smooth domains. In particular, we prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods. Keywords: non-local diffusion; linear problem; metric measure spaces 2010 Mathematics subject classification: Primary 45Axx; 45Cxx; 45M05; 45M20; 47G10 1. Introduction Diffusion is the natural process by which some quantity (for example, heat or matter) is transported from one part of a system to another as a result of random molecular motion. As such, diffusion has a prominent role in such distinct fields as biology, thermodynamics and even economics. In smooth media (for example, an open region in Euclidean space or a smooth manifold) classical diffusion models include differential operators such as the Laplacian, and diffusion problems are usually described in terms of partial differential equations [11]. Since the real world is non-smooth, in the last decade there has been great effort made in developing and applying similar techniques and structures from the realm of differential equations to the analysis of diffusion processes in non-smooth media, including some fractal-like sets (see, for example, [5,20,26]). 833 c 2016 The Royal Society of Edinburgh https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press 834 A. Rodr´ıguez-Bernal and S. Sastre-G´omez There is another approach, however, that allows one to describe and model diffusion processes by means of non-local models (see, for example, [2]), which we apply here in smooth and non-smooth media. Assume then that (Ω,μ) is a measure space and u(x, t) is the density of some population at the point x∈Ωat time t. Also assume that J(x, y) is a non-negative function defined in Ω×Ωthat represents the density of probability of a member of that population to jump from a location yto x. Hence, ΩJ(y,x)dy= 1 for all x∈Ω. Then ΩJ(x, y)u(y,t)dyis the rate at which the individuals arrive at location xfrom all other locations y∈Ω. On the other hand, −ΩJ(y,x)dyu(x, t)=−u(x, t) is the rate at which the individuals are leaving from location xto all other locations y∈Ω. Then the time evolution of the population uin Ωcan be written as ut(x, t)=Ω J(x, y)u(y,t)dy−u(x, t), u(x, 0) = u0(x), x∈Ω, x∈Ω,⎫ ⎬ ⎭ (1.1) where u0is the initial distribution of the population. This problem and variations of it have been previously used to model diffusion processes in, for example, [2,8,13,18] assuming that Ωis an open set in RN. However, non-local diffusion models like (1.1) can be naturally defined in measure spaces since we just need to consider the density of probability of jumping from a location xin Ωto a location yin Ω, given by the function J(x, y). This allows us to study diffusion processes in very different types of spaces like, for example, graphs (which are used to model complicated structures in chemistry, molecular biology or electronics, or they can also represent basic electric circuits in digital computers), compact manifolds, multi-structures composed by several compact sets with different dimensions (for example, a dumbbell domain), or even some fractal sets such as the Sierpinski gasket [5, 21, 26]. Some of these spaces are introduced in §2. Since it is always convenient to speak about continuity, in this paper we consider problems like (1.1) defined in metric measure spaces (Ω, μ, d), which are defined as follows. For more information see [24]. Definition 1.1.Ametric measure space (Ω, μ, d) is a metric space (Ω,d) with a σ-finite regular and complete Borel measure μin Ωthat associates a finite positive measure with the balls of Ω. In this context, we take X=Lp(Ω), 1 ⩽p⩽∞,orX=Cb(Ω) and consider non-local diffusion problems of the form ut(x, t)=KJu(x, t)−h(x)u(x, t), u(x, t0)=u0(x), x∈Ω, t > 0, x∈Ω, (1.2) where u0∈X,h∈L∞(Ω)orinCb(Ω), and the non-local diffusion operator KJu is given by KJu(x, t)=Ω J(x, y)u(y,t)dy. We will not assume, unless otherwise made explicit, that ΩJ(x, y)dy= 1. This is the case, for example, in [1, 2, 8], where it was assumed that Ω⊂RNis an open https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press Linear non-local diffusion problems in metric measure spaces 835 set, Jis defined in RN×RNwith RNJ(x, y)dy= 1 and the diffusion problem can be set as (1.2) with h(x)=Ω J(x, y)dy. This is a particular case that we will pay attention to below. One of the main goals in this paper is to show some similarities and differences between (1.2) and solutions of the classical heat equation. We will show in particular that both models share positivity properties such as the strong maximum principle. However, solutions of (1.2) do not smooth in time, except asymptotically as t→ ∞. This lack of smoothing is a major drawback for the analysis of the behaviour of solutions of (1.2) since if u0belongs to a suitable space X, then u(t)∈X for all times and it is no better. This implies that all possible results become X- dependent, which