Variational characterization of the speed of reaction diffusion fronts for gradient dependent diffusion
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Variational characterization of the speed of reactiondiffusion fronts for gradient dependent diffusion
Rafael D. Benguria · M. Cristina Depassier
Received: date / Accepted: date
Abstract
We study the asymptotic speed of travelling fronts of the scalarreaction diffusion for positive reaction terms and with a diffusion coefficientdepending nonlinearly on the concentration and on its gradient. We restrictour study to diffusion coefficients of the form D ( u, u x ) = mu m − u m ( p − x forwhich existence and convergence to travelling fronts has been established. Weformulate a variational principle for the asymptotic speed of the fronts. Upperand lower bounds for the speed valid for any m ≥ , p ≥ m = 1 , p = 2 the problem reduces to the constant diffusion problemand the bounds correspond to the classic Zeldovich–Frank–Kamenetskii lowerbound and the Aronson-Weinberger upper bound respectively. In the specialcase m ( p −
1) = 1 a local lower bound can be constructed which coincideswith the aforementioned upper bound. The speed in this case is completelydetermined in agreement with recent results.
Keywords
Variational principles · reaction–diffusion equation · gradientdependent diffusion · p–Laplacian Mathematics Subject Classification (2010)
MSC 35 K 57 · MSC 35 K65 · MSC 35 C 07 · MSC 35 K 55 · MSC 58 E 30
R. D. BenguriaInstituto de F´ısica, Pontificia Universidad Cat´olica de Chile, Casilla 306, Santiago 22, ChileE-mail: [email protected]. C. DepassierInstituto de F´ısica, Pontificia Universidad Cat´olica de Chile, Casilla 306, Santiago 22, ChileE-mail: [email protected] Rafael D. Benguria, M. Cristina Depassier
In this work we study the asymptotic propagation of fronts of the scalar reac-tion diffusion equation, ∂ t u = ∂ x ( | ∂ x u m | p − ∂ x u m )+ f ( u ) , f (0) = f (1) = 0 , f ( u ) > , , (1)which reduces to the classical problem [13] when m = 1 , p = 2. The diffusionterm can be seen either as the scalar version of the p-Laplacian acting on u m oras reaction diffusion equation with nonlinear diffusion coefficient D ( u, u x ) = mu m − | u x | m ( p − . Such diffusion coefficients are encountered, for example, inhot plasmas [12,14] and the corresponding processes are referred to as doublynonlinear diffusion processes [4].The classical problem m = 1, p = 2, is fully understood [1,13]. When non-linear diffusion is included several scenarios may arise depending on the preciseform of the diffusion coefficient. The case of a power of concentration diffusioncoefficient of the form D ( u ) = u s has been studied extensively beginning withthe analytical solution found for s = 2. Existence and convergence results areknown for all s . A distinctive feature of density dependent diffusion is theappearance of a finite wave at the asymptotic speed. This is true even in thesimpler p = 2 case when m > m > , p > γ = m ( p − − >
