Asymptotic behavior for the discrete in time heat equation
AASYMPTOTIC BEHAVIOR FOR THE DISCRETE IN TIME HEATEQUATION
LUCIANO ABADIAS AND EDGARDO ALVAREZ
Abstract.
In this paper we investigate the asymptotic behavior and decay of the solutionof the discrete in time N -dimensional heat equation. We give a convergence rate with whichthe solution tends to the discrete fundamental solution, and the asymptotic decay, bothin L p ( R N ) . Furthermore we prove optimal L -decay of solutions. Since the technique ofenergy methods is not applicable, we follow the approach of estimates based on the discretefundamental solution which is given by an original integral representation and also by Mac-Donald’s special functions. As a consequence, the analysis is different to the continuous intime heat equation and the calculations are rather involved. Introduction
The linear heat equation u t = ∆ u is one of the most studied problems in the theory ofpartial differential equations. It was introduced by J. Fourier (see [19]) to model severaldiffusion phenomena. Since then, it has been applied in the study of different processes inmany mathematical areas such as PDEs, functional analysis, harmonic analysis, probability,among others. The nature of this problem is well known and we will not further explain it.One of the aspects of interest, see [14, 27, 29], is the large time behavior of solutions of theheat problem(1.1) (cid:26) u t = ∆ u, t ≥ , x ∈ R N ,u (0) = f ( x ) . If f ∈ L ( R N ) , the solution of (1.1) on L p ( R N ) is u ( t ) = G t ∗ f, where ∗ denotes the classicalconvolution on R N and G t ( x ) = 1(4 πt ) N/ e − | x | t , t > , x ∈ R N , is the heat kernel. It is known that integrating over all of R N we get that the total mass ofsolutions is conserved for all time, that is, (cid:90) R N u ( t, x ) dx = (cid:90) R N u (0 , x ) dx. Mathematics Subject Classification.
Key words and phrases.
Discrete heat equation; Large-time behavior; Decay of solutions; Discrete funda-mental solution.The first named author has been partly supported by Project MTM2016-77710-P, DGI-FEDER, of theMCYTS and Project E26-17R, D.G. Arag´on, Universidad de Zaragoza, Spain. a r X i v : . [ m a t h . A P ] F e b L. ABADIAS AND E. ALVAREZ
This fact leads us to think that the total mass of solutions should have importance in theasymptotic behavior of solutions. Indeed, it is well known that if M = (cid:82) R N u (0 , x ) dx then(1.2) t N (1 − /p ) (cid:107) u ( t ) − M G t (cid:107) p → , as t → ∞ , for 1 ≤ p ≤ ∞ . The previous estimate shows that the difference on L p ( R N ) between thesolution u ( t ) and M G t decays to zero like o ( t N − /p ) ) as t goes to infinity.Also, it is known that the p -norms of the solution vanish as t → ∞ for p > . This fact isknown as that the p -energy is not conservative. More precisely (cid:107) u ( t ) (cid:107) p (cid:46) (cid:107) f (cid:107) q t N (1 /q − /p ) , (cid:107)∇ u ( t ) (cid:107) p (cid:46) (cid:107) f (cid:107) q t N (1 /q − /p )+1 / , (cid:107) ∆ u ( t ) (cid:107) p (cid:46) (cid:107) f (cid:107) q t N (1 /q − /p )+1 , for f ∈ L q ( R N ) , ≤ q ≤ p ≤ ∞ . One can consider the first moment as the vector quantity (cid:82) R n x u ( t, x ) dx. It can be seen thatsuch moment is also conserved in time for the solution of (1.1) whenever (1+ | x | ) f ∈ L ( R N ) . Moreover, under such assumption we are able to improve the convergence (1.2), that is, t N (1 − /p )+1 / (cid:107) u ( t ) − M G t (cid:107) p → , as t → ∞ . However, the second moment (cid:90) R n | x | u ( t, x ) dx = (cid:90) R n | x | f ( x ) dx + 2 N t is not conservative. In fact, it is known that only integral quantities conserved by thesolutions of (1.1) are the mass and the first moment.This type of large-time asymptotic results have been also studied for several diffusion prob-lems. For example in [9, 12, 16, 20, 26] the authors studied large-time behaviour and otherasymptotic estimates for the solutions of different diffusion problems in R N , and similaraspects are studied for open bounded domains in [8, 16]. Estimates for heat kernels on mani-folds have been also studied in [7,18,21]. In [25], the author obtains gaussian upper estimatesfor the heat kernel associated to the sub-laplacian on a Lie group, and also for its first-ordertime and space derivatives.On the other hand, finite differences, sometimes also called discrete derivatives, were intro-duced some centuries ago, and they have been used along the literature in different math-ematical problems, mainly in approximation of derivatives for the numerical solution ofdifferential equations and partial differential equations. The most knowing ones are the for-ward, backward and central differences (the forward and backward differences are associatedto the Euler, explicit and implicit, numerical methods). We denote them in the followingway; let h > , for a function u defined on the mesh Z h := { nh : n ∈ Z } we write δ right u ( nh ) := u (( n + 1) h ) − u ( nh ) h , δ left u ( nh ) := u ( nh ) − u (( n − h ) h , and δ c u ( nh ) := (cid:18) δ right + δ left (cid:19) u ( nh ) = u (( n + 1) h ) − u (( n − h )2 h . SYMPTOTIC BEHAVIOR FOR THE DISCRETE IN TIME HEAT EQUATION 3
In the last years, and taking as a guide the paper [4], several authors have been workingin the context of partial difference-differential equations ( [1, 2, 5, 6, 22, 23]) from an specificpoint of view; in that papers the approach has been focused in mathematical analysis, moreprecisely, harmonic analysis, functional analysis and fractional differences. Particularly in [1]it is shown that the operators δ right and δ left generate markovian C -semigroups on (cid:96) p ( Z ) . Also, in [5], the authors study harmonic properties of the solution of the heat problem onone-dimensional graphs (the mesh Z h ), and the wave equation on graphs is studied in [23].An abstract approach for discrete in time Cauchy problems is given in [22]. Also, non-localproblems in the discrete framework appear in [2, 6].The previous comments motivate the main aim of this paper; let h > , we consider the firstorder Cauchy problem for the heat equation in discrete time(1.3) (cid:40) δ left u ( nh, x ) = ∆ u ( nh, x ) + g ( nh, x ) , n ∈ N , x ∈ R N ,u (0 , x ) = f ( x ) , where ∆ is the classical laplacian on L p ( R N ), u is defined on N h × R N , with N h := { nh : n ∈ N } , f is defined on R and g is defined on N h × R N , with N h := { nh : n ∈ N } . Along the paper we study asymptotic behavior and decay of the solution of (1.3). For thatpurpose, we need to know properties of the fundamental solution of the homogeneous prob-lem associated to (1.3) (when g = 0). In fact, one of the key points to get such asymptoticproperties is an integral representation of the fundamental solution for the associated homo-geneous equation. Furthermore, we describe explicitly this solution in terms of MacDonald’sfunctions which arise naturally from the integral representation of the solution. This repre-sentation is quite original and allows to