Existence of traveling waves for a nonlocal monostable equation: an abstract approach [ with short note at end ]

arXiv:0807.3612v4 [math.AP] 16 Aug 2016 Existence of traveling wave solutions for a nonlocal monostable equation: an abstract approach Hiroki Yagisita Department of Mathematics, Faculty of Science, Kyoto Sangyo University Motoyama, Kamigamo, Kita-Ku, Kyoto-City, 603-8555, Japan Abstract We consider the nonlocal analogue of the Fisher-KPP equation ut=µ∗u−u+f(u), where µis a Borel-measure on Rwith µ(R) = 1 and fsatisfies f(0) = f(1) = 0 and f > 0 in (0,1). The equation may have a standing wave solution (a traveling wave solution with speed 0) whose profile is a monotone but discontinuous function. We show that there is a constant c∗such that it has a traveling wave solution with monotone profile and speed cwhen c≥c∗ while no periodic traveling wave solution with average speed cwhen c < c∗. In order to prove it, we modify an abstract method for monotone semiflows by Weinberger. We note that the semiflow generated by the equation does not have compactness with respect to the compact-open topology. At the end of this paper, we propose a discrete Schrodinger model that describes the measurement process. Keywords: discontinuous profile, convolution model, integro-differential equation, discrete monostable equation, nonlocal evolution equation, Fisher-Kolmogorov equation, multi-species mixture. AMS Subject Classification: 35K57, 35K65, 35K90, 45J05. 1 Introduction In 1930, Fisher [6] introduced the reaction-diffusion equation ut=uxx + u(1−u) as a model for the spatial spread of an advantageous form of a single gene in a population. He [7] found that there is a constant c∗such that the equation has a traveling wave solution with speed cwhen c≥c∗while it does no such solution when c < c∗. Kolmogorov, Petrovsky and Piskunov [14] investigated asymptotic behavior in the model. Since the pioneering 1 works, there have been extensive studies on traveling waves and asymptotic behavior for monostable evolution systems. In this paper, we consider the following nonlocal analogue of the Fisher-KPP equation: ut=µ∗u−u+f(u). Here, µis a Borel-measure on Rwith µ(R) = 1 and the convolution is defined by (µ∗u)(x) = Zy∈R u(x−y)dµ(y) for a bounded and Borel-measurable function uon R. The nonlinearity fis a Lipschitz continuous function with f(0) = f(1) = 0 and f > 0 in (0,1). Then, we would show that there is a constant c∗such that the nonlocal monostable equation has a traveling wave solution with monotone profile and speed cwhen c≥c∗while it does no periodic traveling wave solution with average speed c when c < c∗, if there is a positive constant λsatisfying Ry∈Reλ|y|dµ(y)<+∞. Further, we would also show that there is a smooth and monostable nonlinearity fsuch that the equation has a standing wave solution (a traveling wave solution with speed 0) whose profile is a monotone but discontinuous function, if µsatisfies the