The existence and nonexistence of global solutions for a semilinear heat equation on graphs
TThe existence and nonexistence of global solutionsfor a semilinear heat equation on graphs
Yong Lin Yiting Wu
Abstract
Let G = ( V, E ) be a finite or locally finite connected weighted graph, ∆ be theusual graph Laplacian. Using heat kernel estimate, we prove the existence and nonexistence ofglobal solutions for the following semilinear heat equation on G (cid:26) u t = ∆ u + u α in (0 , + ∞ ) × V , u (0 , x ) = a ( x ) in V .We conclude that, for a graph satisfying curvature dimension condition CDE (cid:48) ( n,
0) and V ( x, r ) (cid:39) r m , if 0 < mα <
2, then the non-negative solution u is not global, and if mα > u provided that the initial value is small enough.In particular, these results are true on lattice Z m . Keywords : Semilinear heat equation, Global existence, Heat kernel estimate : 35A01; 35K91; 35R02; 58J35
The existence or nonexistence of global solutions to a simple system (cid:26) u t = ∆ u + u α ( t > , x ∈ R m ) ,u (0 , x ) = a ( x ) ( x ∈ R m ) , (1.1)has been extensively studied since 1960s. One of the most important results about it is fromFujita [5]. Fujita claimed that, if 0 < mα <
2, then there does not exist a non-negative globalsolution for any non-trivial non-negative initial data. On the other hand, if mα >
2, then thereexists a global solution for a sufficiently small initial data. It is clear that Fujita’s result does notinclude the critical exponent α = m . The nonexistence of global solutions for critical exponentwas proved in [10, 12].Recently, the study of equations on graphs has attracted attention from many researchers invarious fields (see [2, 6, 7, 8, 14, 16] and references therein). Grigoryan et al. [6, 7, 8] establishedexistence results for Yamabe type equations and some nonlinear elliptic equations on graphs.The solutions of heat equation and its variations on graphs have also been investigated by manyauthors due to its wide range of applications ranging from modelling energy flows through anetwork to processing image [3, 4]. Chung et al. [2] considered the extinction and positivity ofthe solutions of the Dirichlet boundary value problem for u t = ∆ u − u q with q > u t = ∆ u + u q with q > q ≤
1, every solutionis global, and if q > u t = ∆ u + u α with α > (cid:26) u t = ∆ u + u α in (0 , + ∞ ) × V , u (0 , x ) = a ( x ) in V . (1.2)1 a r X i v : . [ m a t h . A P ] F e b otivated by [5], we find that the key technical point of proving the existence of globalsolutions is the estimate of heat kernel. In [1], Bauer et al. obtained the Gaussian upper boundfor a graph satisfying CDE ( n, CDE (cid:48) ( n, CDE (cid:48) ( n,
0) and V ( x, r ) (cid:39) r m , the behaviors of the solutions forproblem (1.2) strongly depend on m and α . In particular, for lattice Z m , we have similar resultsof Fujita [5] on Euclidean space R m .The rest of the paper is organized as follows. In Section 2, we introduce some concepts,notations and known results which are essential to prove the main results of this paper. InSection 3, we formally state our main results. In Sections 4 and 5, we respectively prove thenonexistence and existence of global solutions for problem (1.2). In Section 6, we study thebehavior of the solutions for problem (1.2) under the curvature condition CDE (cid:48) . In Section7, we give an example to explain our conclusions intuitively. Meanwhile, we also provide anumerical experiment to demonstrate the example.
Throughout the paper, we assume that G = ( V, E ) is a finite or locally finite connectedgraph and contain neither loops nor multiple edges, where V denotes the vertex set and E denotes the edge set. We write y ∼ x if y is adjacent to x , or equivalently xy ∈ E . For eachvertex x , its degree is defined by deg( x ) = { y ∈ V : y ∼ x } . We allow the edges on the graph to be weighted. Weights are given by a function ω : V × V → [0 , ∞ ), that is, the edge xy has weight ω xy ≥ ω xy = ω yx . Furthermore, let µ : V → R + be a positive finite measure on the vertices of the G . In this paper, all the graphsin our concern are assumed to satisfy D ω := µ max ω min < ∞ and D µ := max x ∈ V m ( x ) µ ( x ) < ∞ , where ω min := inf xy ∈ E ω xy > m ( x ) := (cid:80) y ∼ x ω xy . Let C ( V ) be the set of real functions on V . For any 1 ≤ p < ∞ , we denote by (cid:96) p ( V, µ ) = (cid:40) f ∈ C ( V ) : (cid:88) x ∈ V µ ( x ) | f ( x ) | p < ∞ (cid:41) the set of (cid:96) p integrable functions on V with respect to the measure µ . For p = ∞ , let (cid:96) ∞ ( V, µ ) = (cid:26) f ∈ C ( V ) : sup x ∈ V | f ( x ) | < ∞ (cid:27) . f ∈ C ( V ), the µ -Laplacian ∆ of f is defined by∆ f ( x ) = 1 µ ( x ) (cid:88) y ∼ x ω xy ( f ( y ) − f ( x )) , it can be checked that D µ < ∞ is equivalent to the µ -Laplacian ∆ being bounded on (cid:96) p ( V, µ )for all p ∈ [1 , ∞ ] (see [9]). The special cases of µ -Laplacian operators are the cases where µ ≡ µ ( x ) = (cid:80) y ∼ x ω xy = deg( x ), whichyields the normalized graph Laplacian.The gradient form Γ associated with a µ -Laplacian is defined byΓ( f, g )( x ) = 12 µ ( x ) (cid:88) y ∼ x ω xy ( f ( y ) − f ( x ))( g ( y ) − g ( x )) . We write Γ( f ) = Γ( f, f ).The iterated gradient form Γ is defined by2Γ ( f, g ) = ∆Γ( f, g ) − Γ( f, ∆ g ) − Γ(∆ f, g ) . We write Γ ( f ) = Γ ( f, f ) . Besides, the integration of a function f ∈ C ( V ) is defined by (cid:90) V f dµ = (cid:88) x ∈ V µ ( x ) f ( x ) . The connected graph can be endowed with its graph distance d ( x, y ), i.e. the smallestnumber of edges of a path between two vertices x and y , then we define balls B ( x, r ) = { y ∈ V : d ( x, y ) ≤ r } for any r ≥
0. The volume of a subset A of V can be written as V ( A ) and V ( A ) = (cid:80) x ∈ A µ ( x ), for convenience, we usually abbreviate V (cid:0) B ( x, r ) (cid:1) by V ( x, r ). In addition,a graph G satisfies a uniform volume growth of degree m , if for all x ∈ V , r ≥ V ( x, r ) (cid:39) r m , that is, there exists a constant c (cid:48) ≥
1, such that c (cid:48) r m ≤ V ( x, r ) ≤ c (cid:48) r m . We say that a function p : (0 , + ∞ ) × V × V → R is a fundamental solution of the heatequation u t = ∆ u on G = ( V, E ), if for any bounded initial condition u : V → R , the function u ( t, x ) = (cid:88) y ∈ V p ( t, x, y ) u ( y ) ( t > , x ∈ V )is differentiable in t and satisfies the heat equation, and for any x ∈ V , lim t → + u ( t, x ) = u ( x )holds.For completeness, we recall some important properties of the heat kernel p ( t, x, y ), as follows: Proposition 2.1 (see [11, 15]) . For t, s > x, y ∈ V , we have(i) p ( t, x, y ) = p ( t, y, x ),(ii) p ( t, x, y ) > (cid:80) y ∈ V µ ( y ) p ( t, x, y ) ≤ ∂ t p ( t, x, y ) = ∆ x p ( t, x, y ) = ∆ y p ( t, x, y ),(v) (cid:80) z ∈ V µ ( z ) p ( t, x, z ) p ( s, z, y ) = p ( t + s, x, y ).3n [1], Bauer et al. introduced two slightly different curvature conditions which are called CDE and
CDE (cid:48) . Let us now recall the two definitions.
Definition 2.1.
A graph G satisfies the exponential curvature dimension inequality CDE ( x, n, K ),if for any positive function f : V → R + such that ∆ f ( x ) <
0, we haveΓ ( f )( x ) − Γ (cid:32) f, Γ( f ) f (cid:33) ( x ) ≥ n (∆ f )( x ) + K Γ( f )( x ) , we say that CDE ( n, K ) is satisfied if CDE ( x, n, K ) is satisfied for all x ∈ V . Definition 2.2.
A graph G satisfies the exponential curvature dimension inequality CDE (cid:48) ( x, n, K ),if for any positive function f : V → R + , we haveΓ ( f )( x ) − Γ (cid:32) f, Γ( f ) f (cid:33) ( x ) ≥ n f ( x ) (∆ log f )( x ) + K Γ( f )( x ) , we say that CDE (cid:48) ( n, K ) is satisfied if CDE (cid:48) ( x, n, K ) is satisfied for all x ∈ V .Bauer et al. [1] established a discrete analogue of the Li-Yau inequality and derived a heatkernel estimate under the condition of CDE ( n, Proposition 2.2 (see [1]) . Suppose G satisfies CDE ( n, C such that, for any x, y ∈ V and t > p ( t, x, y ) ≤ C V ( x, √ t ) . (2.1)Furthermore, for any t >
1, there exists constants C and C such that p ( t, x, y ) ≥ C t n exp (cid:32) − C d ( x, y ) t − (cid:33) . (2.2)Although the upper bound in the result of Bauer et al. [1] is formulated with Gaussianform, the lower bound is not quite Gaussian form and is dependent on the parameter n . Basedon this, Horn et al. [11] used CDE (cid:48) to imply volume doubling and derived the Gaussian typeon-diagonal lower bound. Here, we transcribe a relevant result of [11] as follows:
Proposition 2.3 (see [11]) . Suppose G satisfies CDE (cid:48) ( n, x ∈ V and t > , p (2 t , x, x ) ≥ CV ( x, t ) , (2.3)where C >
CDE (cid:48) , Lin et al. [13] only utilized the volumegrowth condition to obtain a on-diagonal lower estimate of heat kernel on graphs for large time,which is enough to prove the nonexistence of global solution of (3.1) stated in Section 3. Werecall it bellow.
Proposition 2.4 (see [13]) . Assume that, for all x ∈ V and r ≥ r , V ( x, r ) ≤ c r m , where r , c , m are some positive constants. Then, for all large enough t , p ( t, x, x ) ≥ V ( x, C t log t ) , (2.4)where C > D µ e . 4 Main results
In this paper, we study whether or not there exist global solutions to the initial valueproblem for the semilinear heat equation (cid:26) u t = ∆ u + u α in (0 , + ∞ ) × V , u (0 , x ) = a ( x ) in V , (3.1)where α is a positive parameter, a ( x ) is bounded, non-negative and not trivial in V . Withoutloss of generality, we can assume that a ( e ) > e ∈ V . Throughout the present paper weshall only deal with non-negative solutions so that there is no ambiguity in the meaning of u α .We shall also fix the vertex e .For convenience, we state relevant definitions firstly. Definition 3.1.