study the decay of solutions for the problem (1.3)when the initial datum belongs to p -integrable Lebesgue spaces. Moreover, both the integralrepresentation and the explicit expression via MacDonald’s functions allow to give a quan-titative rate at which the solution converges to M times the fundamental solution, where M will denote, as in the continuous case, the initial mass of solution. The techniques used toobtain our results differs to the continuous case because we have to deal with the integralrepresentation and asymptotic properties of MacDonald’s special functions. We also noteto the reader that doing the relation t = nh, the asymptotics of G t will be similar to G n,h as t → ∞ or equivalently n → ∞ , where G n,h will denote the fundamental solution of thehomogeneous problem associated to (1.3).In this paper we are not interested in the study of the convergence as h → h ) to the classical heat problem. However, it can be seenas a natural problem studied in semigroup theory via Yosida approximants (see Remark2.1). Also, one can think about the possibility to consider similar problems to (1.3) butconsidering the discrete derivatives δ right or δ c . However, as we explain in Remark 2.2, thefundamental solutions to that problem does not have good properties.The paper is organized as follows. Section 2 is focused in the fundamental solution of thehomogeneous problem associated to (1.3). We introduce an integral representation and theexplicit expression via MacDonald’s functions. We deduce basic properties, we calculate itsgradient and laplacian, and we see that the mass and the first moment of solutions of the
L. ABADIAS AND E. ALVAREZ homogeneous problem are conservative in discrete time nh, and not the second moment.Also some pictures of the continuous and discrete gaussian kernels, with their correspondingcomments, are stated. In Section 3 we give pointwise and L p asymptotic upper bounds forthe fundamental solution G n,h , and we use such estimates to prove in Section 4 that the p -energies of solutions of (1.3) are dissipative. Section 5 is the main part of the paper; we provethe asymptotic behaviour for the discrete in time heat problem (Theorem 5.1). In Section6 we success in proving optimal L -decay estimates for the solution of the homogeneousproblem associated to (1.3). The proof is based on Fourier analysis techniques. Finallywe include an Appendix where we show some basic properties of Gamma and MacDonald’sfunctions, and a technical result about integrability.2. The discrete gaussian fundamental solution
In this section we study the fundamental solution for the homogeneous discrete in time heatinitial value problem on the Lebesgue L p ( R N ) spaces. Let h > , we consider(2.1) (cid:40) δ left u ( nh, x ) = ∆ u ( nh, x ) , n ∈ N , x ∈ R N ,u (0 , x ) = f ( x ) , where u and f are functions defined on N h × R N and R N , respectively. Formally, one canwrite the solution in the following way u ( nh, x ) = 1 h n (1 /h − ∆) − n f ( x ) , n ∈ N , x ∈ R N , whenever the resolvent operator (1 /h − ∆) − has sense. It is well known that the laplacianoperator ∆ associated to the standard heat equation in continuous time on L p ( R N ) for1 ≤ p ≤ ∞ generates the gaussian semigroup with convolution kernel G t ( x ) = 1(4 πt ) N/ e − | x | t , t > , x ∈ R N . From semigroup theory (see [11, Corollary 1.11]) we obtain u ( nh, · ) = 1 h n (1 /h − ∆) − n f ( · ) = 1 h n Γ( n ) (cid:90) ∞ e − t/h t n − ( G t ∗ f )( · ) dt := ( G n,h ∗ f )( · ) , f ∈ L p ( R N ) , where ∗ denotes the classical convolution on R N and(2.2) G n,h ( x ) = 1 h n Γ( n ) (cid:90) ∞ e − t/h t n − G t ( x ) dt, n ∈ N , x ∈ R N \ { } . Remark 2.1.
Note that fixed a positive number t > , the Yosida approximants (1 − tn ∆) − n (see [11, Theorem 3.5]) allow to approximate the gaussian C -semigroup G t as n → ∞ . Writ-ing h = t/n, the previous convergence shows that the gaussian semigroup can be approximatedby the solutions of the discrete in time problems (2.1) as the mesh h → . Remark 2.2.
It is easy to see that if we consider the forward difference δ right on (2.1) , thenformally the solution of the problem would be u ( nh, · ) = h n (1 /h + ∆) n f ( · ) , which is notdefined (bounded) on L p ( R N ) . SYMPTOTIC BEHAVIOR FOR THE DISCRETE IN TIME HEAT EQUATION 5
Also, for the central difference δ c , the fundamental solution would be given by (cid:90) ∞ J n ( t/h ) G t ( x ) dt, where J n are the Bessel functions of first kind. In this case is not difficult to prove that thesolution is bounded on L p ( R N ) , however it does not have as good properties as G n,h satisfies,for example the contractivity on L ( R N ) . These are the main reasons because of we consider the discrete in time heat problem with thebackward difference δ left . Now we will see the explicit expression of the fundamental solution G n,h in terms of specialfunctions. By [17, p.363 (9)] we have G n,h ( x ) = 1 h n Γ( n )(4 π ) N/ (cid:90) ∞ e − ( t/h + | x | / t ) t n − N/ − dt = 2Γ( n )(4 πh ) N/ (cid:18) | x | √ h (cid:19) n − N/ K n − N/ (cid:18) | x |√ h (cid:19) , n ∈ N , x ∈ R N \ { } . (2.3)Here, the functions K ν denote the Bessel functions of imaginary argument, also called Mac-Donald’s functions or modified cylinder functions (see Section 7). Observe that the identityhas not pointwise sense for x = 0 if N/ − n ≥ . In fact, for that values N/ − n ≥
0, taking | x | → G n,h ( x ) → ∞ . However, aswe will see, good properties on L p ( R N ) hold. For the case N/ − n < , by (P4) we have G n,h ( x ) → Γ( n − N/ n )(4 πh ) N/ as | x | → . Remark 2.3.
The gaussian kernel satisfies the semigroup property on time, G t ∗ G s = G t + s .Since G n,h is given by natural powers of the resolvent operator of the laplacian, it satisfiesthe discrete semigroup property. Indeed, we also can prove that property using the expression (2.2) as follows, ( G n,h ∗ G m,h )( x ) = 1 h n + m Γ( n )Γ( m ) (cid:90) R N (cid:18)(cid:90) ∞ (cid:90) ∞ e − t + sh t n − s m − G t ( x − y ) G s ( y ) ds dt (cid:19) dy = 1 h n + m Γ( n )Γ( m ) (cid:90) ∞ (cid:18)(cid:90) ∞ e − t + sh t n − s m − G t + s ( y ) ds (cid:19) dt = 1 h n + m Γ( n )Γ( m ) (cid:90) ∞ (cid:18)(cid:90) ∞ t e − σh t n − ( σ − t ) m − G σ ( x ) dσ (cid:19) dt = 1 h n + m Γ( n )Γ( m ) (cid:90) ∞ e − σh G σ ( x ) (cid:18)(cid:90) σ t n − ( σ − t ) m − dt (cid:19) dσ = 1 h n + m Γ( n )Γ( m ) (cid:90) ∞ e − σh G σ ( x ) σ m + n − B ( n, m ) dσ = G n + m,h ( x ) . Here B ( n, m ) is the Beta function. L. ABADIAS AND E. ALVAREZ
In the following we denote p n,h ( t ) =: 1 h n Γ( n ) e − t/h t n − , n ∈ N . Then we can write(2.4) G n,h ( x ) = (cid:90) ∞ p n,h ( t ) G t ( x ) dt, x (cid:54) = 0 . The above integral representation is a discretization formula for the gaussian semigroup.The case h = 1 was treated in [22] for a general C -semigroup on an abstract context.Next, we refer to the function G n,h as the fundamental solution for the problem (2.1). Thefollowing proposition states some basic properties of it. Proposition 2.4.