is very different to the case of local diffusion (heat-like) models. For example, even the spectrum of the operator KJ−hI could depend on X, and hence the solutions of (1.2) could as well. Hence, we base our strategy on exploiting compactness properties of KJ(see §3.1.1) to give sufficient conditions for the spectrum of KJand KJ−hI to be independent of X(see §3.2.1). With this the asymptotic behaviour of solutions of (1.2) will be analysed in all such spaces X as if we were working in L2(Ω), where Hilbertian techniques are available (see §4.3). The paper is organized as follows. In §2 we present several metric measure spaces in which all the analysis carried out in this paper holds. These include open sets of the Euclidean space, graphs, compact manifolds, multi-structures (sets composed of several compact sets with different dimensions joined together) and even some fractal sets. In §3 we derive a comprehensive study of the linear operator KJ−hI.We will discuss in particular continuity and compactness in different function spaces, including the case of convolution-type operators. We also study the positivity of the diffusive operator KJ. Under the assumption that J(x, y)>0 for all x, y ∈Ωsuch that d(x, y)<R, (1.3) for some R>0, and the geometric condition that Ωis R-connected (see definition 3.10), we show that for a non-negative non-trivial function z, the set of points in Ωwhere KJzis strictly positive is larger than that of z. This will also allow us to use the Kre˘ın–Rutman theorem (see [22]) to obtain that the spectral radius in Cb(Ω) of the operator KJis a positive simple eigenvalue with a strictly positive eigenfunction associated. Condition (1.3) is also shown to be somehow optimal. In the last part of §3 we study similar questions for the non-local operator KJ−hI, with h∈L∞(Ω). In particular, we characterize the spectrum, which is also shown to be independent of the function space. In §4 we analyse the solutions of (1.2), as well as the monotonicity properties of the solutions. In particular, we will show that (1.3) implies that (1.2) has a strong maximum principle. We then show that although solutions of (1.2) do not regularize, because they carry the singularities of the initial data, there is a subtle asymptotic smoothness for large times. In particular, the semigroup S(t) of (1.2) is asymptotically smooth as in [16, p. 4]. Finally, using the techniques of Riesz projections and the fact that the spectrum is independent of the space, we are able to describe the asymptotic behaviour of the solutions of (1.2). https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press 836 A. Rodr´ıguez-Bernal and S. Sastre-G´omez 2. Examples of metric measure spaces In the following sections we will consider a general measure metric space (Ω, μ, d) as in definition 1.1. Below we enumerate some examples to which we can apply the theory developed throughout this paper. 2.1. A subset of RN Let Ωbe a Lebesgue measurable set of RNwith positive measure. A particular case is the one in which Ωis an open subset of RN, which can even be Ω=RN. We consider the metric measure space (Ω, μ, d), where Ω⊆RN,μis the Lebesgue measure on RNand dis the Euclidean metric of RN. 2.2. Graphs We consider a non-empty connected and finite graph in RNdefined by G=(V,E), where V⊂RNis the finite set of vertices and the edge set Econsists of a collection of Jordan curves E={πj:[0,1] →RN|j∈{1,2,3,...,n}}, where πj∈C1([0,1]) is injective with πj(0),π j(1) ∈V. We identify the graph with its associated network G= n  j=1 ej= n  j=1 πj([0,1]) ⊂RN and we assume that any two edges ej=ehsatisfy that the intersection ej∩ehis either empty, one vertex or two vertices. We define the measure structure of this graph. The edges have the associated one-dimensional Lebesgue measure. Hence, a set A⊂eiis measurable if and only if π−1 i(A)⊂[0,1] is measurable, and for any measurable set A⊂eiwe consider the measure μi, defined as μi(A)=π−1 i(A)π i(t)dt. In particular, the length of the edge eiis defined as the length of the curve πi, μi(ei)=1 0π i(t)dt. (2.1) Therefore, a set A⊂Gis measurable if and only if A∩eiis measurable for every i∈{1,2,3,...,n}, and its measure is given by μG(A)= n  i=1 μi(A∩ei). With this, it turns out that a function f:G→Ris measurable if and only if f|ei:ei→Ris measurable if and only if f◦πj:[0,1] →Ris measurable. For 1 ⩽p<∞we set f∈Lp(G)=n i=1 Lp(ei), with norm fLp(G)= n  i=1 fLp(ei)<∞, https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press Linear non-local diffusion problems in metric measure spaces 837 where fLp(ei)=1 0|f(πi(t))|pπ i(t)dt1/p =1 0|f(πi(·))|pdμi1/p . For p=∞,f∈L∞(G)=n i=1 L∞(ei) with norm fL∞(G)= max i=1,...,n fL∞(ei)<∞, where fL∞(ei)= supt∈[0,1] |f(πi(t))|. We now describe the metric associated with the graph. For v,w ∈Gthe geodesic distance from vto w,dg(v,w), is the length of the shortest path from vto w. This distance dgdefines the metric structure associated with the graph G. Observe that since the graph is connected, the path from vto walways exists and, since the graph is finite, the geodesic metric dgis equivalent to the Euclidean metric in RN. With this, a continuous function f:G→Rhas a norm fC(G)= maxi=1,...,n fC(ei)< ∞, where fC(ei)= supt∈[0,1] |f(πi(t))|. Thus, the graph defines a metric measure space (G, μG,d g). 