Assume that
T >
0, a non-negative function u = u ( t, x ) satisfying (3.1) in[0 , T ] × V is called a solution of (3.1) in [0 , T ], if u is bounded and continuous with respect to t . Furthermore, a solution u of (3.1) in [0 , + ∞ ) is a function whose restriction to [0 , T ] × V is asolution of (3.1) in [0 , T ] for any T >
0. A solution u of (3.1) in [0 , + ∞ ) is also called a globalsolution of (3.1) in [0 , + ∞ ). Definition 3.2. F [0 , + ∞ ) is the set of all non-negative continuous(with respect to t ) functions u = u ( t, x ) defined in [0 , + ∞ ) × V satisfying0 ≤ u ( t, x ) ≤ M p ( t + γ, e, x )with some constants M > γ >
0. Furthermore, if u is a solution of (3.1) in [0 , + ∞ ) and u ∈ F [0 , + ∞ ), then u is called a global solution of (3.1) in F [0 , + ∞ ).Our main results are stated in the following theorems. Theorem 3.1.
Assume that, for all x ∈ V and r ≥ r , the volume growth V ( x, r ) ≤ c r m holds,where r , c , m are some positive constants. If 0 < mα <
1, then there is no non-negative globalsolution of (3.1) in [0 , + ∞ ) for any bounded, non-negative and non-trivial initial value. Theorem 3.2.
Assume that G satisfies CDE ( n,
0) and V ( x, r ) ≥ c r m with some positiveconstants c and m . Suppose for any γ >
0, there exists a positive number δ such that 0 ≤ a ( x ) ≤ δp ( γ, e, x ) in V . If mα >
2, then (3.1) has a global solution u = u ( t, x ) in F [0 , + ∞ ),which satisfies 0 ≤ u ( t, x ) ≤ M p ( t + γ, e, x ), for any ( t, x ) ∈ [0 , + ∞ ) × V and some positiveconstants M ( δ ). Corollary 3.1.
Suppose G satisfies CDE (cid:48) ( n,
0) and V ( x, r ) (cid:39) r m for some m > < mα <
2, then there is no non-negative global solution of (3.1) in [0 , + ∞ ) for anybounded, non-negative and non-trivial initial value.(ii) If mα >
2, then there exists a global solution of (3.1) in F [0 , + ∞ ) for a sufficiently smallinitial value. We first introduce a lemma which will be used in the proof of Theorem 3.1.
Lemma 4.1.
Let
T >
0, if u = u ( t, x ) is a non-negative solution of (3.1) in [0 , T ], then we have J − α − u ( t, e ) − α ≥ αt (0 < t ≤ T ) , where J = J ( t ) = (cid:88) x ∈ V µ ( x ) p ( t, e, x ) a ( x ) . roof. Let ε be a positive constant and for any fixed t ∈ (0 , T ], we put v ε ( s, x ) = p ( t − s + ε, e, x ) (0 ≤ s ≤ t, x ∈ V )and J ε ( s ) = (cid:88) x ∈ V µ ( x ) v ε ( s, x ) u ( s, x ) (0 ≤ s ≤ t ) . (i) We prove that J ε is positive for all s ∈ [0 , t ].Since a ( e ) = u (0 , e ) > u ( s, e ) is non-negative in (0 , t ], for all 0 ≤ s ≤ t , it follows that ∂u∂s ( s, e ) − ∆ u ( s, e ) ≥ . (4.1)Note that ∆ u ( s, e ) = 1 µ ( e ) (cid:88) y ∼ e ω ey (cid:0) u ( s, y ) − u ( s, e ) (cid:1) ≥ − µ ( e ) (cid:88) y ∼ e ω ey u ( s, e ) ≥ − D µ u ( s, e ) , then the inequality (4.1) gives ∂u∂s ( s, e ) ≥ − D µ u ( s, e ) , which implies u ( s, e ) ≥ u (0 , e ) exp( − D µ s ) > , s ∈ [0 , t ] . Hence, for all 0 ≤ s ≤ t , we have (cid:88) x ∈ V u ( s, x ) > . In view of the fact that v ε ( s, x ) is positive in [0 , t ] × V , we obtain J ε ( s ) > , t ].(ii) We prove that J ε is differentiable with respect to s and satisfies the following equation dds J ε ( s ) = (cid:88) x ∈ V µ ( x ) v ε ( s, x ) u ( s, x ) α . Case 1.