extra condition Ry∈Rydµ(y)>0. For the nonlocal monostable equation, Schumacher [18, 19] proved that there is the minimal speed c∗and the equation has a traveling wave solution with speed cwhen c≥c∗, if the nonlinearity fsatisfies the extra condition f(u)≤f′(0)u. On the other hand, Coville and Dupaigne [5] proved that the minimal speed c∗is positive, the equation has a traveling wave solution with speed cwhen c≥c∗and the profile of the solution is a smooth function, if the Borelmeasure µsatisfies the extra condition µ((−∞,−y)) ≡µ((+y, +∞)). Further, Schumacher [18, 19] and Carr and Chmaj [2] studied uniqueness of traveling wave solutions for the equation. See, e.g., [4, 8, 9, 10, 11, 12, 13, 16, 20, 21, 23, 24] on traveling waves in various monostable evolution systems, [1, 3] nonlocal bistable equations and [17] Euler equation. In Section 2, we give abstract conditions such that a semiflow satisfying the conditions has a traveling wave solution with speed cwhen c≥c∗while 2 it does no periodic traveling wave solution with average speed c when c < c∗. We also note that it may not be compact with respect to the compact-open topology. In Section 3, we use idea in Weinberger [22] and Li, Weinberger and Lewis [15] to prove abstract theorems mentioned in Section 2. In Section 4, we precisely state our main results for the nonlocal monostable equation. In Section 5, we show that the semiflow generated by the nonlocal monostable equation satisfies the conditions given in Section 2 to prove the main results. At the end of this paper, we propose a discrete Schrodinger model that describes the measurement process. Another interest: validity of Boltzmann equation for multi-species mixture (and/or semi-classical gas). 2 Abstract theorems for monotone semiflows In the abstract, we would treat a monostable evolution system. Put a set of functions on R; M:= {u|uis a monotone nondecreasing and left continuous function on Rwith 0 ≤u≤1}. The followings are our basic conditions for discrete dynamical systems: Hypotheses 1 Let Q0be a map from Minto M. (i) Q0is continuous in the following sense: If a sequence {uk}k∈N⊂ M converges to u∈ M uniformly on every bounded interval, then the sequence {Q0[uk]}k∈Nconverges to Q0[u]almost everywhere. (ii) Q0is order preserving; i.e., u1≤u2=⇒Q0[u1]≤Q0[u2] for all u1and u2∈ M. (iii) Q0is translation invariant; i.e., (Q0[u(·)])(·−x0) = (Q0[u(·−x0)])(·) for all u∈ M and x0∈R. (iv) Q0is monostable; i.e., 0< α < 1 =⇒α < Q0[α] for all constants α. 3 We note that the semiflow generated by a map Q0satisfying Hypotheses 1 may not be compact with respect to the compact-open topology. The following states that existence of suitable super-solutions of the form {vn(x+cn)}∞ n=0 implies that of traveling wave solutions with speed cin the discrete dynamical systems on M: Proposition 2 Let a map Q0:M → M satisfy Hypotheses 1, and c∈R. Suppose there exists a sequence {vn}∞ n=0 ⊂ M with (Q0[vn])(x−c)≤vn+1(x), infn=0,1,2,··· vn(x)6≡ 0and lim infn→∞ vn(x)6≡ 1. Then, there exists ψ∈ M with (Q0[ψ])(x−c)≡ψ(x),ψ(−∞) = 0 and ψ(+∞) = 1. In the discrete dynamical system on Mgenerated by a map Q0satisfying Hypotheses 1, if there is a periodic traveling wave super-solution with average speed c, then there is a traveling wave solution with speed c: Theorem 3 Let a map Q0:M → M satisfy Hypotheses 1, and c∈R. Suppose there exist τ∈Nand φ∈ Mwith (Q0τ[φ])(x−cτ)≤φ(x),φ6≡ 0and φ6≡ 1. Then, there exists ψ∈ M with (Q0[ψ])(x−c)≡ψ(x),ψ(−∞) = 0 and ψ(+∞) = 1. The infimum c∗of the speeds of traveling wave solutions is not −∞, and there is a traveling wave solution with speed cwhen c≥c∗: Theorem 4 Suppose a map Q0:M → M satisfies Hypotheses 1. Then, there exists c∗∈(−∞,+∞]such that the following holds : Let c∈R. Then, there exists ψ∈ M with (Q0[ψ])(x−cτ)≡ψ(x), ψ(−∞) = 0 and ψ(+∞) = 1 if and only if c≥c∗. We add the following conditions to Hypotheses 1 for continuous dynamical systems on M: Hypotheses 5 Let Qtbe a map from Mto Mfor t∈[0,+∞). (i) Qis a semigroup; i.e., Qt◦Qs=Qt+sfor all tand s∈[0,+∞). (ii) Qis continuous in the following sense: Suppose a sequence {tk}k∈N⊂ [0,+∞)converges to 0, and u∈ M. Then, the sequence {Qtk[u]}k∈Nconverges to ualmost everywhere. As we would have Theorems 3 and 4 for the discrete dynamical systems, we would do the following two for the continuous dynamical systems: 4 Theorem 6 Let Qtbe a map from Mto Mfor t∈[0,+∞). Suppose Qt satisfies Hypotheses 1 for all t∈(0,+∞), and QHypotheses 5. Then, the following holds : Let c∈R. Suppose there exist τ∈(0,+∞)and φ∈ M with (Qτ[φ])(x− cτ)≤φ(x),φ6≡ 0and φ6≡ 1. Then, there exists ψ∈ M with ψ(−∞) = 0 and ψ(+∞) = 1 such that (Qt[ψ])(x−ct)≡ψ(x)holds for all t∈[0,+∞). Theorem 7 Let Qtbe a map from Mto Mfor t∈[0,+∞). Suppose Qt satisfies Hypotheses 1 for all t∈(0,+∞), and QHypotheses 5. Then, there exists c∗∈(−∞,+∞]such that the following holds : Let c∈R. Then, there exists ψ∈ M with ψ(−∞) = 0 and ψ(+∞) = 1 such that (Qt[ψ])(x−ct)≡ψ(x)holds for all t∈[0,+∞)if and only if c≥c∗. 3 Proof of the abstract theorems In this section, we would modify the argument in Weinberger [22] and Li, Weinberger and Lewis [15] to prove the theorems stated in Section 2. Lemma 8 Let a sequence {uk}k∈Nof monotone nondecreasing functions on Rconverge to a continuous function uon Ralmost everywhere. Then, {uk}k∈Nconverges to uuniformly on every bounded interval. Proof. Let C∈(0,+∞) and ε∈(0,+∞). Then, there exists δ∈(0,+∞) such that, for any y1and y2∈[−C−1,+C+ 1], |y2−y1|< δ implies |u(y2)−u(y1)|< ε/4. So, we take N∈Nand a sequence {xn}N n=1 such that limk→∞ uk(xn) = u(xn), −C−1≤x1≤ −C,xn< xn+1 < xn+δand +C≤xN≤+C+ 1 hold. Let k∈Nbe sufficiently large. Then, max{|uk(xn)−u(xn)|}N n=1 < ε/4 holds. Let x∈[−C, +C]. There exists nsuch that xn≤x≤xn+1 holds. So, we get |uk(x)−u(x)| ≤ |uk(xn)−u(x)|+|uk(xn+1)−u(x)| ≤ |uk(xn)− u(xx)|+|u(xn)−u(x)|+|uk(xn+1)−u(xn+1)|+|u(xn+1)−u(x)|< ε. The set of discontinuous