0, a uniquemonotonic increasing travelling wave joining the equilibria u = 0 and u = 1exists for speeds c ≥ c ∗ ( m, p ) and none if 0 < c < c ∗ ( m, p ). For c = c ∗ ( m, p )the travelling wave (TW) is finite, whereas for c > c ∗ ( m, p ) the TW is positive(see [4], Theorem 2.1). In the case γ = 0 a unique monotonic increasing travel-ling wave joining the equilibria u = 0 and u = 1 exists for speeds c ≥ c ∗ ( m, p )and none if 0 < c < c ∗ ( m, p ). For c = c ∗ ( m, p ) the travelling wave (TW) ispositive (see [4], Theorem 2.2). Moreover, when γ = 0, an explicit expressionfor the minimal speed is given, c ∗ ( m, p ) | γ =0 ≡ c ( m, p ) = ( m p m +1 f ′ (0)) / ( m +1) . (2)The convergence of suitable initial conditions to the travelling wave of minimalspeed is demonstrated in [4] as well.The purpose of this work is to establish a variational characterization forthe speed c ∗ ( m, p ). The exact value of the speed cannot be determined ingeneral however upper and lower bounds on the speed for general values of m and p can be obtained. The main result of the present work (see Theorem 1below) is the variational expression for the speed c ∗ = sup g " p (cid:18) mp − (cid:19) ( p − /p R u ( m − p − /p h /p f ( p − /p g ( p − /p du R g ( u ) du , (3)from where upper and lower bounds will be constructed. In (3), g ∈ C (0 , g ( u ) ≥
0, with h ( u ) ≡ − g ′ ( u ) > ,
1) and R g ( u ) du finite. ariational Characterization for the speed of fronts 3 We find that for any m > , p > γ ≥
0) the asymptotic speed isbounded by (cid:16) mpp − R u ( m − f ( u ) du (cid:17) ( p − /p ≤ c ∗ ( m, p ) ≤ p (cid:16) mp − (cid:17) ( p − /p sup u (cid:20) u γ (cid:16) fu (cid:17) ( p − (cid:21) /p . (4)The lower bound is a generalization of the Zeldovich-Frank-Kamenetskii (ZFK)bound (see, e.g., [10,9]). Effectively, for m = 1 , p = 2 their classical bound c ∗ ≥ c ZF K = s Z f ( u ) du is recovered. The upper bound, for m = 1 , p = 2 reduces to the Aronson-Weinberger upper bound [1] c ≤ sup u r f ( u ) u . An interesting case arises when γ = 0. As mentioned above the speed can bedetermined exactly [4] and it is given by c ( m, p ) when γ = 0. Here we recoverthis result from the variational principle showing that when γ = 0 a locallower bound can be found choosing an adequate trial function g ( u ). This lowerbound is exactly c ( m, p ). The upper bound given in (4) reduces to c ( m, p )when γ = 0 and f ( u ) satisfies the KPP criterion sup u p f ( u ) /u = f ′ (0) . Inthe following sections we prove the statements made above. Our variationalprinciple reduces to our standard variational principle (see [7], [8], [9]) when p = 2.The rest of this manuscript is organized as follows: In Section 2 we derivethe variational principle, in Section 3 the bounds for general values of γ with m > p > γ ≥ We consider left travelling wave solutions u ( ξ ) with ξ = x + ct so that theTW profile satisfies u ξ > . The TW solution satisfies the ordinary differentialequation (ODE) cu ξ = m p − ddξ (cid:16) u ( m − p − ( u ξ ) p − (cid:17) + f ( u ) . (5)From here on we denote u ′ = u ξ . Following the usual procedure, we introducethe phase space coordinate q ( u ) = u m − u ′ ( u ) Rafael D. Benguria, M. Cristina Depassier in terms of which the ODE for the travelling waves becomes, after dividing by q, cm p − = ddu ( q ( u )) p − + u m − f ( u ) m p − q ( u ) . (6)Here, it is convenient to define F ( u ) = u m − f ( u ) m p − . (7)In what follows, let us define the functional J [ g ] ≡ p m p − ( p − ( p − /p R h ( u ) /p F ( u ) ( p − /p g ( p − /p du R g ( u ) du , (8)which acts on D , the space of functions g ∈ C (0 ,