The function G n,h satisfies:(i) G n,h ( x ) > , n ∈ N , x (cid:54) = 0 . (ii) (cid:90) R N G n,h ( x ) dx = 1.(iii) F ( G n,h )( ξ ) = 1(1 + h | ξ | ) n , ξ ∈ R N . (iv) G n,h ( x ) − G n − ,h ( x ) h = ∆ G n,h ( x ) , n ≥ , x (cid:54) = 0 . (v) (cid:90) R N | x | G n,h ( x ) dx = 2 N nh.
Proof. (i) It is clear by (2.4). (ii) Note that (cid:82) ∞ p n,h ( t ) dt = 1 and (cid:82) R N G t ( x ) dx = 1 , thenthe result follows from the Fubini’s theorem. (iii) It is known that F ( G t )( ξ ) = e − t | ξ | , for ξ ∈ R N , then by (2.4) one gets F ( G n,h )( ξ ) = (cid:90) ∞ p n,h ( t ) F ( G t )( ξ ) dt = 1(1 + h | ξ | ) n . (iv) First of all, observe that ddt p n,h ( t ) = − h ( p n,h ( t ) − p n − ,h ( t )) for n ≥ . Then integratingby parts we get G n,h ( x ) − G n − ,h ( x ) h = (cid:90) ∞ p n,h ( t ) ∂∂t G t ( x ) dt = (cid:90) ∞ p n,h ( t )∆ G t ( x ) dt = ∆ G n,h ( x ) , x (cid:54) = 0 , where we have used that lim t → + and lim t →∞ of p n ( t ) G t ( x ) vanishes. (v) It follows easily bythe second moment of G t ( x ) and the representation (2.2). (cid:3) SYMPTOTIC BEHAVIOR FOR THE DISCRETE IN TIME HEAT EQUATION 7
Remark 2.5.
Observe that one can prove the above properties via the expression (2.3) givenby the MacDonald’s function. For example, from (P1) of Appendix we get the positivity ofthe fundamental solution. Furthermore, by [17, p. 668 (16)] it follows (cid:90) R N G n,h ( x ) dx = 2Γ( n )(4 πh ) / (cid:90) R N (cid:18) | x | √ h (cid:19) n − N/ K n − N/ (cid:18) | x |√ h (cid:19) dx = 2 N − n Γ( n )(4 πh ) / h n − N (cid:32) N π N/ Γ( N + 1) (cid:90) ∞ r n − N/ K n − N/ (cid:18) r √ h (cid:19) dr (cid:33) = N Γ( N ) h N + N h N + N Γ( N + 1) = 1 . Also note that by ∂ | x | ∂x j = x j | x | and (P2) of Appendix, we obtain (2.5) ∂∂x j G n,h ( x ) = − n )(4 πh ) N/ √ h x j | x | (cid:18) | x | √ h (cid:19) n − N/ K n − N/ − (cid:18) | x |√ h (cid:19) , and then derivating once more in the previous expression and taking into account (P3) and(P7) (with ν = n − N − ) of Appendix, we have ∂ ∂x j G n,h ( x ) = − h Γ( n )(4 πh ) N/ (cid:18) | x | √ h (cid:19) n − N − K n − N − (cid:18) | x |√ h (cid:19) + x j h h Γ( n )(4 πh ) N/ (cid:18) | x | √ h (cid:19) n − N − K n − N (cid:18) | x |√ h (cid:19) + N x j h h Γ( n )(4 πh ) N/ (cid:18) | x | √ h (cid:19) n − N − K n − N − (cid:18) | x |√ h (cid:19) − n − x j h h Γ( n )(4 πh ) N/ (cid:18) | x | √ h (cid:19) n − N − K n − N − (cid:18) | x |√ h (cid:19) . Now, since (cid:80) Nj =1 x j h = (cid:16) | x |√ h (cid:17) , we get ∆ G n,h ( x ) = G n,h ( x ) − G n − ,h ( x ) h . Finally observe that the mean square displacement can be also calculated in the followingway; using (2.3) , a change of variables and [17, p.668 (16)], we have (cid:90) R N | x | G n,h ( x ) dx = 2Γ( n )(4 πh ) N/ (2 √ h ) N − / (cid:90) R N | x | n − N/ K n − N/ (cid:18) | x |√ h (cid:19) dx = 2Γ( n )(4 πh ) N/ (2 √ h ) N − / (cid:32) N π N/ Γ( N + 1) (cid:90) ∞ r n + N +1 K n − N/ (cid:18) r √ h (cid:19) dr (cid:33) = 2 N nh.
L. ABADIAS AND E. ALVAREZ
Remark 2.6.
Note that by Proposition 2.4 (i) we have that the total mass of solution of (2.1) is conservative in the discrete time nh, that is, (cid:90) R N u ( nh, x ) dx = (cid:90) R N f ( x ) dx. Moreover, the first moment is also conservative; if (1 + | x | ) f ∈ L ( R N ) one gets (cid:90) R N x ( u ( nh, x ) − u (( n − h, x )) dx = h (cid:90) R N x ∆ u ( nh, x ) dx = 0 , and so (cid:82) R N xu ( nh, x ) dx = (cid:82) R N xf ( x ) dx. However, as in the continuous case holds, by Propo-sition 2.4 (v) it follows that the second moment is (cid:90) R n | x | u ( nh, x ) dx = (cid:90) R n | x | f ( x ) dx + 2 N nh.