points of a monotone function on Ris at most countable. So, if a sequence {uk}k∈Nof monotone functions on Rconverges to a monotone function uon Rat every continuous point of u, then it does almost everywhere. The converse also holds: 5 Lemma 9 Let a sequence {uk}k∈Nof monotone nondecreasing functions on Rconverge to a monotone nondecreasing function uon Ralmost everywhere. Then, limk→∞ uk(x) = u(x)holds for all continuous points x∈Rof u. Proof. We take xn∈(x−2−n, x] and xn∈[x, x + 2−n) satisfying limk→∞ uk(xn) = u(xn) and limk→∞ uk(xn) = u(xn) for n∈N. Then, u(xn)≤ lim infk→∞ uk(x)≤lim supk→∞ uk(x)≤u(xn) holds. Hence, we have limk→∞ uk(x) = u(x) as xis a continuous point of u. Hypotheses 1 imply more continuity than Hypothesis 1 (i): Proposition 10 Let a map Q0:M → M satisfy Hypotheses 1 (i),(ii) and (iii). Suppose a sequence {uk}k∈N⊂ M converges to u∈ M almost everywhere. Then, limk→∞(Q0[uk])(x) = (Q0[u])(x)holds for all continuous points x∈Rof Q0[u]. Proof. We take a cut of function ρ∈C∞(R) with |x| ≥ 1/2 =⇒ρ(x) = 0, |x|<1/2 =⇒ρ(x)>0 and Zx∈R ρ(x)dx = 1. We put smooth functions ρn(·) := 2nρ(2n·), un(·) := (ρn∗u)(·−2−(n+1)) and un(·) := (ρn∗u)(·+ 2−(n+1)) for n∈N. Then, we obtain u(·−2−n)≤un(·)≤u(·)≤un(·)≤u(·+ 2−n). Also, a sequence {min{uk, un}}k∈Nconverges to unalmost everywhere and {max{uk,un}}k∈Nun. Hence, by Lemma 8, the sequence {min{uk, un}}k∈N converges to ununiformly on every bounded interval and {max{uk,un}}k∈N un. Then, by Hypothesis 1 (i), the sequence {Q0[min{uk, un}]}k∈Nconverges 6 to Q0[un] almost everywhere and {Q0[max{uk,un}]}k∈NQ0[un]. From Hypothesis 1 (ii), Q0[min{uk, un}]≤Q0[uk]≤Q0[max{uk,un}] also holds. Therefore, Q0[un]≤lim infk→∞ Q0[uk]≤lim supk→∞ Q0[uk]≤Q0[un] holds almost everywhere. So, by Hypotheses 1 (ii) and (iii), Q0[u](· − 2−n)≤ lim infk→∞ Q0[uk](·)≤lim supk→∞ Q0[uk](·)≤Q0[u](·+ 2−n) holds almost everywhere. Hence, limk→∞ Q0[uk](·) = Q0[u](·) holds almost everywhere, because limn→∞ Q0[u](·−2−n) = limn→∞ Q0[u](·+ 2−n) = Q0[u](·) holds almost everywhere. So, from Lemma 9, limk→∞(Q0[uk])(x) = (Q0[u])(x) holds for all continuous points x∈Rof Q0[u].  Combining Proposition 10 with Helly’s theorem, we can make the argument in Weinberger [22] and Li, Weinberger and Lewis [15] work to prove Proposition 2. Proof of Proposition 2. We put w(·) := limh↓+0 infn=0,1,2,··· vn(· − h), and uk 0:= 2−kw∈ M for k∈N. We also take functions uk n∈ M such that uk n(·) = max{Q0[uk n−1](·−c),2−kw(·)}(3.1) holds for kand n∈N. We show uk n≤uk n+1.(3.2) We have uk 0≤uk 1. As uk n−1≤uk nholds, we get Q0[uk n−1]≤Q0[uk n] and uk n≤uk n+1. In virtue of (3.2), we put uk:= limn→∞ uk n∈ M. Then, by (3.1) and Proposition 10, uk(·) = max{Q0[uk](·−c),2−kw(·)}(3.3) holds. Because limm→∞ Q0[uk(·+m)] = Q0[uk(+∞)] holds from Proposition 10, we have uk(+∞) = lim m→∞ max{Q0[uk](m−c),2−kw(m)} = lim m→∞ max{Q0[uk(·+m)](−c),2−kw(m)} = max{Q0[uk(+∞)],2−kw(+∞)}. Hence, uk(+∞)≥Q0[uk(+∞)] and uk(+∞)≥2−kw(+∞)>0 hold. So, from Hypothesis 1 (iv), we obtain uk(+∞) = 1.