We consider the case where G is a finite connected graph.Since ω xy = ω yx , according to the definition of ∆, for any function f, g ∈ C ( V ), we have (cid:88) x ∈ V µ ( x )∆ f ( x ) g ( x ) = (cid:88) x ∈ V µ ( x ) f ( x )∆ g ( x ) . (4.2)From the property of the heat kernel, we know that ∂∂s v ε = − ∆ v ε . Thus dds J ε ( s ) = (cid:88) x ∈ V (cid:18) µ ( x ) ∂∂s v ε ( s, x ) u ( s, x ) + µ ( x ) v ε ( s, x ) ∂∂s u ( s, x ) (cid:19) = (cid:88) x ∈ V (cid:16) − µ ( x )∆ v ε ( s, x ) u ( s, x ) + µ ( x ) v ε ( s, x ) (cid:16) ∆ u ( s, x ) + u ( s, x ) α (cid:17)(cid:17) = − (cid:88) x ∈ V µ ( x )∆ v ε ( s, x ) u ( s, x ) + (cid:88) x ∈ V µ ( x ) v ε ( s, x )∆ u ( s, x )+ (cid:88) x ∈ V µ ( x ) v ε ( s, x ) u ( s, x ) α = (cid:88) x ∈ V µ ( x ) v ε ( s, x ) u ( s, x ) α . (4.3)6 ase 2. We consider the case where G is a locally finite connected graph.Firstly, we claim that J ε exists if G is locally finite.Since u is bounded, we can assume there exists a constant A > s, x ) ∈ [0 , t ] × V , | u ( s, x ) | ≤ A. Hence, from the property of the heat kernel, we have J ε = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) x ∈ V µ ( x ) v ε ( s, x ) u ( s, x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A (cid:88) x ∈ V µ ( x ) v ε ( s, x ) ≤ A < ∞ . Secondly, we observe that if G is locally finite, the exchange between summation and deriva-tion in the first step of (4.3) is because J ε ( s ) and dds J ε ( s ) both are uniformly convergent.Indeed, when ∆ is a bounded operator, we have P t u ( x ) = e t ∆ u ( x ) = + ∞ (cid:88) k =0 t k ∆ k k ! u ( x ) = (cid:88) y ∈ V µ ( y ) p ( t, x, y ) u ( y ) , (4.4)furthermore, we can prove that the summation (4.4) has a nice convergency when u ( x ) is abounded function. The details are as follows:Assuming that | u ( x ) | ≤ A in V , then | ∆ u ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ ( x ) (cid:88) y ∼ x ω xy (cid:0) u ( y ) − u ( x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ D µ A. By iteration, we obtain for any k ∈ N and x ∈ V , (cid:12)(cid:12) ∆ k u ( x ) (cid:12)(cid:12) ≤ k D kµ A. Thus for any t ∈ (0 , T ] and x ∈ V , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t k ∆ k k ! u ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T k ∆ k k ! u ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ T k k ! 2 k D kµ A. In view of + ∞ (cid:88) k =0 T k k ! 2 k D kµ A = Ae D µ T < ∞ , which shows (cid:80) y ∈ V µ ( y ) p ( t, x, y ) u ( y ) converges uniformly on (0 , T ], when u ( x ) is bounded in V .Since u ( s, x ) and u ( s, x ) α both are bounded, we can obtain that J ε ( s ) and dds J ε ( s ) con-verge uniformly on [0 , t ].Thirdly, we notice that if G is locally finite, (4.2) may not always hold, but for any boundedfunction u , it satisfies (cid:88) y ∈ V µ ( y )∆ p ( t, x, y ) u ( y ) = (cid:88) y ∈ V µ ( y ) p ( t, x, y )∆ u ( y ) . (4.5)7 direct computation yields (cid:88) y ∈ V µ ( y )∆ p ( t, x, y ) u ( y ) = (cid:88) y ∈ V (cid:88) z ∈ V ω yz (cid:0) p ( t, x, z ) u ( y ) − p ( t, x, y ) u ( y ) (cid:1) = (cid:88) y ∈ V (cid:88) z ∈ V ω yz p ( t, x, z ) u ( y ) − (cid:88) y ∈ V (cid:88) z ∈ V ω yz p ( t, x, y ) u ( y )= (cid:88) z ∈ V (cid:88) y ∈ V ω yz p ( t, x, y ) u ( z ) − (cid:88) y ∈ V (cid:88) z ∈ V ω yz p ( t, x, y ) u ( y )= (cid:88) y ∈ V (cid:88) z ∈ V ω yz p ( t, x, y ) u ( z ) − (cid:88) y ∈ V (cid:88) z ∈ V ω yz p ( t, x, y ) u ( y )= (cid:88) y ∈ V µ ( y ) p ( t, x, y )∆ u ( y ) . Note that the summation can be exchanged, since (cid:88) y ∈ V (cid:88) z ∈ V (cid:12)(cid:12) ω yz p ( t, x, y ) u ( z ) (cid:12)(cid:12) ≤ (cid:88) y ∈ V µ ( y ) p ( t, x, y ) (cid:32)(cid:88) z ∈ V ω yz µ ( y ) | u ( z ) | (cid:33) ≤ D µ A. Finally, we state that if G is locally finite, the interchanges of sum in the third step of (4.3)are again because of the convergence of the sums.Noting that | ∆ u ( s, x ) | ≤ D µ A , | u ( s, x ) α | ≤ A α , and (cid:88) x ∈ V µ ( x )∆ v ε ( s, x ) u ( s, x ) = (cid:88) x ∈ V µ ( x ) v ε ( s, x )∆ u ( s, x ) , for any ( s, x ) ∈ [0 , t ] × V , we deduce that (cid:80) x ∈ V µ ( x ) v ε ( s, x )∆ u ( s, x ), (cid:80) x ∈ V µ ( x ) v ε ( s, x ) u ( s, x ) α and (cid:80) x ∈ V µ ( x )∆ v ε ( s, x ) u ( s, x ) all are convergent.(iii) Since v ε > (cid:88) x ∈ V µ ( x ) v ε ( s, x ) ≤ , using the Jensen’s inequality to x α ( α >