To finish this section we show some pictures of the fundamental solution of (2.1). We haveused Mathematica to make them. The objective is that the reader visualizes the convergenceof G n,h to G t as the mesh h → . Figure 1 shows, in the one-dimensional case ( N = 1), the Gauss kernel G and the funda-mental solutions of the discrete problems for several values of h. As we have mentioned, theYosida approximants (which are the fundamental solutions) converge to the gaussian kernelas h → t = nh. Therefore, for the different values of h, we choose n such that nh = 1 . For example for h = 1 / G , / . Also,observe that for N = 1 the fundamental solution G n,h ( x ) is defined on the whole real linesince n − N/ > for all n ∈ N . However by (2.5), and (P6) and (P4) of Appendix we get G (cid:48) ,h ( x ) = C h x | x | / K − / ( | x |√ h ) = C h x | x | / K / ( | x |√ h ) ∼ C h x | x | , x → , where C h is a constant depending on h. This shows that G ,h is not derivable in x = 0 (seeFigure 1 for h = 1). Figure 1. - - t = Gaussianh = = / = / = / = / Figure 2, Figure 3 and Figure 4 show several approximants to the gaussian G in the two-dimensional case ( N = 2). In Figure 2 we observe that G , ( x ) → + ∞ taking x → , as wehave commented previously (since n − N/ n = 1 , N = 2). SYMPTOTIC BEHAVIOR FOR THE DISCRETE IN TIME HEAT EQUATION 9
Figure 2.
Gaussianh = = / In Figure 3 we glimpse that G , / ( x ) is well defined for x = 0 , but it is not differentiable(since n − N/ h decreases, but also that the function is smoother. Figure 3.
Gaussianh = / Figure 4.
Gaussianh = / Estimates for the fundamental solution
In this section we present pointwise and p -norm estimates for the fundamental solution of(2.1). Theorem 3.1.
Let R := | x | nh . (i) If R ≤ , then |G n,h ( x ) | ∼ nh ) N/ , for n − N/ > , and if R ≥ , then |G n,h ( x ) | (cid:46) nh | x | N +2 . (ii) If R ≤ , then |∇G n,h ( x ) | (cid:46) | x | ( nh ) N/ , for n − N/ > , and if R ≥ , then |∇G n,h ( x ) | (cid:46) nh | x | N +3 . (iii) If R ≤ , then | G n,h ( x ) − G n − ,h ( x ) h | (cid:46) nh ) N/ , for n − N/ > , and if R ≥ , then | G n,h ( x ) − G n − ,h ( x ) h | (cid:46) | x | N +2 . Proof. (i) Let R ≤ . By (2.3), (P4) of Appendix and (7.1) we have |G n,h ( x ) | ∼ Γ( n − N/ n )(4 πh ) N/ ∼ nh ) N/ . Now let R ≥ . Along the proof we will use that(3.1) e − | x |√ h ≤ k ! h k/ | x | k , k ∈ N . By (P5) of Appendix and (3.1) for k = n + N + 1 one gets |G n,h ( x ) | (cid:46) n N/ / n nh | x | N +2 (cid:46) nh | x | N +2 . (ii) Let R ≤ . Equation (2.5) implies that(3.2) |∇G n,h ( x ) | = 2Γ( n )(4 πh ) N/ √ h (cid:18) | x | √ h (cid:19) n − N/ K n − N/ − (cid:18) | x |√ h (cid:19) . SYMPTOTIC BEHAVIOR FOR THE DISCRETE IN TIME HEAT EQUATION 11
From (P4) of Appendix and (7.1) we have |∇G n,h ( x ) | ∼ Γ( n − N/ − | x | Γ( n ) h N/ (cid:46) | x | ( nh ) N/ . For R ≥ , by (3.2), (P5) of Appendix and (3.1) for k = n + N + 2 we obtain |∇G n,h ( x ) | (cid:46) hn ( n + 1)( n + 2)( n + 3)2 n | x | (cid:46) nh | x | . (iii) Applying (P3) and (P2) of Appendix in turn, it follows G n,h ( x ) − G n − ,h ( x ) h = − | x | n − N/ − Γ( n )(4 π ) N/ h N/ / (2 √ h ) n − N/ × (cid:18) N K n − N/ − (cid:18) | x |√ h (cid:19) − | x |√ h K n − N/ − (cid:18) | x |√ h (cid:19)(cid:19) := ( I ) + ( II ) . Let R ≤ . By (P4) of Appendix and (7.1), that | ( I ) | (cid:46) Γ( n − N/ − n ) h N/ ∼ nh ) N/ , and | ( II ) | (cid:46) | x | ( nh ) N/ ≤ nh ) N/ . Therefore we conclude the result for R ≤ . Now let R ≥ . Note that from the part (i), we have |G n,h ( x ) | , |G n − ,h ( x ) | (cid:46) hn N/ / n | x | N +2 . Then | G n,h ( x ) − G n − ,h ( x ) h | (cid:46) | x | N +2 . (cid:3) Now we present the asymptotic decay of the fundamental solution G n,h in Lebesgue andSobolev spaces. Theorem 3.2.
Let ≤ p ≤ ∞ , then (i) (cid:107)G n,h (cid:107) p ≤ C p nh ) N (1 − /p ) , n − N (1 − /p ) > . (ii) (cid:107)∇G n,h (cid:107) p ≤ C p nh ) N (1 − /p )+1 / , n − N (1 − /p ) > / . (iii) (cid:107) G n,h −G n − ,h h (cid:107) p ≤ C p nh ) N (1 − /p )+1 , n − N (1 − /p ) > . Here, C p is a constant independent of h and n .Proof. (i) It is well known (see [17, p.334 (3.326)]) that there exist C p (independent of t )such that || G t || p = C p t N − p ) , (cid:107)∇ G t (cid:107) p = C p t N − p )+1 / and (cid:107) ∂∂t G t (cid:107) p ≤ C p t N − p )+1 for t > (cid:107)G n,h (cid:107) p ≤ C p h n Γ( n ) (cid:90) ∞ e − th t n − N (1 − p ) − dt = C p Γ( n − N (1 − p )) h (1 − p ) Γ( n ) ≤ C p ( nh ) N (1 − p ) , where we have applied (7.1). The cases (ii) and (iii) follow in a similar way than (i). (cid:3) Asymptotic L p − L q decay Let h > , and the time mesh N h . Now we consider the non-homogenous problem(4.1) (cid:40) δ left u ( nh, x ) = ∆ u ( nh, x ) + g ( nh, x ) , n ∈ N , x ∈ R N ,u (0 , x ) = f ( x ) , where u, f, g are functions defined on N h × R N , R N and N h × R N respectively.Formally, from (4.1), one gets(4.2) u ( nh, x ) = ( G n,h ∗ f )( x ) + h n (cid:88) j =1 ( G j,h ∗ g (( n − j + 1) h, · ))( x ) , n ∈ N , x ∈ R N . The expression (4.2) gives a classical solution of (4.1) on L p ( R N ) (1 ≤ p ≤ ∞ ) whenever f, g ( nh, · ) ∈ L p ( R N ) , for n ∈ N . For convenience, we write the classical solution as u ( nh, x ) = u c ( nh, x ) + u p ( nh, x ) , where u c ( nh, x ) = ( G n,h ∗ f )( x )and u p ( nh, x ) = h n (cid:88) j =1 ( G n − j +1 ,h ∗ g ( jh, · ))( x ) . Next, let us present a result about the L p − L q asymptotic decay for u c . Theorem 4.1.