2.3. Compact manifolds Let M⊂RNbe a compact manifold that we define as follows. Let Ube an open bounded set of Rd, with d⩽N, and let ϕ:U→RNbe an application such that it defines a diffeomorphism from ¯ Uonto its image ϕ(¯ U). Then we define the compact manifold as M=ϕ(¯ U). A natural measure in Mis the one for which A⊂Mis measurable if and only if ϕ−1(A)⊂Rdis measurable. Hence, for any measurable set A⊂M, we define the measure μas μ(A)=ϕ−1(A) √gdx, (2.2) where g= det(gij) and gij =∂ϕ/∂xi, ∂ϕ/∂xj. Since the compact manifold satisfies M=ϕ(¯ U)⊂RNand U⊂Rd, the measure (2.2) is equal to the d-Hausdorff measure in RNrestricted to M(see [25, p. 48]). To define a natural metric in M, let (c) be the length of a curve cin RNdefined as in (2.1). Then we define the geodesic distance between two points p,qin the manifold Mas dg(p, q):=inf{(c)|c:[0,1] →Ma smooth curve, c(0) = p, c(1) = q}. Since M⊂RNis compact, the geodesic metric dgand the Euclidean metric of RN, d, are equivalent. Thus, we have the metric measure space (M,Hd,d), where Hdis the d-dimen- sional Hausdorff measure in RNand dgis the geodesic metric, which is equivalent to the Euclidean metric of RN. 2.4. Multi-structures We now consider a multi-structure composed of several compact sets with different dimensions. For example, we can consider a piece of plane joined to a curve that https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press 838 A. Rodr´ıguez-Bernal and S. Sastre-G´omez is joined to a sphere in RN, or we can also consider a dumbbell domain (a domain consisting of two open disjoint domains, joined by a line segment). Therefore, we are going to define an appropriate measure and metric for these multi-structures. Consider a collection of metric measure spaces {(Xi,μ i,d i)}i∈{1,...,n}with its respective measures μiand metrics didefined as above. Moreover, we assume that the measure spaces {(Xi,μ i)}i∈{1,...,n}satisfy μi(Xi∩Xj)=μj(Xi∩Xj)=0 for i=jand i, j ∈{1,...,n}. Then we define X= i∈{1,...,n} Xi and we say that E⊂Xis measurable if and only if E∩Xiis μi-measurable for all i∈{1,...,n}. Moreover, we define the measure μXas μX(E)= n  i=1 μi(E∩Xi). Now let us define the metric that we consider in X. We assume that Xi⊂RN is compact for all i∈{1,...,n}, and the metrics diassociated with each Xiare equivalent to the Euclidean metric in RN. Therefore, the metric dthat we consider for the multi-structure is the Euclidean metric in RN. Thus, we have the metric measure space (X,μX,d), which is called the direct sum of metric measure spaces (Xi,μ i,d i), i∈{1,...,n}. 2.5. Spaces with finite Hausdorff measure and geodesic distance There exist examples of compact sets F⊂RNof Hausdorff dimension Hdim(F)= s<Nand finite s-Hausdorff measure, i.e. Hs(F)<∞, which are pathwise connected, with finite length paths. Some of these sets can be constructed as self-similar affine fractal sets, and such an example is provided by the Sierpinski gasket; see, for example, [9,21,23]. For such sets, we can consider the metric measure space (F,Hs,d g), where dgis the geodesic metric, which may not be equivalent to the Euclidean metric in RN. 3. The linear non-local diffusion operator Let (Ω, μ, d) be a metric measure space (not necessarily of finite measure) and consider a linear non-local diffusion operator of the form KJu(x)=Ω J(x, y)u(y)dy, where the function Jis defined in Ωby Ωx→ J(x, ·)⩾0. Hereafter, for 1 ⩽p⩽∞we denote by pits conjugate exponent, that is, satisfying 1 = 1/p+1/p. Note that the dual space of Lp(Ω) is given by (Lp(Ω))=Lp(Ω) for 1 ⩽p<∞, and by (L∞(Ω))=M(Ω) for p=∞, where M(Ω) is the space of Radon measures. For more information see [14, ch. 7]. https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press Linear non-local diffusion problems in metric measure spaces 839 Figure 1. Sierpinski gasket. 3.1. Properties of the operator KJ We begin with the following result. Proposition 3.1. (i) Assume that 1⩽p, q ⩽∞and that J∈Lq(Ω,Lp(Ω)). Then we have KJ∈ L(Lp(Ω),L q(Ω)), the mapping J→ KJis linear and continuous, and KJL(Lp(Ω),Lq(Ω)) ⩽JLq(Ω,Lp(Ω)).(3.1) (ii) Assume that 1⩽p⩽∞, that J∈L∞(Ω,Lp(Ω)) and for any measurable set D⊂Ωsatisfying μ(D)<∞, lim x→x0D J(x, y)dy=D J(x0,y)dy∀x0∈Ω. (3.2) Then we have KJ∈L(Lp(Ω),Cb(Ω)), the mapping J→ KJis linear and continuous, and KJL(Lp(Ω),Cb(Ω)) ⩽JL∞(Ω,Lp(Ω)).