1) such that g ( u ) ≥ h ( u ) = − g ′ ( u ) > ,
1) and R g ( u ) du finite. Here the function F ( u )is given by (7) above. With this notation we state our main result, which isembodied in the following theorem. Theorem 1 (Variational characterization of c ∗ ) Let f ∈ C [0 , with f (0) = f (1) = 0 , f ( u ) > in (0 , , and f ( u ) concave in [0 , . Assume γ = m ( p − − ≥ . Then, c ∗ ( m, p ) = J ≡ sup {J [ g ] (cid:12)(cid:12) g ∈ D} (9) Moreover,i) If γ > , there is a g ∈ D , ˜ g say, such that J = J [˜ g ] . This maximizing ˜ g isunique up to a multiplicative constant, andii) If γ = 0 we construct and explicit maximizing sequence g α ∈ D such that lim α → J [ g α ] = c ∗ ( m, p ) | γ =0 , where c ∗ ( m, p ) | γ =0 is given by (2) above.Proof Let g ( u ) ∈ D . Multiplying (6) by g ( u ) and integrating in u between 0and 1 we obtain after integrating by parts, cm p − Z g ( u ) du = Z du (cid:18) h ( u ) q ( u ) p − + g ( u ) F ( u ) q ( u ) (cid:19) ≡ Z Φ ( u ) du. (10)where h ( u ) ≡ − g ′ ( u ) > g ( u ) is such thatlim u → g ( u ) q ( u ) p − = 0.The integrand of the right side, Φ = h ( u ) q ( u ) p − + g ( u ) F ( u ) q ( u ) , (11)at fixed u can be considered as a function of q . It is clear from (11) that Φ ( q )has a unique positive minimum at ˆ q so that Φ ( q ) ≥ Φ (ˆ q ). A simple calculationyields ˆ q = (cid:20) F g ( p − h (cid:21) /p , (12) ariational Characterization for the speed of fronts 5 and Φ (ˆ q ) = pgh /p F ( p − /p ( p − ( p − /p . It follows from (10) that c ∗ ≥ p m p − ( p − ( p − /p R h /p F ( p − /p g ( p − /p du R g ( u ) du , (13)for every g ∈ D . To establish (9) we need only prove that the supremum ofthe right side of (13) over all g ∈ D is actually c ∗ . we will do this separatelyin the cases γ > γ = 0. i) Case γ > . Below we show that when ˆ q is the solution of (6) (with c = c ∗ ),equality is attained in (13) for some g ∈ D , so that we obtain the variationalcharacterization for the speed c ∗ = sup g p m p − ( p − ( p − /p R h /p F ( p − /p g ( p − /p du R g ( u ) du . (14)We have already proven (see (13) above) that c ∗ ≥ J [ g ] for every g ∈ D . Whatwe will actually show here is that when γ >
0, there exists a g ∈ D , ˜ g say,such that c ∗ = J [˜ g ]. Hence in the case γ > c ∗ = max g ∈D ( J [ g ]) . (15)In the case γ >
0, the existence of a travelling wave for any c ≥ c ∗ was provenin Theorem 2.1 of Reference [4]. Moreover, in the case γ >
0, the solution of(6) satisfies, q ( u ) p − ≈ c ∗ m p − u, (16)in the neighborhood of u = 0. In order to show that the sup is actuallyattained in (9) we have to show that there exists ˜ g ∈ D satisfying (12) whenˆ q is a solution of (6). To construct such a g , let v be the solution of v ′ v = c ∗ m p − q p − ( u ) , (17)where q is a solution of (6). Notice that this v is unique up to a multiplicativeconstant. A simple calculation using (17), (6), and the definition (7) of F ,yields, v ′′ v = c ∗ m p − F ( u ) q p − ( u ) . (18)Choosing ˜ g ( u ) = 1( v ′ ( u )) / ( p − , (19) Rafael D. Benguria, M. Cristina Depassier it follows from (17) and (18) that, − ˜ g ′ ( u ) = p − v ′ ( u )) p/ ( p − v ′′ = p − v ′ ( u )) / ( p − v ′′ v vv ′ = p − g ( u ) F ( u ) q p ( u ) . (20)which is precisely (12). From (16) and (17) we have that v ( u ) ≈ A u and v ′ ( u ) ≈ A, (21)near u = 0. Hence, it follows from (19) that˜ g (0) = A − / ( p − < ∞ . (22)Integrating (17) and using (21) we can write explicitly, v ( u ) = exp (cid:18)Z uu c ∗ m p − q p − ( s ) ds (cid:19) , (23)for some 0 < u <
1. Clearly, the value of A in (21) is determined by the valueof u . Finally, using (17), (19), and (23), we can write,˜ g ( u ) = m q ( u ) c ∗ / ( p − exp (cid:18) p − Z u u c ∗ m p − q p − ( s ) ds (cid:19) . (24)Since, the integrand in (24) is positive, u <
1, and q (1) = 0, it follows from(24) that ˜ g (1) = 0. From all the results above it follows that ˜ g given by (24)is in D , and that c ∗ = J [˜ g ].It is clear from the construction above that ˜ g is unique up to a multi-plicative cosntant. The uniqueness of the maximizing g ∈ D , however, can beseen directly from our variational principle (8). In fact, suppose that there aretwo different maximizers, say g , g ∈ D , with R g ( u ) du = R g ( u ) du = 1.Then, for any α ∈ (0 ,
1) consider now, g α ( u ) = α g ( u ) + (1 − α ) g ( u ) . (25)It is clear from (25) that g α ∈ D and that R g α ( u ) du = 1. Using H¨older’sinequality with exponents p and p ′ = p/ ( p − J [ g α ] > α J [ g ] + (1 − α ) J [ g ] = c ∗ , (26)which is a contradiction with the fact that g and g are the maximizers.Notice that the inequality in (26) is strict if g g . ii) Case γ = 0 . For later purposes it is convenient to denote J g [ f ] = Z [ u m − h ( u ) m f ( u ) g ( u )] / ( m +1) du. (27) ariational Characterization for the speed of fronts 7 It then follows from (7) and (8) that J [ g ] = p m / ( m +1) J g [ f ] (28)in the case γ = 0, when we conveniently normalize g so that R g ( u ) du = 1.Now, choose as a trial function the sequence g α ( u ) = α − α ( u α − − , < α < , with α → . (29)Notice that for each α ∈ (0 , g α ( u ) > g ′ α ( u ) < g α (1) = 0, andlim u → [ u g α ( u )] = 0, so these are appropriate trial functions. Moreover, wehave normalized the g α ’s so that R g α ( u ) du = 1.With this choice we will show that lim α → J g α [ f ] = f ′ (0) / ( m +1) so that J [ g α ] → ( m p m +1 f ′ (0)) / ( m +1) = c ( m, p ) as α →
0. To do so we write J [ f ] = J [ uf ′ (0)] + J [ f ] − J [ uf ′ (0)] and show that J g α [ uf ′ (0)] → f ′ (0) / ( m +1) , J g α [ f ] − J g α [ uf ′ (0)] → , as α → . (30)While the proof of the second limit is given in the Appendix, the proof of thefirst is as follows. Using (27) with g = g α we have, J g α [ uf ′ (0)] = f ′ (0) / ( m +1) α (1 − α ) − / ( m +1) R ( u m ( α − ( u α − − / ( m +1) du = f ′ (0) / ( m +1) α (1 − α ) − ( m +2) / ( m +1) B (cid:16) m +2 m , α − α (cid:17) , (31)where B ( x, y ) denotes the Euler Beta function. Now, B ( t, s ) = Γ ( t ) Γ ( s ) /Γ ( t + s ), hence J g α [ uf ′ (0)] = f ′ (0) / ( m +1) α (1 − α ) − ( m +2) / ( m +1) Γ ( m +2 m ) Γ ( α − α ) Γ ( m +2 m + α − α ) (32)Using lim x → x Γ ( x ) = 1 to evaluate the limit of the right side of (32) when α →