(3.4) 7 We show uk n≤vn.(3.5) We get uk 0≤w≤v0. As uk n−1≤vn−1holds, we have Q0[uk n−1](·−c)≤Q0[vn−1](·−c)≤vn(·) and uk n≤vnbecause of 2−kw≤w≤vn. From (3.5), uk(−∞)≤lim m→∞ lim inf n→∞ vn(−m)<1 (3.6) holds. Also, limm→∞ Q0[uk(· − m)] = Q0[uk(−∞)] holds from Proposition 10. Hence, by (3.3), we have uk(−∞) = lim m→∞ max{Q0[uk](−m−c),2−kw(−m)} ≥ Q0[uk(−∞)]. So, from Hypothesis 1 (iv) and (3.6), we obtain uk(−∞) = 0.(3.7) In virtue of (3.4) and (3.7), there exists xksuch that uk(−xk)≤1/2≤ limh↓+0 uk(−xk+h) for k∈N. We put ψk(·) := uk(·−xk)∈ M. Then, we have ψk(0) ≤1/2≤lim h↓+0 ψk(h) (3.8) and ψk(·) = max{Q0[ψk](·−c),2−kw(·−xk)}(3.9) from (3.3). By Helly’s theorem, there exist a subsequence {k(n)}n∈Nand ψ∈ M such that limn→∞ ψk(n)(x) = ψ(x) holds for all continuous points x∈Rof ψ. So, from (3.8), (3.9) and Proposition 10, ψ(0) ≤1/2≤lim h↓+0 ψ(h) (3.10) and ψ(·) = Q0[ψ](·−c) (3.11) holds. Because ψ(−∞) = Q0[ψ(−∞)] and ψ(+∞) = Q0[ψ(+∞)] also hold by (3.11) and Proposition 10, from Hypothesis 1 (iv) and (3.10), we have ψ(−∞) = 0 and ψ(+∞) = 1.  Proof of Theorem 3. 8 We take functions vn∈ M for n= 0,1,2,··· such that vn+mτ = (Q0n[φ])(·−cn) holds for all n= 0,1,2,···, τ −1 and m= 0,1,2,···. Then, we see vn+1(·)≥Q0[vn](·−c) (3.12) and lim inf n→∞ vn= inf n=0,1,2,··· vn= min n=0,1,2,··· ,τ−1vn.(3.13) We show vn(+∞)>0. We have v0(+∞)>0. As vn−1(+∞)>0 holds, we get vn(+∞)≥Q0[vn−1(+∞)] >0 by (3.12), Proposition 10, Hypotheses 1 (ii) and (iv). Hence, because limm→∞ minn=0,1,2,··· ,τ−1vn(m)>0 holds, from (3.13), we see infn=0,1,2,··· vn6≡ 0. Because minn=0,1,2,··· ,τ−1vn≤φholds, by (3.13) and φ(−∞)<1, we have lim infn→∞ vn6≡ 1. Therefore, by Proposition 2, there exists ψ∈ M with Q0[ψ](·−c) = ψ(·), ψ(−∞) = 0 and ψ(+∞) = 1.  Lemma 11 Let a sequence {uk}k∈Nof monotone nondecreasing functions on Rconverge to a monotone nondecreasing function uon Ralmost everywhere. Then, limk→∞ uk(x−xk) = u(x)holds for all sequences {xk}k∈N⊂R with limk→∞ xk= 0 and continuous points x∈Rof u. Proof. We put yn:= supk=n,n+1,n+2,··· |xk|for n∈N. Then, uk(· − yn)≤uk(· − xk)≤uk(·+yn) holds when k≥n. Hence, u(· − yn)≤ lim infk→∞ uk(·−xk)≤lim supk→∞ uk(·−xk)≤u(·+yn) holds almost everywhere. So, limk→∞ uk(· − xk) = u(·) holds almost everywhere, because limn→∞ u(·−yn) = limn→∞ u(·+yn) = u(·) holds almost everywhere. Hence, from Lemma 9, limk→∞ uk(x−xk) = u(x) holds for all continuous points x∈Rof u. Proof of Theorem 4. [Step 1] Let c∗∈[−∞,+∞] be the infimum of c∈Rsuch that there exists ψ∈ M with Q0[ψ](· − c) = ψ(·), ψ(−∞) = 0 and ψ(+∞) = 1. Then, we have the following: Let c∈R. Then, there exists ψ∈ M with Q0[ψ](·−c) = ψ(·),ψ(−∞) = 0 and ψ(+∞) = 1 only if c≥c∗. [Step 2] In this step, we show the following: Let c∈(c∗,+∞). Then, there exists ψ∈ M with Q0[ψ](·−c) = ψ(·),ψ(−∞) = 0 and ψ(+∞) = 1. 