(3.3) In particular, if J∈C b(Ω,Lp(Ω)), then KJ∈L(Lp(Ω),Cb(Ω)) and KJL(Lp(Ω),Cb(Ω)) ⩽JCb(Ω,Lp(Ω)). (iii) Assume that Ω⊂RNis open, that 1⩽p, q ⩽∞and J∈W1,q(Ω,Lp(Ω)). Then we have KJ∈L(Lp(Ω),W1,q(Ω)), the mapping J→ KJis linear and continuous, and KJL(Lp(Ω),W 1,q (Ω)) ⩽JW1,q (Ω,Lp(Ω)).(3.4) https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press 840 A. Rodr´ıguez-Bernal and S. Sastre-G´omez Proof. (i) Thanks to H¨older’s inequality we have, for 1 ⩽q<∞and 1 ⩽p⩽∞, KJuq Lq(Ω)=ΩΩ J(x, y)u(y)dy q dx ⩽uq Lp(Ω)ΩJ(x, ·)q Lp(Ω)dx =uq Lp(Ω)Jq Lq(Ω,Lp(Ω)). For q=∞and 1 ⩽p⩽∞, for each x∈Ω, |KJu(x)|=Ω J(x, y)u(y)dy ⩽uLp(Ω)J(x, ·)Lp(Ω), and taking the supremum in x∈Ω, we obtain the result. (ii) Note that since J∈L∞(Ω,Lp(Ω)), from part (i) with q=∞, we have that KJ∈L(Lp(Ω),L ∞(Ω)). Also note that hypothesis (3.2) can be written as lim x→x0Ω J(x, y)χD(y)dy=Ω J(x0,y)χD(y)dy∀x0∈Ω, where χDis the characteristic function of D⊂Ωwith μ(D)<∞, which means that KJ(χD) is continuous and bounded in Ω. Since μ(D)<∞,wehaveχD∈Lp(Ω) for 1 ⩽p⩽∞. Moreover, the space V= span[χD;D⊂Ωwith μ(D)<∞] is dense in Lp(Ω) for 1 ⩽p⩽∞and KJ:V→C b(Ω), and then KJ(Lp(Ω)) = KJ(¯ V)⊂KJ(V)⊂C b(Ω) and we get (3.3). In particular, if J∈Cb(Ω,Lp(Ω)), then hypothesis (3.2) is satisfied. (iii) As a consequence of Fubini’s theorem, and since Ωis open, we have that for all u∈Lp(Ω) and i=1,...,N the weak derivative of KJuis given by ∂ ∂xi KJu, ϕ=−KJu, ∂xiϕ=−ΩΩ J(x, y)u(y)∂xiϕ(x)dydx =−J(·,y),∂ xiϕ,u =∂xiJ(·,y),ϕ,u =ΩΩ ∂xiJ(x, y)u(y)ϕ(x)dydx =K∂J/∂xiu, ϕ(3.5) for all ϕ∈C∞ c(Ω). Therefore, ∂ ∂xi KJu=K∂J/∂xiu. (3.6) https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press Linear non-local diffusion problems in metric measure spaces 841 Since J∈W1,q(Ω,Lp(Ω)), and from part (i) and (3.6), we have that for i= 1,...,N,     ∂ ∂xi KJ   L(Lp(Ω),Lq(Ω)) =K∂J/∂xiL(Lp(Ω),Lq(Ω)) ⩽    ∂J ∂xi   Lq(Ω,Lp(Ω)) .(3.7) Hence, KJ∈L(Lp(Ω),W1,q(Ω)) for all 1 ⩽p, q ⩽∞, and from part (i) and (3.7) we have (3.4). The following result collects cases in which KJ∈L(X,X), with X=Lp(Ω)or X=Cb(Ω). Corollary 3.2. (i) If J∈Lp(Ω,Lp(Ω)), then KJ∈L(Lp(Ω),L p(Ω)). (ii) If J∈C b(Ω,L1(Ω)), then KJ∈L(Cb(Ω),Cb(Ω)). (iii) If μ(Ω)<∞and J∈L∞(Ω,L∞(Ω)), then KJ∈L(Lp(Ω),L p(Ω)) for all 1⩽p⩽∞. Proof. (i) From proposition 3.1 we have the result. (ii) If J∈C b(Ω,L1(Ω)), then, thanks to proposition 3.1, we have that KJbelongs to L(L∞(Ω),Cb(Ω)). Moreover, since Cb(Ω)⊂L∞(Ω), we have that KJ∈ L(Cb(Ω),Cb(Ω)). (iii) From proposition 3.1 we have that KJ∈L(L1(Ω),L ∞(Ω)). Moreover, since μ(Ω)<∞, we have that Lp(Ω)→L1(Ω)KJ −−→ L∞(Ω)→Lp(Ω). The particular case in which the non-local diffusion term is given by a convolution in Ω=RNwith a function J0:RN→R, i.e. J(x, y)=J0(x−y) and KJu=J0∗u, has been widely considered; see, for example [3,8,10] and references therein. Hence, we consider here such a type of operator. For this, let Ω⊂RNbe a measurable set (it can be Ω=RNor just a subset Ω⊂RN) and consider the kernel J(x, y)=J0(x−y),x,y∈Ω, (3.8) where J0is a function in Lp(RN) for 1 ⩽p⩽∞. Straight from proposition 3.1, we get the following corollary. Corollary 3.3.For 1⩽p⩽∞let Ω⊆RNbe a measurable set, let J0∈Lp(RN) and let Jbe defined as in (3.8). Then KJ∈L(Lp(Ω),L ∞(Ω)). In particular, if μ(Ω)<∞, then KJ∈L(Lp(Ω),L q(Ω)) for 1⩽q⩽∞. https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press 848 A. Rodr´ıguez-Bernal and S. Sastre-G´omez (0,0) (1,0) (0,1) (1/2,1/2) Figure 2. The shaded areas are the points (x, y) where Jis strictly positive, R=1 4. Now, we consider a function z0:Ω→R,z0⩾0, such that P(z0)⊂[1 2,1]. Since z0(y) = 0 in [0,1 2], we have that KJz0(x)=Ω J(x, y)z0(y)dy=1 1/2 J(x, y)z0(y)dy. Moreover, from (3.16), we have that for ˜x∈[0,1 2), J(˜x, y) = 0 for all y∈[1 2,1] (see figure 2). Hence, KJz0(˜x) = 0 in [0,1 2), and therefore P(KJz0)⊂[1 2,1]. Iterating this argument, we obtain that P(Kn J(z0)) ⊂[1 2,1] for all n∈N. 3.1.3. Spectrum of KJ We will now prove that under certain hypotheses on KJthe spectrum σX(KJ)is independent of Xwith X=Lp(Ω), 1 ⩽p⩽∞,orX=Cb(Ω). We also characterize the spectrum of KJwhen KJis self-adjoint in L2(Ω) and prove that, under the same hypothesis on the positivity of Jin proposition 3.15, the spectral radius of KJ in Cb(Ω) is a simple eigenvalue that has a strictly positive associated eigenfunction. The proposition below is for a general compact operator K, not only for the integral operator KJ(see proposition 3.7 to check compactness for operators with kernel KJ). Proposition 3.17.Let (Ω, μ, d)be a metric measure space with μ(Ω)<∞and let 1⩽p0<p 1⩽∞. Assume that one of the following cases hold. (i) K∈L(Lp0(Ω),L p1(Ω)) and K∈L(Lp0(Ω),L p0(Ω)) is compact. Write then X=Lp(Ω)for p∈[p0,p 1). (ii) K∈L(Lp0(Ω),L p1(Ω)) is compact. Write then X=Lp(Ω)for p∈[p0,p 1]. (iii) K∈L(Lp0(Ω),Cb(Ω)) is compact. Write then X=Lp(Ω)for p∈[p0,∞]or X=Cb(Ω). Then K∈L(X,X)and σX(K)is independent of X. https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press Linear