Let ≤ q ≤ p ≤ ∞ . If f ∈ L q ( R N ) , then the solution u of (4.1) satisfies (i) (cid:107) u c ( nh ) (cid:107) p ≤ C p nh ) N (1 /q − /p ) (cid:107) f (cid:107) q . (ii) (cid:107)∇ u c ( nh ) (cid:107) p ≤ C p nh ) N (1 /q − /p )+1 / (cid:107) f (cid:107) q . (iii) (cid:107) δ left u c ( nh ) (cid:107) p ≤ C p nh ) N (1 /q − /p )+1 (cid:107) f (cid:107) q . Here, C p is a constant independent of h and n . SYMPTOTIC BEHAVIOR FOR THE DISCRETE IN TIME HEAT EQUATION 13
Proof.
Take r ≥ /p = 1 /q + 1 /r, and applying Young’s inequality we get (cid:107) u c ( nh ) (cid:107) p = (cid:107)G n,h ∗ f (cid:107) p ≤ (cid:107)G n,h (cid:107) r (cid:107) f (cid:107) q . From Theorem 3.2 (i) follows the case (i). The other cases are similar. (cid:3)
Now, assuming certain conditions on the function g we get an asymptotic decay for u p . Theorem 4.2.
Let ≤ q ≤ p ≤ ∞ , and g ( nh, · ) ∈ L q ( R N ) with (cid:107) g ( nh, · ) (cid:107) q (cid:46) nh ) γ , for γ > . (i) If γ, N (1 /q − /p ) (cid:54) = 1 , then the solution u of (4.1) satisfies (cid:107) u p ( nh ) (cid:107) p (cid:46) ( nh ) − min { ,γ } ( nh ) min { , N (1 /q − /p ) } . (ii) If γ = 1 and N (1 /q − /p ) (cid:54) = 1 , then the solution u of (4.1) satisfies (cid:107) u p ( nh ) (cid:107) p (cid:46) log nh ( nh ) min { , N (1 /q − /p ) } . (iii) If γ (cid:54) = 1 and N (1 /q − /p ) = 1 , then the solution u of (4.1) satisfies (cid:107) u p ( nh ) (cid:107) p (cid:46) log nh ( nh ) min { ,γ } . (iv) If γ, N (1 /q − /p ) = 1 , then the solution u of (4.1) satisfies (cid:107) u p ( nh ) (cid:107) p (cid:46) log nhnh . Proof.
Let r ≥ /p = 1 /q + 1 /r. By Young’s inequality and Theorem 3.2 (i)one gets (cid:107) u p ( nh ) (cid:107) p ≤ h n (cid:88) j =1 (cid:107)G n − j +1 ,h (cid:107) r (cid:107) g ( jh, · )) (cid:107) q (cid:46) h n (cid:88) j =1 n − j + 1) h ) N (1 /q − /p ) jh ) γ = h (cid:18) [ n/ (cid:88) j =1 + n (cid:88) j =[ n/ (cid:19) n − j + 1) h ) N (1 /q − /p ) jh ) γ = I + I . On one hand, for 1 ≤ j ≤ [ n/
2] we have n/ ≤ n − j + 1 , which in turn implies that I (cid:46) nh ) N (1 /q − /p ) [ n/ (cid:88) j =1 jh ) γ (cid:46) ( nh ) − min { ,γ } ( nh ) N (1 /q − /p ) , γ (cid:54) = 1 , and I (cid:46) log nh ( nh ) N /q − /p ) when γ = 1 . On the other hand, I (cid:46) nh ) γ n (cid:88) j =1 jh ) N (1 /q − /p ) (cid:46) ( nh ) − min { , N (1 /q − /p ) } ( nh ) γ , γ (cid:54) = 1 , and I (cid:46) log nh ( nh ) γ when N (1 /q − /p ) = 1 . (cid:3) Large-time behaviour of solutions
In the following we study the asymptotic behavior of solution of (4.1), more precisely we willprove as the solution u c + u p converges asymptotically to a lineal combination of the massof the initial data f and the mass of the non-homogeneity g. Moreover, we will able to statethe rate of the convergence. Along the section we will assume the following:(a) f ∈ L ( R N ).(b) There exists γ > max { , N (1 − /p ) } such that (cid:107) g ( jh, · ) (cid:107) (cid:46) j γ , j ∈ N . Set also M c = (cid:90) R N f ( x ) dx, M p = ∞ (cid:88) j =1 (cid:90) R N g ( jh, x ) dx. Taking into account the previous notation, we present the next theorem.
Theorem 5.1.
Let ≤ p ≤ ∞ . Assume the conditions ( a ) - ( b ) , and suppose that u is theclassical solution of (4.1) . (i) Then ( nh ) N (1 − p ) (cid:107) u c ( nh ) − M c G n,h (cid:107) p → , as n → ∞ , and ( nh ) N (1 − p ) (cid:107) u p ( nh ) − hM p G n,h (cid:107) p → , as n → ∞ . (ii) Suppose in addition that | x | f ∈ L ( R ) , then ( nh ) N (1 − p ) (cid:107) u c ( nh ) − M c G n,h (cid:107) p (cid:46) ( nh ) − / . Proof.