0) and owing to its convexity, we obtain (cid:80) x ∈ V µ ( x ) v ε ( s, x ) u ( s, x ) α (cid:80) x ∈ V µ ( x ) v ε ( s, x ) ≥ (cid:18) (cid:80) x ∈ V µ ( x ) v ε ( s, x ) u ( s, x ) (cid:80) x ∈ V µ ( x ) v ε ( s, x ) (cid:19) α , that is, (cid:32)(cid:88) x ∈ V µ ( x ) v ε ( s, x ) u ( s, x ) (cid:33) α ≤ (cid:32)(cid:88) x ∈ V µ ( x ) v ε ( s, x ) u ( s, x ) α (cid:33) (cid:32)(cid:88) x ∈ V µ ( x ) v ε ( s, x ) (cid:33) α ≤ (cid:88) x ∈ V µ ( x ) v ε ( s, x ) u ( s, x ) α . It follows that dds J ε ≥ J αε . J ε (0) − α − J ε ( t ) − α ≥ αt. (4.6)According to (4.4), we can assert that for any bounded function u ,lim t → + P t u ( x ) = lim t → + (cid:88) y ∈ V µ ( y ) p ( t, x, y ) u ( y ) = u ( x ) , from which we will get J ε ( t ) → u ( t, e ) ( ε → + ) . (4.7)Moreover, it is not difficult to find that J ε (0) → J ( ε → + ) . (4.8)In fact, if G is a finite connected graph, the (4.8) is obvious. If G is a locally finite connectedgraph, because of the uniform convergence of J ε ( s ), we can exchange limitation with summationand obtain (4.8).Combining (4.7) and (4.8) into (4.6), for any t ∈ (0 , T ], we have J − α − u ( t, e ) − α ≥ αt. This completes the proof of Lemma 4.1.
Proof of Theorem 3.1.
Based on the above Lemmas, we prove Theorem 3.1 by contradiction.Suppose that there exists a non-negative global solution u = u ( t, x ) of (3.1) in [0 , + ∞ ),according to Lemma 4.1, we have for any t > J − α ≥ u ( t, e ) − α + αt ≥ αt. Since V ( x, r ) ≤ c r m , ( r ≥ r ), from Proposition 2.4, we have for all large enough t , p ( t, e, e ) ≥ c C m ( t log t ) − m ( C > D µ e ) . Hence, for all sufficiently large t , J = (cid:88) x ∈ V µ ( x ) p ( t, e, x ) a ( x ) ≥ µ ( e ) a ( e ) p ( t, e, e ) ≥ C ( t log t ) − m , where C = µ ( e ) a ( e )4 c C m > C > D µ e .Combining J − α ≥ αt and J ≥ C ( t log t ) − m , for all large enough t , we get( t log t ) mα ≥ αC α t. (4.9)However, if 0 < mα <
1, we will get a contradiction for large enough t .This completes the proof of Theorem 3.1. 9 Proof of Theorem 3.2
Before proving Theorem 3.2, we consider the following integral equations associated with(3.1) and obtain its solution u ( t, x ) in F (0 , + ∞ ). u ( t, x ) = u ( t, x ) + (cid:90) t (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) u ( s, y ) α ds in (0 , + ∞ ) × V ,u ( t, x ) = (cid:88) y ∈ V µ ( y ) p ( t, x, y ) a ( y ) in (0 , + ∞ ) × V , (5.1)where α > a ( y ) is bounded, non-negative, not trivial and satisfying 0 ≤ a ( y ) ≤ δp ( γ, e, y ) in V with some constants γ > δ >
0. We fix γ here and will determine δ later.For any function v ( t, x ) with | v | ∈ F (0 , + ∞ ), we can define its norm || v || = sup t> ,x ∈ V | v ( t, x ) | ρ ( t, x ) , (5.2)where ρ ( t, x ) = p ( t + γ, e, x ).Let (Φ u )( t, x ) = (cid:90) t (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) u ( s, y ) α ds. We first prove some lemmas which are essential to prove the Theorem 3.2.
Lemma 5.1. If G satisfies CDE ( n,
0) and V ( x, r ) ≥ c r m with some positive constants c and m . Let mα >
2, then Φ ρ ∈ F (0 , + ∞ ) and || Φ ρ || ≤ (cid:101) C, where (cid:101) C is a positive constant. Proof.
For any ( t, x ) ∈ (0 , + ∞ ) × V ,(Φ ρ )( t, x ) = (cid:90) t (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) ρ ( s, y ) α ds = (cid:90) t (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) p ( s + γ, e, y ) p ( s + γ, e, y ) α ds. Obviously, Φ ρ is non-negative and continuous with respect to t .According to Proposition 2.2, for γ > s ≥
0, there exists a constant C such that p ( s + γ, e, y ) ≤ C V (cid:0) e, √ s + γ (cid:1) . Since V (cid:0) e, √ s + γ (cid:1) ≥ c ( s + γ ) m , we obtain p ( s + γ, e, y ) ≤ C c − ( s + γ ) − m . (5.3)Hence, (Φ ρ )( t, x ) ≤ (cid:90) t (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) p ( s + γ, e, y )( C c − ) α ( s + γ ) − mα ds ≤ ( C c − ) α (cid:90) t ( s + γ ) − mα (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) p ( s + γ, e, y ) ds = ( C c − ) α p ( t + γ, e, x ) (cid:90) t ( s + γ ) − mα ds. (cid:90) t ( s + γ ) − mα ds ≤ (cid:90) + ∞ ( s + γ ) − mα ds = − γ − mα γ − mα , (5.4)it is worth noting that the existence of the integral in (5.4) is based on the assumption mα > t, x ) ∈ (0 , + ∞ ) × V ,(Φ ρ )( t, x ) ≤ (cid:101) Cp ( t + γ, e, x ) , (5.5)where (cid:101) C = − γ − mα ( C c − ) α γ − mα > ρ ∈ F (0 , + ∞ ) and || Φ ρ || ≤ (cid:101) C. This completes the proof of Lemma 5.1.