non-local diffusion problems in metric measure spaces 849 Proof. (i) Thanks to proposition 3.9, we have that K∈L(Lp(Ω),L p(Ω)) is compact for all p∈[p0,p 1). Thus, its spectrum is composed of zero and a discrete set of eigenvalues of finite multiplicity (see [7, ch. 6]). So, we now prove that the set of eigenvalues is independent of p∈[p0,p 1]. If λ∈σLp1(K) is an eigenvalue, with an associated eigenfunction Φ∈Lp1(Ω)→ Lp(Ω) for all p∈[p0,p 1], then λ∈σLp(K) for all p∈[p0,p 1]. Conversely, if for p∈[p0,p 1), λ∈σLp(Ω)(K) is an eigenvalue, with an associated eigenfunction Φ∈Lp(Ω)→Lp0(Ω) since K:Lp0(Ω)→Lp1(Ω), then KΦ =λΦ ∈Lp1(Ω). Hence, λ∈σLp1(Ω)(K). (ii) Since K∈L(Lp0(Ω),L p1(Ω)) is compact and Lp1(Ω)→Lp0(Ω)K −→ Lp1(Ω)→Lp0(Ω), it follows that K∈L(Lp(Ω),L p(Ω)) is compact for p∈[p0,p 1]. Arguing as in (i), we have the result. (iii) Since K∈L(Lp0(Ω),Cb(Ω)) is compact and Cb(Ω)→Lr(Ω)→Lp0(Ω)K −→C b(Ω)→Lr(Ω)→Lp0(Ω), it follows that K∈L(X,X) is compact for X=Cb(Ω)orX=Lp(Ω) with p∈[p0,∞]. Hence, following the arguments in (i), we get the result. The following proposition gives more details about the spectrum of KJ. Proposition 3.18.Let (Ω, μ, d)be a metric measure space with μ(Ω)<∞.We assume that KJ∈L(Lp0(Ω),Cb(Ω)) is compact for some p0⩽2.LetX=Lp(Ω) with p∈[p0,∞]or let X=Cb(Ω), and assume that Jsatisfies J(x, y)=J(y,x). Then KJ∈L(X, X)and σX(KJ)\{0}is a real sequence of eigenvalues of finite multiplicity, independent of X, that converges to 0. Moreover, if we consider m= inf u∈L2(Ω), uL2(Ω)=1 KJu, uL2(Ω)and M= sup u∈L2(Ω), uL2(Ω)=1 KJu, uL2(Ω),(3.17) then σX(KJ)⊂[m, M]⊂R,m∈σX(KJ)and M∈σX(KJ). In particular, L2(Ω) admits an orthonormal basis consisting of eigenfunctions of KJ. Proof. Thanks to proposition 3.5, KJis self-adjoint in L2(Ω), so σL2(KJ)\{0}is a real sequence of eigenvalues of finite multiplicity that converges to 0 (see [7, ch. 6]). Furthermore, from proposition 3.17 we have that σX(KJ) is independent of X. Thus, the result. On the other hand, we have that σX(KJ)⊂[m, M]⊂Rwith m∈σX(KJ) and M∈σX(KJ), where mand Mare given by (3.17), and thanks to the spectral theorem (see [7, ch. 6]), we know that L2(Ω) admits an orthonormal basis consisting of eigenfunctions of KJ. Now, we apply the Kre˘ın–Rutman theorem [22] in Cb(Ω) to prove that the spectral radius of the operator Kis a simple eigenvalue with a strictly positive associated eigenfunction. https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press 850 A. Rodr´ıguez-Bernal and S. Sastre-G´omez Proposition 3.19.Let (Ω, μ, d)be a compact connected metric measure space. We assume that Jsatisfies J(x, y)=J(y,x) and J(x, y)>0∀x, y ∈Ωsuch that d(x, y)<Rfor some R>0, and KJ∈L(Lp(Ω),Cb(Ω)) for some 1⩽p⩽∞is compact. Then we have that KJ∈L(Cb(Ω),Cb(Ω)) is compact, the spectral radius r(KJ) = sup |σ(K)|is a positive simple eigenvalue, and its associated eigenfunction can be taken strictly positive. Proof. Since Ωis compact and connected, from proposition 3.15 we obtain that there exists n0∈Nsuch that for any non-trivial non-negative u∈C b(Ω), Ω= Pn R(u) for all n⩾n0(see definition 3.14), and for all n∈N,P(Knu)⊃Pn R(u). Therefore, Ω=Pn R(u)⊂P(Knu) for all n⩾n0, i.e. for any non-negative u∈C b(Ω), Kn Ju>0inΩfor all n⩾n0. Hence, KJis strongly positive in Cb(Ω). Moreover, KJ:Cb(Ω)→Lp(Ω)→C b(Ω) is compact. Hence, the result follows from the Kre˘ın–Rutman theorem (see [22]). A similar result was proved by Bates and Zhao [4] for an open set Ω⊂RNunder the stronger assumption that J(x, y)>0 for all x, y ∈Ω. 3.2. Properties of K−hI Let (Ω, μ, d) be a metric measure space. We will always assume below that h∈ L∞(Ω). In the case in which we work in the space X=Cb(Ω), we will furthermore assume that h∈C b(Ω). The following result collects some properties of multiplication operators. Note that below we denote by R(h) the range of the function h:Ω→R, and by R(h) its closure. Proposition 3.20.Let hbe as above and consider the multiplication operator hI that maps u(x)→ h(x)u(x). Then the resolvent set and spectrum of the multiplication operator are independent of Xand are given by ρX(hI)=C\R(h)and σX(hI)=R(h), respectively. Moreover, for X=Lp(Ω)the eigenvalues associated with hI have infinite multiplicity and satisfy EV(hI)={α;μ({x∈Ω;h(x)=α})>0}. On the other hand, for X=Cb(Ω)the eigenvalues have infinite multiplicity and satisfy EV(hI)⊃{α;{x∈Ω;h(x)=α}has non-empty interior}. https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press Linear non-local diffusion problems in metric measure spaces 851 Proof. If X=Lp(Ω), consider f∈Lp(Ω) and u∈Lp(Ω) such that h(x)u(x)− λu(x)=f(x), that is, u(x)= f(x) h(x)−λ=1 h(x)−λf(x). Then we have that λ∈ρLp(Ω)(hI) if and only if (hI −λI)−1∈L(Lp(Ω)) if and only if (1/(h−λ)) ∈L∞(Ω), and this happens if and only if λ∈ R(h). Thus, ρLp(Ω)(h)=C\R(h). If λis an eigenvalue, then for some Φ∈Lp(Ω) with Φ≡ 0wehave h(x)Φ(x)=λΦ(x), and this only happens if there exists a set A⊂Ω, with μ(A)>0, such that h(x)=λ for all x∈A⊂Ω. Hence, we have that Ker(hI −λI)=Lp(A). Thus, the result. If X=Cb(Ω), the arguments run as above. Just note that if {λ;{x∈Ω;h(x)=λ}} has non-empty interior A, then Ker(hI −λI)={Φ∈C b(Ω): Φ(x) = 0 for all x∈Ω\A}. 