We start proving the assertion ( ii ). Since f, | x | f ∈ L ( R N ) , by Decomposition Lemma7.1 there exists φ ∈ L ( R N ; R N ) such that f = M c δ + div φ in the distributional sense, and (cid:107) φ (cid:107) ≤ C (cid:90) R N | x || f ( x ) | dx < ∞ . SYMPTOTIC BEHAVIOR FOR THE DISCRETE IN TIME HEAT EQUATION 15
Then u c ( nh, x ) = ( G n,h ∗ ( M c δ + div φ ( · )))( x )= M c G n,h ( x ) + ( ∇G n,h ∗ φ )( x ) , which implies (cid:107) u c ( nh ) − M c G n,h (cid:107) p ≤ C (cid:107)∇G n,h (cid:107) p (cid:107)| x | f ( x ) (cid:107) ≤ K p,f nh ) N (1 − /p )+1 / , where we have used part ( ii ) of Theorem 3.2. This implies(5.1) (cid:107)G n,h ∗ f − M c G n,h (cid:107) p ≤ K p,f nh ) N (1 − /p )+ . To prove the first part of assertion ( i ), we choose a sequence ( η j ) ⊂ C ∞ ( R N ) such that (cid:82) R N η j ( x ) dx = M c for all j, and η j → f in L ( R N ). For each j , by Theorem 3.2 and (5.1)we get (cid:107) u c ( nh ) − M c G n,h (cid:107) p = (cid:107)G nh ∗ f − M c G n,h (cid:107) p ≤ (cid:107)G n,h ∗ ( f − η j ) (cid:107) p + (cid:107)G n,h ∗ η j − M c G n,h (cid:107) p ≤ (cid:107)G n,h (cid:107) p (cid:107) f − η j (cid:107) + (cid:107)G n,h ∗ η j − M c G n,h (cid:107) p ≤ C p nh ) N (1 − /p ) (cid:107) f − η j (cid:107) + K p,η j nh ) N (1 − /p )+ . It follows that ( nh ) N (1 − /p ) (cid:107) u c ( nh ) − M c G n,h (cid:107) p ≤ C p (cid:107) f − η j (cid:107) + K p,η j nh ) , which implies lim sup n →∞ ( nh ) N (1 − /p ) (cid:107) u c ( nh ) − M c G n,h (cid:107) p ≤ C p (cid:107) f − η j (cid:107) . The assertion follows by letting j → ∞ .Next, let us prove the second part of ( i ). We can write M p = n (cid:88) j =1 (cid:90) R N g ( jh, x ) + ∞ (cid:88) j = n +1 (cid:90) R N g ( jh, x ) dx. It follows from Theorem 3.2 that( nh ) N (1 − p ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) G n,h ( · ) ∞ (cid:88) j = n +1 (cid:90) R N g ( jh, x ) dx (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ ( nh ) N (1 − p ) (cid:107)G n,h ( · ) (cid:107) p ∞ (cid:88) j = n +1 (cid:90) R N | g ( jh, x ) | dx → , n → ∞ . Therefore it is enough to show the following( nh ) N (1 − p ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h n (cid:88) j =1 ( G n − j +1 ,h ∗ g ( jh, · ))( · ) − h G n,h ( · ) n (cid:88) j =1 (cid:90) R N g ( jh, y ) dy (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p → , n → ∞ . Ir order to prove the assertion, we fix 0 < δ < . In particular, this implies that 0 < δ < <
12 and δ − δ < . Next, we decompose the set { , , , .., n } × R N into two partsΩ := { , , ..., (cid:100) nδ (cid:101)} × { y ∈ R N : | y | ≤ ( δnh ) / } , Ω := { , , ..., n } × R N \ Ω . Let us start with the set Ω . By the integral form of the Minkowski inequality we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) ( j,y ) ∈ Ω (cid:90) ( h G n − j +1 ,h ( · − y ) − h G n,h ( · )) g ( jh, y ) dy (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ h (cid:88) ( j,y ) ∈ Ω (cid:90) (cid:107)G n − j +1 ,h ( · − y ) − G n,h ( · ) (cid:107) p | g ( jh, y ) | dy. Note that in this set the following inequalities hold(5.2) n ≥ n − j + 1 ≥ n (1 − δ ) > n , where the second inequality follows from δn −(cid:100) δn (cid:101) ≥ − . Now, when ( j, y ) ∈ Ω , we considerthe following subsets over R N A = { x ∈ R N : | x − y | ≤ δnh ) / } , B := { x ∈ R n : | x − y | > δnh ) / } , and we write the p -norm over Ω in the following way (cid:107)G n − j +1 ,h ( · − y ) − G n,h ( · ) (cid:107) p ≤ (cid:18)(cid:90) A |G n − j +1 ,h ( x − y ) − G n,h ( x ) | p dx (cid:19) /p + (cid:18)(cid:90) B |G n − j +1 ,h ( x − y ) − G n,h ( x ) | p dx (cid:19) /p . Let us estimate on Ω the part A of the p -norm. First we write (cid:18)(cid:90) A |G n − j +1 ,h ( x − y ) − G n,h ( x ) | dx (cid:19) /p ≤ (cid:18)(cid:90) A |G n − j +1 ,h ( x − y ) | p dx (cid:19) /p + (cid:18)(cid:90) A |G n,h ( x ) | p dx (cid:19) /p =: I + I . SYMPTOTIC BEHAVIOR FOR THE DISCRETE IN TIME HEAT EQUATION 17
For ( j, y ) ∈ Ω and x ∈ A we have that | x − y | ( n − j + 1) h ≤ δ (1 − δ ) < . Since we want to estimate the solution for large values of n , we can assume that n > N .Thus, (5.2) implies that n − j + 1 > n/ > N/ . It follows from Theorem 3.1 (i) that |G n − j +1 ,h ( x − y ) | ∼ n − j + 1) h ) N/ . Then ( nh ) N (1 − p ) hI (cid:46) ( nh ) N (1 − p ) h (( n − j + 1) h ) N/ (cid:18)(cid:90) A dx (cid:19) /p = C p ( nh ) N (1 − p ) h ( δnh ) N/ p (( n − j + 1) h ) N/ ≤ C p hδ N/ p , where in the last inequality we have used (5.2). Analogously, for ( j, y ) ∈ Ω and x ∈ A wehave | x | nh ≤ ( | x − y | + | y | ) nh ≤ δ < , which implies that |G n,h ( x ) | ∼ nh ) N/ . Therefore ( nh ) N (1 − p ) hI (cid:46) C p hδ N/ p . Since (cid:80) ∞ j =1 (cid:107) g ( jh, · ) (cid:107) < ∞ , we get( nh ) N/ − /p ) h (cid:88) ( j,y ) ∈ Ω (cid:90) (cid:18)(cid:90) A |G n − j +1 ,h ( x − y ) −G n,h ( x ) | p dx (cid:19) /p | g ( jh, y ) | dy ≤ C p hδ N/ p → , δ → . Now we consider on Ω the part B of the p -norm. We write (cid:18)(cid:90) B |G n − j +1 ,h ( x − y ) − G n,h ( x ) | p dx (cid:19) /p ≤ (cid:18)(cid:90) B |G n − j +1 ,h ( x − y ) − G n − j +1 ,h ( x ) | p dx (cid:19) /p + (cid:18)(cid:90) B |G n − j +1 ,h ( x ) − G n,h ( x ) | p dx (cid:19) /p =: I + I . First, let us estimate I . By mean value theorem there exists ˜ x between x − y and x ( x denote the integration variable) such that I = | y | (cid:18)(cid:90) B |∇G n − j +1 ,h (˜ x ) | p dx (cid:19) /p . Since | y | ≤ ( δnh ) / < | x − y | then(5.3) | ˜ x | ≥ | x − y | − | ˜ x − ( x − y ) | ≥ | x − y | − | y | ≥ | x − y | , and(5.4) | ˜ x | ≤ | x − y | + | y | ≤ | x − y | + | x − y | | x − y | . Equations (5.3) and (5.4) show that | ˜ x | and | x − y | are comparable. Also, by (5.2) and (5.3)we obtain(5.5) | ˜ x | (( n − j + 1) h ) / ≥ | x − y | n − j + 1) h ) / > δ / . Now we will use the asymptotics of |∇G n − j +1 ,h (˜ x ) | , so we divide I in two parts, I and I depending on whether | ˜ x | (( n − j +1) h ) / is less or greater than 1 respectively (we are assuming δ enough small).In I , when | ˜ x | (( n − j +1) h ) / ≥
