In this article we are concerned with the existence of traveling wave solutions of a general class of nonlocal wave equations: utt - a^2uxx = (beta* u^p)xx, p > 1. Members of the class arise as mathematical models for the propagation of waves in a wide variety of situations. We assume that the kernel beta is a bell-shaped function satisfying some mild differentiability and growth conditions. Taking advantage of growth properties of bell-shaped functions, we give a simple proof for the existence of bell-shaped traveling wave solutions

Erkip, Albert

Ramadan, Abba Ibrahim

Sabanci University Research Database

Existence of Traveling Waves for a Class of NonlocalNonlinear Equations with Bell Shaped KernelsA. Erkip∗, A. I. RamadanFaculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, TurkeyAbstractIn this article we are concerned with the existence of traveling wave solutions of a generalclass of nonlocal wave equations: utt − a2uxx = (β ∗ up)xx, p > 1. Members of theclass arise as mathematical models for the propagation of waves in a wide variety ofsituations. We assume that the kernel β is a bell-shaped function satisfying some milddifferentiability and growth conditions. Taking advantage of growth properties of bell-shaped functions, we give a simple proof for the existence of bell-shaped traveling wavesolutions.Keywords: Solitary waves, Bell-shaped functions, Nonlocal wave equations, Variationalmethods.2000 MSC: 74H20, 74J30, 35Q51, 35A151. IntroductionThe present paper is concerned with the existence of traveling wave solutions u(x, t) =u(x− ct) of a general class of nonlocal nonlinear wave equations of the formutt − a2uxx = (β ∗ up)xx, x ∈ R, t > 0 (1.1)where a is some constant, c 6= 0 is the wave velocity, p > 1 is an integer, and(β ∗ v)(x) =∫Rβ(x− y)v(y)dy∗Corresponding author. Tel: +90 216 483 9508 Fax: +90 216 483 9550Email addresses: albert@sabanciuniv.edu (A. Erkip ), aramadan@sabanciuniv.edu (A. I.Ramadan )Preprint submitted to Elsevier December 22, 2016denotes the convolution in the x variable for a given bell-shaped kernel function β. Asthe name suggests, a function is bell-shaped if it is even, non-negative, and nonincreasingon [0,∞).The problem (1.1) is closely related to the well known and much investigated questionof solitary wave pulses propagating in an infinite chain of beads. The equation governingpropagation of wave pulses turns out to be of the formr′′n = V′(rn+1)− 2V ′(rn) + V ′(rn−1), n ∈ Zwhere rn(t) denotes the distance between the centers of the (n− 1)-th and n-th balls. Inthe continuum limit, for traveling waves, one obtains the stationary differential-differenceequationr′′ = ∆discV ′(r), (1.2)with the discrete Laplacian ∆discw(x) = w(x+ 1)− 2w(x) +w(x− 1). Letting Λ be thetriangular kernel defined as Λ(x) = (1 − |x|)χ[−1,1](x) with the characteristic functionχ[−1,1], it then follows that (1.2) can be written asr′′ = (Λ ∗ V ′(r))′′.We note that when V (r) = 1p+1rp+1 this is the equation for a traveling wave solution of(1.1) for the particular kernel Λ. In this sense, our problem generalizes the chain problem(1.2) to bell shaped kernels.Equation (1.1) is a particular case of the nonlocal double dispersive equationutt − Luxx = (β ∗ g(u))xx, (1.3)where L is a general constant coefficient pseudodifferential operator (Fourier multiplier),studied in [1]. The case a = 0 with a general nonlinearity f(u) was first proposed in[2] as a general equation governing the propagation of nonlinear strain waves in a one-dimensional, nonlocally elastic medium. Local well posedness of the Cauchy problemsare characterized by the smoothness of β, or equivalently the decay condition on theFourier transform|β̂(ξ)| ≤ c21(1 + ξ2)−r/2.2This decay condition means that the convolution operator has a smoothing effect of orderr. It follows from [1] and [2] that the Cauchy problem for (1.1) with a 6= 0 is locallywell posed for r ≥ 1, while in the case of a = 0, further smoothness as r ≥ 2 is required.A traveling wave u = u(x− ct) of (1.1) will satisfy(c2 − 1)u = β ∗ up,when u and u′ vanish at infinity. The usual method of proving existence of nontrivial solu-tions to such stationary equations is via calculus of variations where one obtains a solutionas an optimizer u0 of a suitable energy functional. To pass from an optimizing sequence{un} to an actual optimizer u0 often requires some compactness argument. In our caseand similar problems, one difficulty rises from compactness criteria in Lp(R); Sobolevembeddings are not compact unless one has tail control in the sense of the Kolmogorovcompactness criteria. In terms of the optimizing sequence {un}, this tail control meanslimR→∞∫|x|≥R|un(x)|pdx = 0,uniformly in n. Another difficulty is due to the translation invariance of the problem; if{un} is a minimizing sequence, then so is any shift {u˜n} with u˜n(x) = un(x−xn). This,in particular, means that to achieve tail control, one should first align the functions un ,i.e. choose the correct shifts xn so that the sequence {u˜n} is uniformly small for |x| ≥ R.The Concentration Compactness Principle of Lions [7] settles these two issues at once,yet it involves heavy techniques. Moreover it does not give any extra information on theshape of the traveling wave.On the other hand, symmetries of the problem may offer an easier alternative. Re-stricting the optimization problem to bell shaped functions, whenever possible, leads toa more direct approach. The decay of a bell-shaped function can be easily estimated byits Lp norm, which in turn will yield tail control. In other words, bell-shaped functionsare already aligned in the sense we mentioned above. Moreover, this approach will yielda bell-shaped traveling wave, as is in the case of many examples where explicit solutionsare known.Traveling waves for (1.3) were studied in [4] where their existence was proved using theConcentration Compactness Principle, for kernels β that satisfy the elipticity conditionc22(1 + ξ2)−r/2 ≤ β̂(ξ) ≤ c21(1 + ξ2)−r/2. (1.4)3Their results imply, in particular, existence of traveling waves for (1.1). In the presentwork we are able to extend their existence result to kernels that are not necessarilyelliptic.Existence of traveling waves for (1.2) was first proved by Friesecke and Wattis in1994 [5] using the Concentration Compactness Principle. In 2005, English and Pegoinvestigated decay properties of the solitary waves [3]. In a series of papers, Stefanovand Kevrekides studied traveling waves for Hertzian chains with precompression usingbell-shaped functions [8], [9]. In [8], they consider the simpler precompression-free case,namely the equationutt = ∆disc(up). (1.5)Starting with a variational approach, they prove the existence of a maximizer usinggrowth estimates related to bell-shapedness. In [9], the authors summarise an alternativeproof for (1.5) relying again on growth estimates but this time employing the LebesgueDominated Convergence Theorem rather than compactness. We recall that (1.5) can bewritten as utt = (Λ ∗ up)xx, where Λ is the triangular kernel introduced above.In this paper we adapt the approach in [8], [9] to give a simple proof for the existenceof traveling waves for (1.1). Our proof works for bell shaped kernels satisfying mildconditions, but we do not require the ellipticity condition (1.4).The plan of the paper is as follows: In Section 2, we give the necessary preliminar-ies about bell-shaped rearrangements. In Section 3, we define a related maximizationproblem and prove the existence of a maximizer. In Section 4, via the Euler Lagrangeequation and the regularity result, we prove that maximizers are indeed