Lemma 5.2.
Under the condition of Lemma 5.1 and u ∈ F (0 , + ∞ ), we haveΦ u ∈ F (0 , + ∞ ) and || Φ u || ≤ (cid:101) C || u || α . Proof.
Since u ∈ F (0 , + ∞ ), we can define its norm and then have u ( t, x ) ≤ || u || ρ ( t, x ) for any( t, x ) ∈ (0 , + ∞ ) × V .A simple calculations show that0 ≤ (Φ u )( t, x ) = (cid:90) t (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) u ( s, y ) α ds ≤ || u || α (cid:90) t (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) ρ ( s, y ) α ds = || u || α (Φ ρ )( t, x ) . (5.6)Combining (5.6) with (5.5), we getΦ u ∈ F (0 , + ∞ ) and || Φ u || ≤ (cid:101) C || u || α . This completes the proof of Lemma 5.2.
Lemma 5.3.
Under the condition of Lemma 5.1, we suppose that u and v are in F (0 , + ∞ ) andsatisfy || u || ≤ M and || v || ≤ M with a positive number M . Then we have || Φ u − Φ v || ≤ (cid:101) C (1 + α ) M α || u − v || . Proof.
Since u, v ∈ F (0 , + ∞ ), for any ( t, x ) ∈ [0 , ∞ ) × V , we get (cid:12)(cid:12) u ( t, x ) − v ( t, x ) (cid:12)(cid:12) ≤ | u ( t, x ) | − | v ( t, x ) | ≤ M ρ ( t, x ) , which implies | u − v | ∈ F (0 , + ∞ ).By using of the elementary inequality | p α − q α | ≤ (1 + α ) | p − q | max { p α , q α } ( q ≥ , p ≥ ,
11e have (cid:12)(cid:12) u ( s, y ) α − v ( s, y ) α (cid:12)(cid:12) ≤ (1 + α ) | u ( s, y ) − v ( s, y ) | max { u ( s, y ) α , v ( s, y ) α }≤ (1 + α ) M α ρ ( s, y ) α (cid:12)(cid:12) u ( s, y ) − v ( s, y ) (cid:12)(cid:12) ≤ (1 + α ) M α ρ ( s, y ) α || u − v || ρ ( s, y )= (1 + α ) M α ρ ( s, y ) α || u − v || . Case 1.
When G is a finite connected graph, for any ( t, x ) ∈ (0 , + ∞ ) × V , we find that (cid:12)(cid:12) Φ u ( t, x ) − Φ v ( t, x ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) (cid:16) u ( s, y ) α − v ( s, y ) α (cid:17) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) t (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) (cid:12)(cid:12)(cid:12) u ( s, y ) α − v ( s, y ) α (cid:12)(cid:12)(cid:12) ds ≤ (1 + α ) M α || u − v || (cid:90) t (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) ρ ( s, y ) α ds =(1 + α ) M α || u − v || (Φ ρ )( t, x ) ≤ (1 + α ) M α || u − v || (cid:101) Cρ ( t, x ) , (5.7)thus || Φ u − Φ v || ≤ (cid:101) C (1 + α ) M α || u − v || . (5.8) Case 2.
When G is a locally finite connected graph, we shall make an annotation on theabove calculation.Since u ∈ F (0 , + ∞ ) and || u || ≤ M , we have0 ≤ u ( t, x ) ≤ M p ( t + γ, e, x ) . By (5.3), we know that p ( t + γ, e, x ) ≤ C c − γ − m , hence for any ( t, x ) ∈ (0 , + ∞ ) × V , we deduce that0 ≤ u ( t, x ) ≤ A, where A = M C c − γ − m .Similarly, v also satisfies 0 ≤ v ( t, x ) ≤ A .Hence, (cid:80) y ∈ V µ ( y ) p ( t − s, x, y ) u ( s, y ) α and (cid:80) y ∈ V µ ( y ) p ( t − s, x, y ) v ( s, y ) α both are con-vergent, which shows that (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) u ( s, y ) α − (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) v ( s, y ) α = (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) (cid:16) u ( s, y ) α − v ( s, y ) α (cid:17) . Based on the above discussion, we verify the validity of inequalities (5.7) and (5.8) underthe condition that G is locally finite.The proof of Lemma 5.3 is complete. 12 roof of Theorem 3.2. (i) We construct the solution of (5.1) in F (0 , + ∞ ).Setting a iteration relation u n +1 = u + Φ u n ( n = 0 , , · · · ) (5.9)with u given by (5.1) and u n ∈ F (0 , + ∞ ), ( n = 1 , , · · · ).Since 0 ≤ a ( y ) ≤ δp ( γ, e, y ), for any ( t, x ) ∈ (0 , + ∞ ) × V , we have0 ≤ u ( t, x ) ≤ δ (cid:88) y ∈ V µ ( y ) p ( t, x, y ) p ( γ, e, y )= δp ( t + γ, e, x ) , which shows u ∈ F (0 , + ∞ ) and || u || ≤ δ .According to Lemma 5.2, we obtain the inequalities || u n +1 || ≤ || u || + || Φ u n || ≤ || u || + (cid:101) C || u n || α , that is, || u n +1 || ≤ δ + (cid:101) C || u n || α , ( n = 0 , , · · · ) . It is easy to observe that lim δ → δ α (cid:0) δ α (cid:1) α δ α = 0 , so there exist some δ < δ α (cid:0) δ α (cid:1) α < δ α .Setting A ≡ (cid:110) δ : 0 < δ < , δ α (cid:0) δ α (cid:1) α < δ α and (cid:101) Cδ α < (cid:111) . For any δ ∈ A , we have || u || ≤ δ, || u || ≤ δ + (cid:101) Cδ α < δ + δ α , || u || ≤ δ + (cid:101) C (cid:0) δ + δ α (cid:1) α < δ + δ α (cid:0) δ α (cid:1) α < δ + δ α (cid:0) δ α (cid:1) α , || u || ≤ δ + (cid:101) C (cid:2) δ + δ α (cid:0) δ α (cid:1) α (cid:3) α < δ + δ α (cid:2) δ α (cid:0) δ α (cid:1) α (cid:3) α < δ + δ α (cid:0) δ α (cid:1) α , · · ·|| u n || < δ + δ α (cid:0) δ α (cid:1) α , · · · thus we can assume that || u n || ≤ M , ( n = 0 , , · · · ) with a constant M = M ( δ ) satisfying M ( δ ) → + ( δ → + ) . (5.10)From (5.10), we can choose a constant δ ∈ A such that κ ≡ (cid:101) C (1 + α ) M ( δ ) α < . Note that u n +2 − u n +1 = Φ u n +1 − Φ u n ,