1, by (5.4) one gets(( n − j + 1) h ) / ≤ | ˜ x | ≤ | x − y | . For this reason, the integration region in I is contained in { x ∈ R N : (( n − j + 1) h ) / ≤| x − y |} . From Theorem 3.1 (ii), the fact that | y | ≤ ( δnh ) / , (5.2) and (5.3), we have I ≤ C ( δnh ) / (cid:32)(cid:90) | x − y |≥ (( n − j +1) h ) / (cid:18) ( n − j + 1) h | ˜ x | N +3 (cid:19) p dx (cid:33) /p ≤ C ( δnh ) / (cid:32)(cid:90) | x − y |≥ (( n − j +1) h ) / (cid:18) nh | x − y | N +3 (cid:19) p dx (cid:33) /p = C p δ / ( nh ) / (( n − j + 1) h ) N/ (2 p ) − ( N +3) / ≤ C p δ / ( nh ) − N/ − /p ) . Consequently, ( nh ) N/ − /p ) hI ≤ C p hδ / . For I , by (5.5) note that the set of integration is contained in { x ∈ R N : 1 ≥ | x − y | n − j +1) h ) / >δ / } . Then from Theorem 3.1 (ii) it follows I ≤ C ( δnh ) / (cid:90) δ / ≤ | x − y | (( n − j +1) h )1 / ≤ (cid:18) | ˜ x | (( n − j + 1) h ) N/ (cid:19) p dx /p ≤ C δ / ( nh ) / ( nh ) N/ (cid:90) δ / ≤ | x − y | (( n − j +1) h )1 / ≤ | x − y | p dx /p ≤ C p δ / ( nh ) − N/ − /p ) (1 − δ ( N + p ) / ) /p SYMPTOTIC BEHAVIOR FOR THE DISCRETE IN TIME HEAT EQUATION 19 ≤ C p δ / ( nh ) − N/ − /p ) , which is equivalent to ( nh ) N/ − /p ) hI ≤ C p hδ / . Next, let us estimate I . From discrete mean value theorem (see [3, Corollary 2]), there exist˜ n ∈ { n − j + 2 , ..., n } (whenever j ≥
2) and
C > I ≤ C ( j − h (cid:18)(cid:90) B |G ˜ n,h ( x ) − G ˜ n − ,h ( x ) | dx (cid:19) p = C ( j − h (cid:18)(cid:90) B | ∆ G ˜ n,h ( x ) | dx (cid:19) p . Recall that in Ω we have n − j +1 ≤ ˜ n ≤ n, which implies by (5.2) that nh (1 − δ ) ≤ ˜ nh ≤ nh. Also, in Ω and B we have | x | = | x + y − y | ≥ | x − y | − | y | ≥ ( δnh ) / , so ˜ z := | x | (˜ nh ) / ≥ | x | ( nh ) / ≥ ( δnh ) / ( nh ) / = δ / , and we have again two cases. We denote by I and I depending on whether ˜ z ≤ z ≥ I , since ˜ z ≤ | x | ≥ ( δnh ) / the set of integration is contained in { x ∈ R N :( δnh ) / ≤ | x | ≤ ( nh ) / } . Then, from Theorem 3.1 (iii) and the fact that we are in Ω , wehave I ≤ C ( j − h (cid:18)(cid:90) ( δnh ) / ≤| x |≤ ( nh ) / nh ) ( N/ p dx (cid:19) /p ≤ Cδnh ( nh (1 − δ )) N/ (cid:18)(cid:90) ( δnh ) / ≤| x |≤ ( nh ) / dx (cid:19) /p = C p δ (1 − δ N/ ) /p ( nh ) − N/ − /p ) (1 − δ ) N/ . Consequently, ( nh ) N/ − /p ) hI ≤ C p hδ (1 − δ ) N/ . For I we have 1 ≤ ˜ z = | x | (˜ nh ) / ≤ | x | ( nh (1 − δ )) / , which implies that the set of integration is contained in { x ∈ R N : | x | ≥ ( nh (1 − δ )) / } . Then I ≤ C ( j − h (cid:18)(cid:90) | x |≥ ((1 − δ ) nh ) / | x | ( N +2) p dx (cid:19) /p = C p δ ( nh ) − N/ − /p ) (1 − δ ) N/ − /p )+1 . Consequently, ( nh ) / − /p ) hI ≤ C p hδ (1 − δ ) N/ − /p )+1 . Collecting all above terms over B we get( nh ) N/ − /p ) h (cid:88) ( j,y ) ∈ Ω (cid:90) (cid:18)(cid:90) B |G n − j +1 ,h ( x − y ) − G n,h ( x ) | p dx (cid:19) /p | g ( jh, y ) | dy ≤ C p δ η n (cid:88) j =1 (cid:90) R N | g ( jh, y ) | dy for some positive number η . The upper bound tends to zero as δ → nh .Now, we consider the set Ω . Then( nh ) N/ − /p ) h (cid:88) ( j,y ) ∈ Ω (cid:90) (cid:107)G n − j +1 ,h ( · − y ) − G n,h ( · ) (cid:107) p | g ( jh, y ) | dy ≤ ( nh ) N/ − /p ) h (cid:88) ( j,y ) ∈ Ω (cid:90) (cid:107)G n − j +1 ,h ( · − y ) (cid:107) p | g ( jh, y ) | dy + ( nh ) N/ − /p ) h (cid:88) ( j,y ) ∈ Ω (cid:90) (cid:107)G n,h ( · ) (cid:107) p | g ( jh, y ) | dy =: I + I . By Theorem 3.2 (i) one gets I ≤ C p (cid:88) ( j,y ) ∈ Ω (cid:90) | g ( jh, y ) | dy. As n → ∞ , Ω → N × R N . This implies that Ω has measure zero, and since (cid:80) ∞ j =1 (cid:82) R N | g ( jh, y ) | dy < ∞ , then (cid:80) ( j,y ) ∈ Ω (cid:82) | g ( jh, y ) | dy → n → ∞ . It follows that I → n → ∞ .For I we have two possibilities: either j ≤ (cid:100) δn (cid:101) or j > (cid:100) δn (cid:101) . Thus, we divideΩ = { , ..., (cid:100) δn (cid:101)} × { y ∈ R : | y | > ( δnh ) / } ∪ {(cid:100) δn (cid:101) + 1 , ..., n } × R N . Then I ≤ ( nh ) N/ − /p ) h (cid:100) δn (cid:101) (cid:88) j =1 (cid:90) | y | > ( δnh ) / (cid:107)G n − j +1 ,h ( · − y ) (cid:107) p | g ( jh, y ) | dy + ( nh ) N/ − /p ) h n (cid:88) j = (cid:100) δn (cid:101) +1 (cid:90) R N (cid:107)G n − j +1 ,h ( · − y ) (cid:107) p | g ( jh, y ) | dy =: I + I . SYMPTOTIC BEHAVIOR FOR THE DISCRETE IN TIME HEAT EQUATION 21
Let us start with I . Recall that for j ∈ { , ..., (cid:100) δn (cid:101)} the expression (5.2) holds. So, byTheorem 3.2 (i) we have that I ≤ C p h (cid:100) δn (cid:101) (cid:88) j =1 (cid:90) | y | > ( δnh ) / | g ( jh, y ) | dy → n → ∞ . Next, for I , again by Theorem 3.2 (i), (5.2) and the fact that γ > I ≤ C p ( nh ) N/ − /p ) h n (cid:88) j = (cid:100) δn (cid:101) +1 n − j + 1) h ) N (1 − /p ) j γ . Thus, if N (1 − /p ) ∈ [0 ,
1) and γ > , then I ≤ C p ( nh ) N/ − /p ) h ( (cid:100) δn (cid:101) + 1) γ h γ n −(cid:100) δn (cid:101) (cid:88) j =1 jh ) N (1 − /p ) ≤ C p h ( nh ) N/ − /p ) ( nh − (cid:100) δn (cid:101) h ) − N/ − /p ) ( δnh ) γ → , as n → ∞ . Also, if γ > N (1 − /p ) >
1, then I ≤ C p ( nh ) N/ − /p ) h ( (cid:100) δn (cid:101) + 1) γ h γ n −(cid:100) δn (cid:101) (cid:88) j =1 jh ) N (1 − /p ) ≤ C p h ( nh ) N/ − /p ) ( δnh ) γ → , n → ∞ . The case γ > N (1 − /p ) = 1 implies, similarly to the previous one, that I ≤ C p h ( nh ) log( n )( δnh ) γ → , n → ∞ . (cid:3) Optimal L -decay for solutions In this section we prove that the decay rate of the solution u c of (2.1) given in Theorem 4.1(i) is optimal. Theorem 6.1.