3.2.1. Spectrum Now we describe the spectrum of K−hI ∈L(X,X), where X=Lp(Ω) with 1⩽p⩽∞or X=Cb(Ω), and we prove that, under certain conditions on the operator K, it is independent of X. For this, we start by introducing some definitions used in the following theorems. Definition 3.21.If Tis a linear operator in a Banach space Y,anormal point of Tis any complex number that is in the resolvent set, or is an isolated eigenvalue of Tof finite multiplicity. Any other complex number is in the essential spectrum of T. To describe the spectrum of K−hI, we use the following theorem, which can be found in [17, p. 136]. Theorem 3.22.Suppose that Yis a Banach space, that T:D(T)⊂Y→Yis a closed linear operator, that S:D(S)⊂Y→Yis linear with D(S)⊃D(T)and that S(λ0−T)−1is compact for some λ0∈ρ(T).LetUbe an open connected set in Cconsisting entirely of normal points of T, which are points of the resolvent of T, or isolated eigenvalues of Tof finite multiplicity. Then either Uconsists entirely of normal points of T+Sor entirely of eigenvalues of T+S. Remark 3.23.If S:Y→Yis compact, theorem 3.22 implies that the perturbation Scan not change the essential spectrum of T. We then have the following theorem. Theorem 3.24.If K∈L(X,X)is compact (see proposition 3.7), then σ(K−hI)=R(−h)∪{μn}M n=1 with M∈N∪{∞}, where {μn}M n=1 are eigenvalues of K−hI with finite multiplicity. If M=∞, then {μn}∞ n=1 accumulates in R(−h). https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press 852 A. Rodr´ıguez-Bernal and S. Sastre-G´omez Proof. With the notation of theorem 3.22, we consider the operators S=Kand T=−hI. First of all, we prove that C\R(−h)⊂ρ(K−hI). We choose the set Uin theorem 3.22 as U=ρ(T)=C\R(−h), which is an open connected set. Hence, every λ∈Uis a normal point of T. On the other hand, if λ0∈ρ(T), then (T−λ0)−1∈L(X, X) and S=Kis compact. Then we have that S(λ0−T)−1∈L(X, X) is compact. Thus, all the hypotheses of theorem 3.22 are satisfied. Now, thanks to theorem 3.22, we have that U=C\R(−h) consists entirely of eigenvalues of T+S=K−hI or U consists entirely of normal points of T+S=K−hI.IfU=C\R(−h) consists entirely of eigenvalues of T+S=K−hI, we arrive at a contradiction, because the spectrum of K−hI is bounded. So U=C\R(−h) has to consist entirely of normal points of T+S. Then they are points of the resolvent or isolated eigenvalues of T+S=K−hI. Since any set of isolated points in Cis a finite set, or a numerable set, we have that the isolated eigenvalues are {μn}M n=1 with M∈Nor M=∞. Moreover, since the spectrum of K−hI is bounded, if M=∞, then {μn}∞ n=1 is a sequence of eigenvalues of K−hI with finite multiplicity that accumulates in R(−h). Now we prove that R(−h)⊂σ(K−hI). We argue by contradiction. Suppose that there exists a ˜ λ∈R(−h) that belongs to ρ(K−hI). Since the resolvent set is open, there exists a ball Bε(˜ λ) centred in ˜ λthat is in the resolvent of K−hI. Then U=Bε(˜ λ) is an open connected set that consists of normal points of K−hI. With the notation of theorem 3.22, we consider the operators T=K−hI and S=−K and the open connected set U=Bε(˜ λ). Arguing as in the previous case, if λ0∈ρ(T), we have that S(λ0−T)−1is compact, and thus the hypotheses of theorem 3.22 are satisfied. Hence, U=Bε(˜ λ) consists entirely of eigenvalues of T+S=−hI or U=Bε(˜ λ) consists entirely of normal points of T+S=−hI.IfU=Bε(˜ λ) consists entirely of eigenvalues of T+S=−hI, we would arrive at a contradiction because the eigenvalues of −hI are only inside R(−h) and the ball Bε(˜ λ)isnot inside R(−h). So U=Bε(˜ λ) has to consist of normal points of T+S=−hI,so they are points of the resolvent of −hI or isolated eigenvalues of finite multiplicity of −hI. Since ρ(−hI)=C\R(−h) and ˜ λ∈R(−h), we have that ˜ λhas to be an isolated eigenvalue of −hI with finite multiplicity. But from proposition 3.20 we know that the eigenvalues of −hI have infinite multiplicity. Thus, we arrive at a contradiction. Hence, we have proved that R(−h)⊂σ(K−hI). With this, we have finished the proof of the theorem. In the following proposition we give conditions that guarantee that the spectrum of K−hI is independent of X=Lp(Ω) with 1 ⩽p⩽∞or X=Cb(Ω). https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press Linear non-local diffusion problems in metric measure spaces 853 Proposition 3.25.Let μ(Ω)<∞, let h∈L∞(Ω)and assume either one of the following cases holds. (i) For some 1⩽p0<p 1⩽∞and K∈L(Lp0(Ω),L p1(Ω)), we have that K∈L(Lp0(Ω),L p0(Ω)) is compact and h∈L∞(Ω). Write then X=Lp(Ω) for p∈[p0,p 1). (ii) For some 1⩽p0<p 1⩽∞, we