smooth bell-shaped traveling waves and present some examples of kernels for which our results apply.In what follows, W k,p(R) denotes the Sobolev space of k times weakly differentiablefunctions with derivatives in Lp(R), W∞,p(R) = ∩k∈NW k,p(R).2. Rearrangement and Bell Shaped FunctionsA function f : R → R is said to be bell-shaped if it is nonnegative, even and non-increasing on [0,∞). In this section we will consider bell-shaped rearrangements. Wefollow the presentation in [9] and refer to [6] for further information.4For a measurable function f : R→ R, the distribution function of f is defined asdf (s) = m({x ∈ R : |f(x)| > s})where m stands for the Lebesgue Measure. It follows that for a C1 function ϕ one hasthe identity ∫ ∞−∞ϕ(|f(x)|)dx =∫ ∞0ϕ′(s)df (s)ds,whenever the first integral exists. The non-increasing rearrangement of f is defined onR+ as,f∗(t) = inf{s > 0 : df (s) ≤ t};and its bell-shaped rearrangement f# on R, as f#(t) = f∗(2|t|). Clearly f∗ is anonnegative and nonincreasing measurable function and f# is bell shaped. Moreover, fora bell shaped function we have f# = f. Both rearrangements have the same distributionfunction as f , namely df = df∗ = df# . The integral identity above then implies:Lemma 2.1. Suppose φ ∈ C1. Then,∫ ∞−∞ϕ(|f(x)|)dx =∫ ∞0φ(|f∗(x)|)dx =∫ ∞−∞φ(f#(x))dx,whenever the first integral is finite. In particular for f ∈ Lp(R), 1 ≤ p ≤ ∞, we have‖f‖Lp(R) = ‖f∗‖Lp(R+) = ‖f#‖Lp(R).Lemma 2.2. (Riesz’s Rearrangement Inequality) Let f, g and h be nonnegativemeasurable functions on a real line, vanishing at infinity. Then∫ ∞−∞∫ ∞−∞f(x)g(x− y)h(y)dxdy ≤∫ ∞−∞∫ ∞−∞f#(x)g#(x− y)h#(y)dxdy.We will also make use of Young’s inequality:Lemma 2.3. (Young’s Inequality) Suppose f ∈ Lp(R), g ∈ Lq(R) and 1p + 1q = 1r +1,1 ≤ p, q, r ≤ ∞. Then‖f ∗ g‖r ≤ ‖f‖p‖g‖q.53. Setting the Variational Problem and Constructing a MaximizerThroughout this paper we will assume that the the kernel β ∈W 1, p+12 is bell shaped,and satisfies the decay condition at infinity, β(x) = O(|x|−s) for some s > 1. Thenthe Sobolev embedding theorem implies that β ∈ L∞; while the growth condition givesβ ∈ L1.A traveling wave solution u = u(x− ct) of (1.1) satisfies the equation(c2 − a2)u′′ = (β ∗ up)′′.Assuming that u, u′ vanish at infinity we can integrate twice to get,(c2 − a2)u = β ∗ up. (3.6)When u is nonnegative, letting v = u1p , (3.6) can be expressed as(c2 − a2)v 1p = β ∗ v. (3.7)Considering the related Euler Lagrange equation, this leads to the functional J(v) =〈β ∗ v, v〉 and the constraint ‖v‖L1+ 1p= 1. We thus define the constrained optimizationproblemJmax = sup{J(v) : ‖v‖Lq = 1}, (3.8)where q = 1 + 1p and 〈β ∗ v, v〉 denotes the (Lq∗, Lq) duality with 1q +1q∗ = 1. We notethat q = p+1p , and q∗ = p+ 1.Our aim in this section is to prove that there is a bell-shaped maximizer of (3.8).This will be done through the following steps.For v with ‖v‖Lq = 1, since 2p+1 = 1p+1 − pp+1 + 1 = 1q∗ − 1q + 1, by Young’s inequality,we haveJ(v) = 〈β ∗ v, v〉 ≤ ‖β ∗ v‖Lq∗‖v‖Lq (3.9)≤ ‖β‖Lp+12‖v‖2Lq = ‖β‖L p+12 .This shows that Jmax exists. We next consider the more restricted optimization problem,J#max = sup{J(v) : ‖v‖Lq = 1, v is bell-shaped}.6Clearly J#max ≤ Jmax. Conversely, let v ∈ Lq with ‖v‖Lq = 1. Then for the bell-shapedrearrangement v# = |v|# we have ‖v#‖Lq = 1. By Riesz’s rearrangement inequalityJ(w) = 〈β ∗ v, v〉 ≤ 〈β ∗ |v|, |v|〉 ≤ 〈β ∗ v#, v#〉 = J(v#).Hence Jmax ≤ J#max, so that Jmax = J#max. Based on this, in the first maximizationproblem, we can reduce the set of allowable v to the set of bell-shaped functions. Wenext obtain some growth estimates for bell-shaped functions [9].Lemma 3.1. Suppose that v ∈ Lq is bell-shaped. Then for x 6= 0,v(x) ≤ 2− 1q ‖v‖Lq |x|− 1q .Proof. By bell-shapedness, for |y| ≤ |x| we have v(y) ≥ v(x). Then2|x|v(x)q ≤∫|y|≤|x||v(y)|qdy ≤ ‖v‖qLq ,which implies the required estimate.Lemma 3.2. There is some C so that for all bell-shaped v with ‖v‖Lq = 1,β ∗ v(x) ≤ Θ(x) = C 1 for |x| < 2(|x| − 1)−1q for |x| ≥ 2Proof. Since‖β ∗ v‖L∞ ≤ ‖β‖Lq∗‖v‖Lq ≤ ‖β‖12L∞‖β‖12Lp+12‖v‖Lq ≤ ‖β‖12L∞‖β‖12Lp+12,β ∗ v is bounded. For the case |x| ≥ 2, as β ∗ v is even it suffices to consider x ≥ 2.