9 exist a subsequence nkand ψ∈ M such that limk→∞ vnk(x) = ψ(x) almost everywhere in x. Then, ku(t, x)−ψ(x)kL1([−C,+C]) ≤limk→∞(ku(t, x)− vnk(x)kL1([−C,+C]) +kvnk(x)−ψ(x)kL1([−C,+C])) = 0 holds for all C∈(0,+∞). Hence, we obtain ku(t, x)−ψ(x)kL∞(R)= 0.  Proposition 19 Let a Borel-measure µhave λ∈(0,+∞)satisfying (4.2), and T∈(0,+∞). Suppose a sequence {un}∞ n=0 ⊂C1([0, T], L∞(R)) of solutions to (4.1) with supn∈N,x∈R|un(0, x)−u0(0, x)| ≤ 1satisfies lim n→∞ sup x∈[−I,+I]|un(0, x)−u0(0, x)|= 0 for all I∈(0,+∞). Then, lim n→∞ sup t∈[0,T ]kun(t, x)−u0(t, x)kL∞([−J,+J]) = 0 holds for all J∈(0,+∞). Proof. Let J∈(0,+∞) and ε∈(0,+∞). We take K∈[0,+∞) such that K≥Zy∈R eλ|y|dµ(y)−1 + sup h>0,u∈R f(u+h)−f(u) h. Put positive constants δ:= min{εe−(KT +λJ),1}and I:= 1 λlog(2 δ). Let n∈N be sufficiently large. Then, we have sup x∈[−I,+I]|un(0, x)−u0(0, x)| ≤ δ. (5.7) We consider the following two functions v(t, x) := u0(t, x)−eKtw(x) and v(t, x) := u0(t, x) + eKtw(x), where w(x) := min{δeλx+e−λx 2,1}. We see (µ∗w)(x) ≤min    δRy∈Re−λydµ(y)eλx +Ry∈Reλydµ(y)e−λx 2, µ(R)   16 ≤Zy∈R eλ|y|dµ(y)w(x). So, vis a super-solution to (4.1), because of dv dt −(µ∗v−v+f(v)) = (K+ 1)eKtw−eKt(µ∗w) + f(u0+eKtw)−f(u0)≥0 almost everywhere in x. We can also see that vis a sub-solution. Because of w(0) = δ,w(±I) = 1 and (5.7), we get v(0, x)≤un(0, x)≤v(0, x). Hence, by Proposition 18, v(t, x)≤un(t, x)≤v(t, x) holds almost everywhere in x. So, we have kun(t, x)−u0(t, x)kL∞([−J,+J]) ≤eKT w(±J)≤ε. In virtue of Propositions 13, 18 and 19, if µhas a constant λ∈(0,+∞) satisfying (4.2), then Qt(t∈(0,+∞)) satisfies Hypotheses 1 and Q5 for the semiflow Q={Qt}t∈[0,+∞)on the set Mgenerated by (4.1). So, Theorems 6 and 7 can work for this semiflow. Proof of Theorem 14. Put monotone nondecreasing functions ϕ(x) := max{α∈R|α≤u(0, y) holds almost everywhere in y∈(x, +∞)}and φ(x) := limh↓+0 ϕ(x−h). Then, φ∈ M,φ(−∞)<1 and φ(+∞) = 1 hold. We take a cut of function ρ∈C∞(R) with |x+ 1/2| ≥ 1/2 =⇒ρ(x) = 0, |x+ 1/2|<1/2 =⇒ρ(x)>0 and Zx∈R ρ(x)dx = 1. As we put vn(x) := Zy∈R 2nρ(2n(x−y))u(0, y)dy for n∈N, we see φ≤vn. Let N∈N. Because of limn→∞ kvn(x)− u(0, x)kL1([−N,+N]) = 0, there exists a subsequence nksuch that limk→∞ vnk(x) =u(0, x) almost everywhere in x∈[−N, +N]. Therefore, we have φ(x)≤ u(0, x) almost everywhere in x∈R. So, by Proposition 18, we obtain Qτ[φ](x−cτ)≤u(τ, x −cτ) = u(0, x) almost everywhere in x. Hence, because Qτ[φ](x−cτ)≤ϕ(x) holds, we get Qτ[φ](x−cτ)≤φ(x). Therefore, 17 by Theorem 6, there exists ψ∈ M with ψ(−∞) = 0 and ψ(+∞) = 1 such that Qt[ψ](x−ct)≡ψ(x) holds for all t∈[0,+∞).  Proof of Theorem 15. By Theorem 7, there exists c∗∈(−∞,+∞] such that the following holds: Let c∈R. Then, there exists ψ∈ M with ψ(−∞) = 0 and ψ(+∞) = 1 such that {ψ(x+ct)}t∈Ris a solution to (4.1) if and only if c≥c∗. We show c∗6= +∞. Take K∈[0,+∞) such that K≥max Zy∈R e−λydµ(y), µ(R)−1 + sup h>0 f(h) h. As we put φ(x) := min{eλx,1} ∈ M, we see (µ∗φ)(x)≤min Zy∈R e−λydµ(y)eλx, µ(R) ≤max Zy∈R e−λydµ(y), µ(R)φ(x). So, eKtφ(x) is a super-solution to (4.1), because of eKt(µ∗φ)−eKtw+f(eKtφ)≤KeKtφ. Hence, by Proposition 18, we obtain Q1[φ](x)≤eKφ(x)≤eλ(x+K λ), and Q1[φ](x−K λ)≤φ(x). Therefore, from Theorem 6, there exists ψ∈ M with ψ(−∞) = 0 and ψ(+∞) = 1 such that Qt[ψ](x−K λt)≡ψ(x) holds for all t∈[0,+∞). So, c∗≤K λholds.  