have that K∈L(Lp0(Ω),L p1(Ω)) is compact and h∈L∞(Ω). Write then X=Lp(Ω)for p∈[p0,p 1]. (iii) For some 1⩽p0⩽∞, we have that K∈L(Lp0(Ω),Cb(Ω)) is compact and write then X=Lp(Ω)for p∈[p0,∞]or X=Cb(Ω)(in which case we assume h∈C b(Ω)). Then K∈L(X,X)and σX(K)is independent of X. Proof. Following the same arguments in proposition 3.17, we have that in any of the cases (i), (ii) or (iii), K∈L(X, X) is compact. Then, from theorem 3.24 we just have to prove that the eigenvalues λ∈σX(K−hI) such that λ/∈R(−h) are independent of X. For such an eigenvalue and an associated eigenfunction Φ,we have Φ(x)= 1 h(x)+λKΦ(x) (3.18) and 1/(h(·)+λ)∈L∞(Ω) since λ/∈R(−h). For cases (i) and (ii), if λ∈σLp1(K− hI), then Φ∈Lp(Ω) and λ∈σLp(K−hI) for all p∈[p0,p 1]. Conversely, if λ∈σLp(Ω)(K−hI), then 1 h(·)+λKΦ =Φ∈Lp1(Ω). Case (iii) is analogous using that 1/(h(·)+λ)∈C b(Ω). Note that if 1 ⩽p0⩽2, proposition 3.25 allows us to analyse the spectrum of K−hI in the L2(Ω) setting for which we have conditions to be self-adjoint using proposition 3.18. In such a case, and as will be seen in §4, it will be important to determine when the largest point on the spectrum is a simple eigenvalue. For ha constant function this is easily derived from proposition 3.19. However, for non-constant hthe situation could be more involved. A simple particular situation occurs for the function h0(x)=Ω J(x, y)dy, for which we assume that J∈L∞(Ω,L1(Ω)), and hence h0∈L∞(Ω). Note that in this case, λ= 0 is an eigenvalue with constant eigenfunctions. The result below gives the condition for λ= 0 being the largest eigenvalue and being simple. Corollary 3.26.Under the assumptions of proposition 3.25, assume furthermore that J(x, y)=J(y,x), J∈L∞(Ω,L1(Ω)), and h(x)=h0(x)=ΩJ(x, y)dy∈L∞(Ω)satisfies that h0(x)⩾α>0for all x∈Ω. https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press 854 A. Rodr´ıguez-Bernal and S. Sastre-G´omez Then σX(KJ−h0I)is non-positive and 0is an isolated eigenvalue. Moreover, if Jsatisfies that J(x, y)>0∀x, y ∈Ωsuch that d(x, y)<R and Ωis R-connected, then 0is a simple eigenvalue with only constant eigenfunctions. Proof. From proposition 3.25, the spectrum σX(K−hI) is independent of X. Hence, we prove the result in L2(Ω) and in this space K−h0Iis self-adjoint. Using the symmetry of Jand Fubini’s theorem, we get, for any u∈L2(Ω), (KJ−h0)u, uL2,L2=−1 2ΩΩ J(x, y)(u(x)−u(y))2dydx⩽0.(3.19) From this, as in (3.17), we get σL2(Ω)(KJ−h0)⩽sup u∈L2(Ω), uL2(Ω)=1 (KJ−h0)u, uL2(Ω),L2(Ω)⩽0. Also observe that constant functions satisfy (K−h0)u= 0 and, since 0 ∈ R(−h0), 0 is an isolated eigenvalue with finite multiplicity. If, moreover, Jsatisfies the sign assumption in the statement and ϕis an eigenfunction of 0, from (3.19) we get 0=(K−h0I)ϕ, ϕL2(Ω),L2(Ω)=−1 2ΩΩ J(x, y)(ϕ(y)−ϕ(x))2dydx. Since J(x, y)>0 for x, y ∈Ωwith d(x, y)<R, for all x∈Ω,ϕ(x)=ϕ(y) for any y∈B(x, R). Thus, since Ωis R-connected, ϕis a constant function in Ω. Therefore, 0 is a simple eigenvalue. 4. The linear evolution equation Let (Ω, μ, d) be a metric measure space. The problem we are going to work with in this section is ut(x, t)=(KJ−hI)u(x, t)=Lu(x, t), u(x, 0) = u0(x), x∈Ω, t > 0, x∈Ω, (4.1) with KJu(x)=ΩJ(x, y)u(y)dy,J⩾0, h∈L∞(Ω), u0∈Xand KJ∈L(X,X), where either X=Lp(Ω), with 1 ⩽p⩽∞,orX=Cb(Ω) (in which case we assume that h∈C b(Ω)). First, since KJ∈L(X,X), problem (4.1) has a unique strong solution u∈ C∞(R,X) given by u(t)=e Ltu0. The mapping Rt→ u(t)=e Ltu0∈X is analytic. Moreover, the mapping (t, u0)→ eLtu0is continuous. https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press Linear non-local diffusion problems in metric measure spaces 855 We denote by SK,h the group associated with the operator L=KJ−hI to highlight the dependence on KJand h. Hence, the solution of (4.1) is u(t, u0)=SK,h(t)u0=e Ltu0. 4.1. Maximum principles We now prove that the solutions of (4.1) satisfy maximum principles. For this, let ube the solution to (4.1). We take the function v(t)=e h(·)tu(t) for t⩾0, which satisfies vt(x, t)=e h(x)tKJu(x, t) and v(x, 0) = u0(x). Hence, integrating in time we get u(x, t)=e −h(x)tu0(x)+t 0 e−h(x)(t−s)KJu(x, s)ds. (4.2) Let X=Lp(Ω) with 1 ⩽p⩽∞,orletX=Cb(Ω). For every ω0∈Xand T>0 we consider the mapping Fω0:C([0,T]; X)→C([0,T]; X) defined as Fω0(ω)(x, t)=e −h(x)tω0(x)+t 0 e−h(x)(t−s)KJ(ω)(x, s)ds. Then we have the following immediate result. Lemma 4.1.If ω0,z 0∈Xand ω,z ∈XT=C([0,T]; X), then there exist two constants C1and C2depending on hand T, such that |Fω0(ω)−Fz0(z)| ⩽C1(T)ω0−z0X+C2(T)|ω−z|,(4.3) where C1(T)=e h−L∞(Ω)T,C2(T)=CTeh−L∞(Ω)T,C2:[0,∞)→Ris increasing and continuous, and C2(T)→0as T→0. With this and the standard Picard iterations, we can prove the following proposition. Proposition 4.2 (weak maximum principle).For every non-negative u0∈Xthe solution to problem (4.1) is non-negative for all t⩾0. Proof. Thanks to (4.2), we