0, we finally conclude, J g α [ uf ′ (0)] → f ′ (0) / ( m +1) as α → J [ g α ] → ( m p m +1 f ′ (0)) / ( m +1) = c ( m, p ) as α →
0, which concludes the proof of the Theorem. γ ≥ In this section we derive from our variational principle (i.e., from Theorem 1above) an explicit upper bound on the speed of fronts. In order to do this werewrite (14) as c ∗ = sup g " p m p − ( p − ( p − /p R [ hF ( p − /g ] /p gdu R g ( u ) du . (33) Rafael D. Benguria, M. Cristina Depassier
Since the mapping x → x /p is concave for p >
1, defining the probabilitymeasure dν = g ( u ) du/ R gdu , and using Jensen’s inequality we get, c ∗ ≤ sup g p m p − ( p − ( p − /p (cid:20) R hF ( p − du R g ( u ) du (cid:21) /p ≤ p m p − ( p − ( p − /p (cid:20) sup u (cid:16) F p − u (cid:17) sup g R h ( u ) u du R g ( u ) du (cid:21) /p . (34)Integrating R h ( u ) u du by parts, using g (1) = 0, and lim u → ug ( u ) = 0 itfollows from (34) that c ∗ ≤ p m p − ( p − ( p − /p (cid:20) sup u F p − u (cid:21) /p . Replacing the expression for F ( u ) in terms of f ( u ) we finally obtain the upperbound c ∗ ≤ p (cid:18) mp − (cid:19) ( p − /p sup u " u γ (cid:18) f ( u ) u (cid:19) ( p − /p (35)with γ = m ( p − − γ = 0 the expression abovereduces to c ∗ | γ =0 ≤ p ( m ) / ( m +1) sup u (cid:18) f ( u ) u (cid:19) / ( m +1) . (36)In particular, when m = 1 (i.e., p = 2 since γ = 0), (36) is the classical upperbound of Aronson and Weinberger [1]. Notice that for the reaction profilesconsidered here (i.e., f ( u ) positive and concave in [0 , f (0) = f (1) = 0 and f ∈ C [0 , u ∈ [0 , f ( u ) /u = f ′ (0), and in fact wehave equality in (36). From the variational characterization lower bounds can be constructed choos-ing specific values for the trial function g ( u ). In this section we construct alower bound which involves the integrals of the reaction term as the Zeldovich–Frank–Kamenetskii classical bound [15,10,9]. Our ZFK type bound is embod-ied in the following lemma. Lemma 1
For any m > , p > , γ ≥ and f satisfying the hypothesis ofTheorem 1, we have that c ∗ ≥ (cid:18) mpp − (cid:19) ( p − /p (cid:20)Z u m − f ( u ) du (cid:21) ( p − /p . (37) ariational Characterization for the speed of fronts 9 Proof
Choose as a trial function of our variational principle (9) the function g ( u ) = (cid:18)Z u F ( u ′ ) du ′ (cid:19) /p . It is simple to verify that g ∈ C (0 , h = − g ′ > g (1) = 0, and g ( u ) ≥ , g is decreasing, R g ( u ) du ≤ g (0) = (cid:16)R F ( u ′ ) du ′ (cid:17) /p < ∞ . Hence, g ∈ D . A simple calculation yields h ( u ) = F ( u ) p (cid:18)Z u F ( u ′ ) du ′ (cid:19) /p − and hg p − = F ( u ) /p . It follows then from (14) that c ∗ ≥ p m p − ( p − ( p − /p (cid:18) p (cid:19) /p R F ( u ) du R g ( u ) du . Now, since g ( u ) is a decreasing positive function, R g ( u ) du ≤ g (0). Hence, c ∗ ≥ p m p − ( p − ( p − /p (cid:18) p (cid:19) /p (cid:18)Z F ( u ) du (cid:19) ( p − /p . (38)If we express the right side of (38) in terms of the original reaction term f ( u )we get (37) which proves the lemma. Acknowledgements
This work was supported by Fondecyt (Chile) projects 114–1155,116–0856 and by Iniciativa Cient´ıfica Milenio, ICM (Chile), through the Millennium NucleusRC–120002.