Let u c be the solution of (2.1) . Assume that f ∈ L ( R N ) ∩ L ( R N ) and (cid:82) R N f ( x ) dx (cid:54) = 0 . Then (cid:107) u c ( nh, · ) (cid:107) ∼ C ( nh ) N/ , nh ≥ . Proof.
Let ρ > , we have by Proposition 2.4 (iii) that (cid:107) u c ( nh, · ) (cid:107) = (cid:107)F u c ( n, · ) (cid:107) = (cid:90) R N |F G n,h ( ξ ) | |F f ( ξ ) | d ξ ≥ (cid:90) B (0 ,ρ ) h | ξ | ) n |F f ( ξ ) | d ξ ≥ h | ρ | ) n (cid:90) B (0 ,ρ ) |F f ( ξ ) | d ξ. (6.1) By Plancherel Theorem and the Riemann-Lebesgue Lemma we have that F f ∈ C ( R N ) ∩ L ( R N ). By the Lebesgue differentiation theorem, we may choose ρ small enough such that1 ρ N (cid:90) B (0 ,ρ ) |F f ( ξ ) | d ξ ≥ |F (0) | for all ρ ∈ (0 , ρ ] . Substituting the previous inequality in (6.1) we have that for all ρ ∈ (0 , ρ ] (cid:107) u c ( nh, · ) (cid:107) ≥ ρ N h | ρ | ) n |F (0) | . We choose ρ := ρ ( nh ) / . For n enough large, nh ≥ ρ belongs to (0 , ρ ). Hence ρ N (1 + h | ρ | ) n = ρ N ( nh ) N/ (cid:18) ρ n (cid:19) n ≥ ρ N ( nh ) N/ e ρ = C ( nh ) N/ , nh ≥ , and then we get the first assertion of the result.Next, let us prove the upper bound. By Plancherel’s Theorem and the Riemann-LebesgueLemma we have (cid:107) u c ( nh, · ) (cid:107) = (cid:90) R N h | ξ | ) n |F f ( ξ ) | d ξ ≤ (cid:107)F f (cid:107) ∞ (cid:90) R N h | ξ | ) n d ξ ≤ (cid:107) f (cid:107) (cid:90) R N nh | ξ | ) d ξ = C (cid:107) f (cid:107) ( nh ) N/ (cid:90) R N | ξ | ) d ξ = C ( nh ) N/ . (cid:3) Appendix
Here, we present some useful facts which are needed in order to obtain our results.First, we recall the following asymptotic behavior of the Gamma function. Let α, z ∈ C ,then(7.1) Γ( z + α )Γ( z ) = z α (cid:18) α ( α + 1)2 z + O ( | z | − ) (cid:19) , | z | → ∞ , whenever z (cid:54) = 0 , − , − , . . . , and z (cid:54) = − α, − α − , . . . , see [13].Next, we recall the definition of Bessel functions and some basic results which are used inthis work. See [17, 24, 28] for more information about this topic. SYMPTOTIC BEHAVIOR FOR THE DISCRETE IN TIME HEAT EQUATION 23
Let ν ∈ R . The Modified Bessel functions of the first kind are defined by I ν ( x ) = ∞ (cid:88) n =0 n + ν + 1) n ! (cid:16) x (cid:17) n + ν . Such functions allow to define, for ν ∈ R a non entire number, the Modified Bessel functionsof second kind or MacDonald’s functions as follows K ν ( x ) = π I ν ( x ) − I − ν ( x )sin( νx ) . For the case µ ∈ Z they are defined by K µ ( x ) = lim ν → µ K ν ( x ) = lim ν → µ π I ν ( x ) − I − ν ( x )sin( νx ) . These functions arise as the solutions for the ODE d dz u ( z ) = (cid:18) ν z (cid:19) u ( z ) − z ddz u ( z ) . Some properties of the MacDonald’s functions used along the paper are the following ones:(P1) K ν ( z ) = (cid:82) ∞ e − z cosh t cosh( νt ) dt, | arg ( z ) | < π or (cid:60) z = 0 and ν = 0.(P2) z ddz K ν ( z ) + νK ν ( z ) = − zK ν − ( z ).(P3) z ddz K ν ( z ) − νK ν ( z ) = − zK ν +1 ( z ).(P4) When 0 < z (cid:28) √ ν + 1, we have K ν ( z ) ∼ Γ( ν )2 (cid:18) z (cid:19) ν , if ν (cid:54) = 0 . (P5) K ν ( z ) = (cid:18) π z (cid:19) / e − z (cid:18) O (1 /z ) (cid:19) , z → ∞ . (P6) K ν = K − ν . (P7) zK ν − ( z ) − zK ν +1 ( z ) = − νKν ( z ) . We also need in this paper the following decomposition lemma (see [10]).
Lemma 7.1.
Suppose f ∈ L ( R N ) such that (cid:82) R N | x || f ( x ) | dx < ∞ . Then there exists F ∈ L ( R N ; R N ) such that f = (cid:18)(cid:90) R N f ( x ) dx (cid:19) δ + div F in the distributional sense and (cid:107) F (cid:107) L ( R N ; R N ) ≤ C d (cid:90) R N | x || f ( x ) | dx. Acknowledgments.
The authors would like to thank to Jorge Gonz´alez-Camus by his helpand advice with the pictures along the paper.
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Departamento de Matem´aticas, Instituto Universitario de Matem´aticas y Apli-caciones, Universidad de Zaragoza, 50009 Zaragoza, Spain.
Email address : [email protected] (E. Alvarez) Universidad del Norte, Departamento de Matem´aticas y Estad´ıstica, Barran-quilla, Colombia
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