know that the solution to (4.1) can be written as u(x, t)=e −h(x)tu0(x, t)+t 0 e−h(x)(t−s)KJu(x, s)ds=Fu0u(x, t). We choose Tsmall enough such that C2(T) in lemma 4.1 satisfies that C2(T)<1. Hence, by (4.3) we have that Fu0(·) is a contraction in XT=C([0,T]; X). We consider the sequence of Picard iterations un+1(x, t)=Fu0(un)(x, t)∀n⩾1,x∈Ω, 0⩽t⩽T. Then the sequence unconverges to uin XT. We take u1(x, t)=u0(x)⩾0, and then, for t⩾0, u2(x, t)=Fu0(u1)(x, t)=e −h(x)tu0(x)+t 0 e−h(x)(t−s)KJ(u0)(x)ds https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press 856 A. Rodr´ıguez-Bernal and S. Sastre-G´omez is non-negative, because KJis a positive operator. Thus, u2(x, t)⩾0 for all t⩾0. Repeating this argument for all un, we get that un(x, t) is non-negative for every n⩾1 for t⩾0. As unconverges to uin XT, we have that uis non-negative. Since T>0 does not depend on the initial data, if we consider again the same problem with initial data u(·,T), then the solution u(·,t) is non-negative for all t∈[T,2T]. Since (4.1) has a unique solution, we have proved that the solution of (4.1) satisfies u(x, t)⩾0 for all t∈[0,2T]. Repeating this argument, we have that the solution of (4.1) is non-negative for all t⩾0. We now show that with the assumptions in proposition 3.15 we have in fact the strong maximum principle. Theorem 4.3 (strong maximum principle).Assume that KJ∈L(X,X)and J⩾ 0satisfies J(x, y)>0for all x, y ∈Ωsuch that d(x, y)<R for some R>0, and Ωis R-connected. Then for every non-trivial u0⩾0in X, the solution u(t)of (4.1) is strictly positive for all t>0. Proof. Thanks to proposition 4.2, we know that u⩾0inΩfor all t⩾0. We take v(t)=e h(·)tu(t) and then, recalling the definition of the essential support in definition 3.13, we have P(u(t)) = P(v(t)) for all t⩾0. From the argument above (4.2), we know that v satisfies vt(t)=e h(·)tKJ(u(t)) ⩾0∀t⩾0.(4.4) Integrating (4.4) over [s, t], we obtain v(t)=v(s)+t s vt(r)dr⩾v(s) for any t⩾s⩾0.(4.5) Then P(v(t)) ⊃P(v(s)) for all t⩾s. Moreover, since v(t)=e h(·)tu(t) and thanks to (4.5), we obtain u(t)⩾e−h(·)(t−s)u(s). This implies that P(u(t)) ⊃P(u(s)) for all t⩾s. As a consequence of (4.5), we have that for any subset D⊂Ω, v|D(t)=v|D(s)+t s (eh(·)rKJ(u(r)))|Ddr. (4.6) Since P(v(t)) ⊃P(v(s)) for all t⩾s, and from (4.6), we have that P(u(t)) ∩D=P(v(t)) ∩D⊃P(KJu(r)) ∩Dfor all r∈[s, t].(4.7) Moreover, applying proposition 3.15 to u(s), we have P(KJu(r)) ⊃P(KJ(u(s))) ⊃P1 R(u(s)) =  x∈P(u(s)) B(x, R) for all r∈[s, t]. (4.8) https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press Linear non-local diffusion problems in metric measure spaces 857 Hence, if we consider the set D=P1 R(u(s)), from (4.7) and (4.8) we have that P(u(t)) ⊃P1 R(u(s)) for all t>s. (4.9) Hence, the essential support of the solution at time tcontains the balls of radius R centred at the points in the support of the solution at time s<t.Wefixt>0, let C⊂Ωbe a compact set and then proposition 3.15 implies that there exists n0∈N such that C⊂Pn(u0) for all n⩾n0. We consider the sequence of times t=tn,t n−1=t(n−1) n, ..., t j=tj n, ..., t 1=t n,t 0=0. Therefore, thanks to (4.9), we have that the essential support at time tcontains the balls of radius Rcentred at the points in the essential support at time tn−1, P1 R(u(tn−1)), which contains the balls of radius Rcentred at the points in the essential support at time tn−2,P2 R(u(tn−2)). Hence, repeating this argument, we have P(u(t)) = P(u(tn)) ⊃P1 R(u(tn−1)) ⊃P2 R(u(tn−2)) ⊃···⊃Pn R(u0)⊃C. Thus, we have proved that u(t) is strictly positive for every compact set in Ωfor all t>0. Therefore, u(t) is strictly positive in Ωfor all t>0. We also get the following immediate consequence. Corollary 4.4.Under the assumptions of theorem 4.3, if u0⩾0, not identically zero, with P(u0)=Ω, then the solution to (4.1) has to be sign changing in Ωfor all t<0. 4.2. Asymptotic regularizing effects In general, the group associated with (4.1) has no regularizing effects. However, we will prove that there exists a part of the group, which we call S2(t), that is compact, so it somehow regularizes. Moreover, there exists another part of the group, which we call S1(t), that does not regularize, i.e. it carries the singularities of the initial data, but it decays to zero exponentially as tgoes to ∞if h⩾0. Thus, we have asymptotic smoothness as defined in [16, p. 4]. Theorem 4.5.Let μ(Ω)<∞.For1⩽p⩽q⩽∞, let X=Lq(Ω)or Cb(Ω).If KJ∈L(Lp(Ω),X)is compact (see proposition 3.7) and hsatisfies h(x)⩾α>0for all x∈Ω and u0∈Lp(Ω), then the group associated with problem (4.1) satisfies that u(t)=SK,h(t)u0=S1(t)u0+S2(t)u0 with (i) S1(t)∈L(Lp(Ω)) for all t>0, and S1(t)L(Lp(Ω),Lp(Ω)) →0exponentially as t→∞; (ii) S2(t)∈L(Lp(Ω),X)is compact for all t>0. Therefore, SK,h(t)is asymptotically smooth. https://doi.org/10.1017/S0308210515000724 Published online by Cambridge University Press