13t follows Lemma 5.3 that || u n +2 − u n +1 || ≤ κ || u n +1 − u n || ( n = 0 , , · · · ) . (5.11)Since κ <
1, the inequality (5.11) implies that u n converges with respect to the norm || · || .Moreover, for any Cauchy sequence { u n } in F (0 , ∞ ), one can easily conclude that { u n } isconvergent.Hence, there exists a function u ∈ F (0 , + ∞ ) such that || u n − u || → n → ∞ ) . (5.12)Utilizing (5.9) and (5.12) leads us to the assertion that u is a solution of (5.1) in F (0 , + ∞ ).(ii) We prove that the solution u ( t, x ) of (5.1) constructed above satisfies (3.1).For any T >
0, since u ∈ F (0 , T ], we derive from (5.3) that u is bounded and continuouswith respect to t in (0 , T ] × V .Taking a small positive number ε , we put(Φ ε u )( t, x ) = (cid:90) t − ε (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) u ( s, y ) α ds, where 0 < ε < t ≤ T and x ∈ V .Obviously, Φ ε u tends to Φ u in [ σ, T ] × V as ε → + , here σ is an arbitrary positive numberand σ > ε . Case 1. If G is a finite connected graph, recalling an important property of heat kernel: p t ( t, x, y ) = ∆ x p ( t, x, y ) = ∆ y p ( t, x, y ) , (5.13)we have ∂∂t (Φ ε u ) = (cid:88) y ∈ V µ ( y ) p ( ε, x, y ) u ( t − ε, y ) α + (cid:90) t − ε (cid:88) y ∈ V µ ( y ) p t ( t − s, x, y ) u ( s, y ) α ds = (cid:88) y ∈ V µ ( y ) p ( ε, x, y ) u ( t − ε, y ) α + (cid:90) t − ε (cid:88) y ∈ V µ ( y )∆ x p ( t − s, x, y ) u ( s, y ) α ds ≡ I + I . (5.14)Owing to the boundedness of u α , it is immediate from (4.4) that I tends to u ( t, x ) α in[ σ, T ] × V as ε → + . On the other hand, when ε → + , I converges in [ σ, T ] × V to a function ϕ ( t, x ) = (cid:90) t (cid:88) y ∈ V µ ( y )∆ x p ( t − s, x, y ) u ( s, y ) α ds. Letting ε → + in (5.14), we obtain for any ( t, x ) ∈ [ σ, T ] × V , ∂∂t (Φ u )( t, x ) = u ( t, x ) α + ϕ ( t, x )= u ( t, x ) α + ∆ x (Φ u )( t, x ) . (5.15)Since u = u + Φ u and ∂∂t u = ∆ u , for any ( t, x ) ∈ [ σ, T ] × V , we conclude that u t = ∆ u + ∂∂t Φ u = ∆ u + ∆(Φ u ) + u α = ∆ u + u α . (5.16)14ecause of the arbitrariness of σ , the (5.16) is true for all ( t, x ) ∈ (0 , T ] × V .Furthermore, we can prove that the initial-value condition is satisfied in the sense that u ( t, x ) → a ( x ) ( t → + ) , from which we can extend u ( t, x ) to t = 0 and set u (0 , x ) = a ( x ).By the arbitrariness of T , we can deduce that the solution u ( t, x ) of (5.1) constructed aboveis the required global solution of (3.1) in F [0 , + ∞ ). Case 2. If G is a locally finite connected graph, as before, some facts need to be verified:(a) ∂∂t (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) u ( s, y ) α = (cid:88) y ∈ V µ ( y ) p t ( t − s, x, y ) u ( s, y ) α ;(b) (cid:90) t (cid:88) y ∈ V µ ( y )∆ x p ( t − s, x, y ) u ( s, y ) α ds = ∆ (cid:90) t (cid:88) y ∈ V µ ( y ) p ( t − s, x, y ) u ( s, y ) α ds . Note that u is bounded and D µ < ∞ , we deduce that ∆ u α is bounded too.Following (4.5) and (5.13), we find that (cid:88) y ∈ V µ ( y ) p t ( t − s, x, y ) u ( s, y ) α = (cid:88) y ∈ V µ ( y )∆ y p ( t − s, x, y ) u ( s, y ) α = (cid:88) y ∈ V µ ( y ) p ( t − s, x, y )∆ u ( s, y ) α , thus (cid:80) y ∈ V µ ( y ) p t ( t − s, x, y ) u ( s, y ) α converges uniformly, from which we can see that the fact(a) is valid.Similar to the proof of (4.5), the fact (b) is also true due to the absolute convergence ofsums.In view of (a) and (b), for a locally finite graph we will have the same conclusion as a finitegraph.This completes the proof of Theorem 3.2. Proof of Corollary 3.1.