Appendix
In this appendix we show that J g α [ f ] − J g α [ uf ′ (0)] → α → , (39)where g α is given by (29). We defined J g α [ f ] = Z [ u m − h mα f ( u ) g α ( u )] / ( m +1) du, so that | J g α [ f ] − J g α [ uf ′ (0)] | ≤ R ( u m − h mα g α ( u )) / ( m +1) | f ( u ) / ( m +1) − ( uf ′ (0)) / ( m +1) | du. (40) Since for m ≥
0, 1 / ( m + 1) ≤
1, it is not difficult to verify the inequality | a / ( m +1) − b / ( m +1) | ≤ | a − b | / ( m +1) for all a ≥ , b ≥ , m ≥ . In thepresent case, we have | f ( u ) / ( m +1) − ( uf ′ (0)) / ( m +1) | ≤ | f ( u ) − uf ′ (0) | / ( m +1) . (41)If f ( u ) and its derivative are continuous in [0 , d > , k > | f ( u ) − uf ′ (0) | u < d u k . (42)Using (41) and (42) in (40), together with the explicit form of g α we have that | J g α [ f ] − J g α [ uf ′ (0)] | ≤ α (1 − α ) / ( m +1) Z d / ( m +1) u N ( α ) du, where N ( α ) = α − k ( m + 1) . Since α > k > N ( α ) > − u N ( α ) is integrable. Performing theintegral we finally find | J g α [ f ] − J g α [ uf ′ (0)] | ≤ m + 1 α ( m + 1) + k αd / ( m +1) (1 − α ) / ( m +1) → α → . References
1. D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising inpopulation genetics , Adv. Math., , 33–76 (1978).2. D. G. Aronson, Density–dependent interaction–diffusion systems . In: Proc. Adv. Seminaron Dynamics and Modeling of Reactive System, W. Stewart et al. (eds.), Academic Press,New York, 1161–1176 (1980).3. A. Audrito
Bistable and monostable reaction equations with doubly nonlinear diffusion ,arXiv:1707.01240v1(2017).4. A. Audrito and J. L. V´azquez
The Fisher-KPP problem with doubly nonlinear diffusion ,arXiv:1601.05718 (2016).5. R. D. Benguria and M. C. Depassier,
A variational principle for the asymptotic speedof fronts of the density dependent diffusion–reaction equation , Physical Review E, ,3285–3287 (1995).6. R. D. Benguria, J. Cisternas and M. C. Depassier, Variational calculations for thermalcombustion waves
Phys. Rev. E, , 4410–4413 (1995).7. R. D. Benguria and M. C. Depassier, Variational characterization of the speed of propa-gation of fronts for the nonlinear diffusion equation , Commun. Math. Phys. , 221–227(1996).8. R. D. Benguria and M. C. Depassier,
Speed of fronts of the reaction–diffusion equation ,Phys. Rev. Lett., , 1171–1173 (1996).9. R. D. Benguria and M. C. Depassier, A Variational method for nonlinear eigenvalueproblems , in Advances in Differential Equations and Mathematical Physics, Eric Carlen,Evans Harrell, and Michael Loss, Eds., Amer. Math. Soc., Contemporary Mathematics, , 1–17 (1998).10. H. Berestycki and L. Nirenberg,
Travelling fronts in cylinders , Ann. Inst. HenriPoincar´e , Analyse non lin´eaire, , 497–572 (1992).ariational Characterization for the speed of fronts 1111. A. Gavioli, L. Sanchez, A variational property of critical speed to travelling waves inthe presence of nonlinear diffusion , Applied Mathematics Letters, , 47–54 (2015).12. S. C. Jardin, G. Bateman, G. W. Hammett and L. P. Ku, On 1–d diffusion problemswith a gradient–dependent diffusion coefficient , J. Comp. Phy., , 8769–8775 (2008).13. A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov,
A study of the diffusion equa-tion with increase in the amount of substance, and its applications to a biological problem ,in Selected Works of A. N. Kolmogorov, V. M. Tikhomirov (Ed.), Kluwer Academic Pub-lishers, Dordrecht, The Netherlands (1991).14. H. Wilhelmsson and E. Lazzaro,
Reaction-diffusion problems in the physics of hot plas-mas . IOP Publ., Bristol and Philadelphia (2001).15. Y. B. Zeldovich and D. A. Frank-Kamenetskii,