β ∗ v(x) =∫ x+1x−1β(x− y)v(y)dy +∫|y−x|>1β(x− y)v(y)dy = I + IIBy Lemma 3.1, v(x) ≤ |x|− 1q ; since 0 < x− 1, so we haveI =∫ x+1x−1β(x− y)v(y)dy ≤ v(x− 1)∫β(x− y)dy ≤ C‖β‖L1(|x| − 1)−1q .For II, we first show that |x| 1q β ∈ Lq∗ . But(|x| 1q β(x))q∗ ≤ ‖β‖q∗L∞ |x|−q∗(s− 1q ) = O(|x|−q∗(s−1)−1).7Since −q∗(s− 1)− 1 < −1 , and β is bounded, this shows that |x| 1q β ∈ Lq∗ . Then,II =∫|x|>1|x− y|−1q |x− y| 1q β(x− y)v(y)dy≤ (x− 1)−1q∫ ∞−∞|x− y| 1q β(x− y)v(y)dy≤ (x− 1)−1q ‖|x| 1q β‖Lq∗ ‖v‖Lq ≤ (x− 1)−1q ‖|x| 1q β‖Lq∗ .Adding up I and II gives the estimate for |x| ≥ 2.We now pick a bell shaped maximizing sequence {vn}, that is lim J(vn) = Jmaxand ‖vn‖Lq = 1. The next lemma provides the crucial step where we pass from themaximizing sequence to the actual maximizer.Lemma 3.3. The constrained variational problem (3.8) has a bell shaped maximizer.Proof. We start with some bell-shaped maximizing sequence {vn}. Since ‖vn‖Lq = 1, byAlaoglu’s theorem there is a weakly convergent subsequence such that vnk ⇀ v for somev ∈ Lq. Then for each x ∈ R,limβ ∗ vnk(x) = lim〈β(x− ·, vnk〉 = 〈β(x− ·), v〉 = β ∗ v(x).In other words, β ∗ vnk(x) converges pointwise to β ∗ v(x). By Lemma 3.3, we have|β ∗ vn(x)|q∗ ≤ Θ(x)q∗ , so for |x| ≥ 2|Θ(x)|q∗ ≤ C(|x| − 1)−q∗q .Recall that q = 1 + 1p with p > 1. Then q < 2 < q∗, thus q∗q > 1. This shows that|Θ|q∗ ∈ L1, and by the Lebesgue Dominated Convergence Theorem β ∗ vnk converges toβ ∗ v in Lq∗ . Then|J(vnk)− J(v)| ≤ |〈β ∗ vnk − β ∗ v, , vnk〉|+ |〈β ∗ v, vnk − v〉|.The first term is bounded by ‖β ∗ vnk − β ∗ v‖Lq∗ hence converges to zero. The secondterm also converges to zero since v nk ⇀ v. Adding up we getJmax = lim J(vnk) = J(v).By the lower semicontinuity of weak convergence, we have ‖v‖Lq ≤ 1. As J(v) = Jmax 6=0, we have v 6= 0. Letting λ = ‖v‖Lq then ‖λ−1v‖Lq = 1, hence λ−2Jmax = λ−2J(v) =8J(λ−1v) ≤ Jmax, which shows that ‖v‖Lq = 1 and v is a maximizer. Finally, if v is notbell-shaped, then v# will be a bell-shaped maximizer.4. Proof of the Main ResultWe have obtained a maximizer v, the next step is to derive the corresponding Euler-Lagrange equation. Consider perturbations of v in the form v + tϕ, where ϕ is a fixednonnegative C∞0 (R) function. Clearly, ‖v + tϕ‖−1Lq (v + tϕ) will satisfy the constraint.Thusg(t) = J(‖v + tϕ‖−1Lq (v + tϕ)) = ‖v + tϕ‖−2Lq J(v + tϕ)will have a maximum at t = 0. Setting g′(0) = 0, gives2∫(β ∗ v − Jmaxvq−1)ϕdx = 0.Since this holds for all nonnegative test functions ϕ ≥ 0, we get the inequality EulerLagrange equationβ ∗ v − Jmaxvq−1 ≤ 0.Multiplying by v and integrating over R givesJmax = J(v) ≤ Jmax∫vqdx = Jmax,so that we have the Euler Lagrange equation β∗v = Jmaxvq−1. With u = v 1p , as q = 1+ 1p ,this implies u ∈ Lp+1 andJmaxu = β ∗ up.Then for any c2 > a2, uc = (√c2−a2Jmax)2p−1u will satisfy the traveling wave equation (3.6).Clearly uc and β ∗ upc are in Lp+1.Finally, we will consider the regularity of these traveling waves. Since β ∈ W 1, p+12we have β′ ∈ L p+12 . By Young’s inequality, both β ∗ up and Dx(β ∗ up) = β′ ∗ up arein Lp+1; that is β ∗ up ∈ W 1,p+1; so uc = c−2β ∗ upc ∈ W 1,p+1. Then we will haveD2x(β ∗ upc) = β′ ∗ (upc)′ ∈ Lp+1, hence uc = c−2β ∗ upc ∈ W 2,p+1. Repeating the samesteps, this boot-strap argument shows that uc ∈ W∞,p+1, in other words uc ∈ C∞ andall its derivatives decay at infinity. We can then integrate (3.6) and obtain the mainresult:9Theorem 4.1. Suppose β ∈W 1, p+12 is bell shaped and β(x) = O(|x|−s) as |x| → ∞ forsome s > 1. Then for any c2 > a2, the equation (1.1) has a bell shaped traveling wavesolution u ∈W∞,p+1 with velocity c.Remark 4.2. It is possible to have an alternative proof using the Kolmogorov compact-ness criteria. The condition β ∈ W 1, p+12 implies that the sequence {β ∗ vnk} is boundedin W 1,q∗. The uniform decay estimates on β ∗ vnk will then enable using Kolmogorov’scompactness criteria to obtain a subsequence that converges in Lq∗. This alternative proofis in fact closer in spirit to the one provided in [8].Remark 4.3. Replacing the nonlinear term up by |u|p−1u, or |u|p, the approach abovecan be extended to all real values of p > 1. The whole proof works except one has to payattention to regularity. When p is not an integer f(x) = |x|p−1x or f(x) = |x|p is onlydifferentiable k times with the greatest integer k ≤ p; we then have uc ∈W k,p+1.Remark 4.4. As the equation for traveling waves is the same, our approach also showsthat the nonlocal unidirectional wave equationut − aux = (β ∗ up)xhas bell-shaped traveling wave solutions with velocity c > a. We note that the BBMequation is an example for this form.We now give some examples of kernels. Clearly they are all bell-shaped, in W 1,1 ∩W 1,∞ and O(x−s) for some s > 1.1. The exponential kernel β(x) = 12e−|x|.2. The Gaussian kernel β(x) = e−x2.3. β(x) = 11+x2 .4. The triangular kernel Λ(x) = (1− |x|)χ[−1,1].5. β(x) = (1− x2)(1− |x|)χ[−1,1].Note that the exponential kernel is in fact the Green’s function for the operator 1−D2xwith decay conditions at infinity. The corresponding equation (1.1) can also be writtenas the Improved Boussinesq type equationutt − uttxx = (up)xx.10The triangular kernel leads to the differential-difference equation considered in [8] .Theother kernels will yield integro-differential equations. Clearly examples 4 and 5 can begeneralized to any sufficiently smooth bell-shaped kernel in with compact support. Wealso note that except for the exponential kernel, none of the others satisfies the ellipticitycondition (1.4) so that the results in [4] will not provide the existence of traveling wavesolutions.Acknowledgement: This paper is an extension of part of the second author’s M.S.thesis (Sabanci University, August 2016). Abba I. Ramadan was supported by TUBITAKGrant 2215 - Graduate Scholarship Programme for International Students.References[1] C. Babaoglu, H.A. Erbay, A. Erkip, Global existence and blow-up of solutions for a general class ofdoubly dispersive nonlocal nonlinear wave equations, Nonlinear Anal. 77 (2013) 82-93.[2] N. Duruk, H.A. Erbay, A. Erkip, Global existence and blow-up for a class of nonlocal nonlinearCauchy problems arising in elasticity, Nonlinearity 23 (2010) 107-118.[3] J.M. English, R.L. Pego, On the solitarywave pulse in a chain of beads, Proc. AMS, 133, (2005),1763-1768[4] H.A. Erbay, S. Erbay, A. Erkip, Existence and stability of traveling waves for a class of nonlocalnonlinear equations, J. Mathematical Analysis and Applications, 425 (2015) 307-336[5] G. Friesecke, J.A.D. Wattis, Existence theorem for solitary waves on lattices, Commun. Math. Phys.161, (1994) 391-418[6] E.H. Lieb, M. Ross, Analysis, Volume 14, Graduate Studies in Mathematics, AMS, 2001.[7] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally com-pact case, Part 1. Ann. Inst. Henri Poincare´ 1, (1984) 109-145.[8] A. Stefanov, P. Kevrekidis, On the existence of solitary traveling waves for generalized Hertzianchains, Journal of Nonlinear Science 22 (2012), 327–349.[9] A. Stefanov, P. Kevrekidis, Traveling waves for monomer chains with precompression, Nonlinearity26 (2013), 539 –349.11

Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels

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Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels

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