As a direct consequence of Theorem 3.1, if we add a curvature condition
CDE (cid:48) ( n,
0) to G , then we conclude that there is no global solution to problem (3.1) for the caseof 0 < mα < t > max (cid:8) , r (cid:9) , we have p ( t, e, e ) ≥ c − C (cid:18) t (cid:19) − m . Hence J ≥ C (cid:48) t − m , where C (cid:48) = 2 m c − Cµ ( e ) a ( e ) > t > max (cid:8) , r (cid:9) .Combining J − α ≥ αt and J ≥ C (cid:48) t − m , for any t > max (cid:8) , r (cid:9) , we have t mα ≥ αC (cid:48) α t. (6.1)15owever, if 0 < mα <
2, the above inequality (6.1) is distinctly not true for a sufficientlylarge t . This proves the assertion in the first part of Corollary 3.1.On the other hand, Horn et al. [11] have concluded that CDE (cid:48) ( n,
0) implies
CDE ( n, CDE ( n, CDE (cid:48) ( n,
0) in Theorem 3.2.This completes the proof of Corollary 3.1.
In this section, we give an example to illustrate our result asserted in Corollary 3.1It is well known that the integer grid Z m admits the uniform volume growth of degree m . In[1], Bauer et al. proved that Z m satisfies CDE (2 m,
0) and
CDE (cid:48) (4 . m,
0) for the normalizedgraph Laplacian, from which we can deduce the existence and non-existence of global solutionto problem (3.1) in Z m with the normalized graph Laplacian, as follows: Proposition 7.1
Let G be Z m with µ ( x ) ≡ deg( x ).(i) If 0 < mα <
2, then there is no non-negative global solution of (3.1) in [0 , + ∞ ) for anybounded, non-negative and non-trivial initial value.(ii) If mα >
2, then there exists a global solution of (3.1) in F [0 , + ∞ ) for a sufficiently smallinitial value.For example, we consider a circle C n (as shown in Figure 1) which satisfies CDE (cid:48) ( n (cid:48) ,
0) forsome number n (cid:48) related to n. And then the problem (3.1) can be written as u t ( t, x ) = (cid:0) u ( t, x ) + u ( t, x ) (cid:1) − u ( t, x ) + u ( t, x ) α ,u t ( t, x ) = (cid:0) u ( t, x ) + u ( t, x ) (cid:1) − u ( t, x ) + u ( t, x ) α ,u t ( t, x ) = (cid:0) u ( t, x ) + u ( t, x ) (cid:1) − u ( t, x ) + u ( t, x ) α ,u t ( t, x ) = (cid:0) u ( t, x ) + u ( t, x ) (cid:1) − u ( t, x ) + u ( t, x ) α ,u t ( t, x ) = (cid:0) u ( t, x ) + u ( t, x ) (cid:1) − u ( t, x ) + u ( t, x ) α ,u t ( t, x ) = (cid:0) u ( t, x ) + u ( t, x ) (cid:1) − u ( t, x ) + u ( t, x ) α ,u (0 , x ) = a ( x ) ,u (0 , x ) = a ( x ) ,u (0 , x ) = a ( x ) ,u (0 , x ) = a ( x ) ,u (0 , x ) = a ( x ) ,u (0 , x ) = a ( x ) , (7.1)where we take µ ( x ) = (cid:80) y ∼ x ω xy = deg( x ) = 2.Figure 1: C n If we choose α = 1 , a ( x ) = 1 , a ( x ) = 2 , a ( x ) = 3 , a ( x ) = 4 , a ( x ) = 5 , a ( x ) = 6,respectively. It is easy to verify that the above choices satisfy the condition of non-existence ofglobal solution to the equations (7.1). The numerical experiment result is shown in Figure 2.Besides, if we choose α = 3 , a ( x ) = 1 × − , a ( x ) = 2 × − , a ( x ) = 3 × − , a ( x ) =4 × − , a ( x ) = 5 × − , a ( x ) = 6 × − , respectively. Then the above choices satisfy the16ondition of existence of global solution to the equations (7.1). The numerical experiment resultis shown in Figure 3.Figure 2: Non-existence of global solution to the equations (7.1)Figure 3: Existence of global solution to the equations (7.1)17 cknowledgments This research is supported by the Fundamental Research Funds for the Central Universities,and the Research Funds of Renmin University of China under Grant 17XNH106.
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