Proof of Proposition 16. Suppose ψis a continuous function. We take a interval (a, b)⊂(0,1) such that inf u∈(a,b)fu(u)>1.(5.8) Let x∈ψ−1((a, a+b 2)) and y∈ψ−1((a+b 2, b)). Then, because of x < y and (4.8), we have (µ∗ψ)(x)−ψ(x) + f(ψ(x)) ≤(µ∗ψ)(y)−ψ(x) + f(ψ(x)) <(µ∗ψ)(y)−ψ(y) + f(ψ(y)). It is a contradiction, as ψ−1((a, a+b 2)) and ψ−1((a+b 2, b)) are open intervals.  18 Proof of Proposition 17. We have g(0) = 1 and g′(0) = −Ry∈Rydµ(y)<0 for g(ζ) := Ry∈Re−ζydµ(y) (ζ∈[−λ, +λ]). Hence, there exists ξ∈(0,+∞) with Ry∈Re−ξydµ(y)<1. Then, we take γ∈(0,1−Ry∈Re−ξydµ(y)). We consider the equation ut=µ∗u−u+˜ f(u) (5.9) instead of (4.1), where ˜ f(u) := γu in (−∞,0), f(u) in [0,1], −2(u−1) in (1,2) and −uin [2,+∞). Also, we put K:= Ry∈Re−ξydµ(y)−1+γ∈(−1,0). Then, we show that the function v(t, x) := 2eK(t−1) min{eξx,1}is a super-solution to (5.9) on t∈[0,1]. For x∈(−∞,0], we can see µ∗v−v+˜ f(v)≤dv dt from (µ∗v)(t, x)≤2eK(t−1)(Ry∈Re−ξydµ(y))eξx,˜ f(u)≤γu and dv dt (t, x) = 2KeK(t−1)eξx. For x∈[0,+∞), we can also see it from (µ∗v)(t, x)≤2eK(t−1), ˜ f(v(t, x)) = −v(t, x) and dv dt (t, x) = 2KeK(t−1). Hence, by Proposition 18, we obtain Q1[φ](x)≤v(1, x)≤2eξx, as we put φ(x) := min{2eξx−K,1} ∈ M. So, because Q1[φ](x−K ξ)≤φ(x) holds, by Theorem 6, there exists ψ∈ M with ψ(−∞) = 0 and ψ(+∞) = 1 such that {ψ(x+K ξt)}t∈Ris a solution to (5.9). Because it is also one to (4.1), by Theorem 15, we have c∗≤K ξ<0.  A discrete Schrodinger model: We propose a discrete Schrodinger model that describes the measurement process. Let Xbe the Hilbert space of state vectors of a one-particle system of spin 1 2on the one-dimensional discrete grid Z. That is, X=L2(Z;C2). Let natural numbers L0and N0be large. Put I={n∈Z|L0≤ |n| ≤ L0+N0}. We assume that Iis the place where two detectors exist. Suppose that for n∈I,V(n) is a 2 ×2 Hermitian random matrix. Suppose that for n∈I, if Uis a 2 ×2 unitary matrix, then the distributions of (U−1)(V(n))U and V(n) are the same. Suppose that for n∈I, the density function of V(n), roughly speaking, is smooth and compact supported. Suppose that {V(n)}n∈Iis independent and identically distributed. Suppose that for n6∈ I,V(n) is the 2 ×2zero matrix. Then, we propose a discrete Schrodinger model on the Hilbert space X( = L2(Z;C2) ) √−1du dt (n) = −1 2m( (u(n−1) −u(n)) + (u(n+ 1) −u(n)) ) + V(n)u(n). 19 Acknowledgments. I thank Professor Hiroshi Matano, Mr. Xiaotao Lin and Mr. Masahiko Shimojo for their discussion. It was partially supported by Grant-in-Aid for Scientific Research (No.19740092) from Ministry of Education, Culture, Sports, Science and Technology, Japan. REFERENCES [1] P. W. Bates, P. C. Fife, X. Ren and X. 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