Exponential Stability of the Traveling Fronts for a Pseudo-Para. Pseudo-Parabolic Fisher-KPP Equation

Transcription

1 Exponential Stability of the Traveling Fronts for a Pseudo-Parabolic Fisher-KPP Equation Based on joint work with Xueli Bai (Center for PDE, East China Normal Univ.) and Yang Cao (Dalian University of Technology)

2 1 Introduction of problem and known results 2

3 1 Introduction of problem and known results 2

4 Introduction u t k u t = u + f (u), x R n, t > 0, (1) If k = 0, (1) is just the semi-linear reaction-diffusion equation; If k > 0, (1) is called to be pseudo-parabolic equation. T.W.Ting, Parabolic and pseudo-parabolic partial differential equations, J.Math.Soc.Japan, 1969; R.E. Showalter, T.W.Ting, Pseudo-parabolic partial differential equations, SIAM J. Math. Anal, 1970.

5 Introduction u t k u t = u + f (u), x R n, t > 0, (1) If k = 0, (1) is just the semi-linear reaction-diffusion equation; If k > 0, (1) is called to be pseudo-parabolic equation. T.W.Ting, Parabolic and pseudo-parabolic partial differential equations, J.Math.Soc.Japan, 1969; R.E. Showalter, T.W.Ting, Pseudo-parabolic partial differential equations, SIAM J. Math. Anal, 1970.

6 Introduction Pseudo-parabolic equations describe a variety of important physical and biological processes, such as the seepage of homogeneous fluids through a fissured rock (where k is a characteristic of the fissured rock); G.Barenblat, I.Zheltov, I.Kochiva, J.Appl.Math.Mech, 1960; the unidirectional propagation of nonlinear, dispersive, long waves (where u is typically the amplitude or velocity); T.B.Benjamin, J.L.Bona, J.J.Mahony, Philos.Trans.R.Soc.Lond.Ser.A, 1972; the aggregation of populations (where u represents the population density); V.Padron, Trans.Amer.Math.Soc, 2004.

7 Introduction Pseudo-parabolic equations describe a variety of important physical and biological processes, such as the seepage of homogeneous fluids through a fissured rock (where k is a characteristic of the fissured rock); G.Barenblat, I.Zheltov, I.Kochiva, J.Appl.Math.Mech, 1960; the unidirectional propagation of nonlinear, dispersive, long waves (where u is typically the amplitude or velocity); T.B.Benjamin, J.L.Bona, J.J.Mahony, Philos.Trans.R.Soc.Lond.Ser.A, 1972; the aggregation of populations (where u represents the population density); V.Padron, Trans.Amer.Math.Soc, 2004.

8 Introduction Pseudo-parabolic equations describe a variety of important physical and biological processes, such as the seepage of homogeneous fluids through a fissured rock (where k is a characteristic of the fissured rock); G.Barenblat, I.Zheltov, I.Kochiva, J.Appl.Math.Mech, 1960; the unidirectional propagation of nonlinear, dispersive, long waves (where u is typically the amplitude or velocity); T.B.Benjamin, J.L.Bona, J.J.Mahony, Philos.Trans.R.Soc.Lond.Ser.A, 1972; the aggregation of populations (where u represents the population density); V.Padron, Trans.Amer.Math.Soc, 2004.

9 Introduction Consider the following equation u t = εu xxt + u xx + u(1 u), x R, t > 0, (2) with ε > 0 small enough, and the initial data u(x, 0) = u 0 (x). (3)

10 Existence of TW with ε = 0 When ε = 0, (2) becomes the classical Fisher-KPP equation u t = u xx + u(1 u), x R, t > 0. (4) For each fixed c 2, equation (4) has a monotone increasing traveling wave U 0 c (z) which is unique up to a shift and satisfies U zz + cu z + U(1 U) = 0, z = x ct, (5) and U( ) = 0, U(+ ) = 1. (6)

11 Existence of TW with ε = 0

12 Existence of TW with ε = 0 By phase-plane analysis, we have 1 Uc 0 (z) exp(c 0 + z), as z +, for c 2; Uc 0 (z) exp(c0 z), as z, for c < 2; (7) Uc 0 (z) exp(c0 z), as z, for c = 2; where c ± 0 = c c 2 ±4 2, c 0 = 1.

13 Stability of the TW with ε = 0 Sattinger (1972): Local exponential stability of traveling waves with noncritical speeds in some exponentially weighted spaces; M.Bramson (1983), K.S. Lau (1985) and Uchiyama (1995), etc: for more general initial values in [0; 1] decaying to zero exponentially at x, the asymptotic behavior of the solution is determined by the exponential decaying rate of the initial value at x. F. Hamel and L. Roques (2010): the asymptotic speed of level set of solution becomes infinity if the initial values decay more slowly than any exponential functions at x.

14 Introduction In this paper, we are interested in the asymptotic stability (nonlinear exponential stability) of the waves for equation (2) with small ε > 0. Traveling waves are solutions of form U ε c (z), z = x ct, which satisfy εcu zzz + U zz + cu z + U(1 U) = 0, (8) and U( ) = 0, U(+ ) = 1. (9)

15 1 Introduction of problem and known results 2

16 1 Introduction of problem and known results 2

17 We can prove that the exponential stability of the traveling fronts for the equation with ε = 0 is also valid for the equation with small ε > 0. u t = εu xxt + u xx + u(1 u), x R, t > 0, u t = u xx + u(1 u), x R, t > 0.

18 First, for later proof of spectral stability of traveling fronts, we first need to prove the detailed existence results on the perturbation and spacial decay of the waves when ε > 0 is small, which is not available from the phase plane analysis. By using geometric singular perturbation method, we prove the existence and perturbation of traveling waves (U ε c (x ct) with ε > 0 small enough.

19 Introduction of problem and known results Theorem (Perturbation of Traveling Waves for Small ε > 0 ) There exists small ε 0 > 0 such that for each fixed 0 < ε < ε 0 and c 2, equation (2) has a monotone increasing traveling wave U ε c (z) which is unique up to a shift and satisfies (U ε c (z), (U ε c ) z (z)) ( U 0 c (z), (U 0 c ) z (z) ) L (R) 0, as ε 0, (10) where U 0 c (z) satisfies (5) and (6). Furthermore, 1 U ε c (z) exp(c + ε z), as z +, for c 2; U ε c (z) exp(c ε z), as z, for c < 2; U ε c (z) exp(c ε z), as z, for c = 2; (11) where c ± ε c 0 ±, c ε c0 as ε 0, with c 0 ± = c c 2 ±4 2, 2

20 Introduction of problem and known results Theorem (Nonlinear Exponential Stability of the Waves with c < 2) Let Uc ε (x ct) be the traveling wave solution of (2) satisfying (9). For small ε > 0 and each fixed c < 2, there exist α > 0 satisfying c c < α < c+ c 2 4 2, and δ 0 > 0 such that if u 0 (z) U ε c (z) H 2 α (R) δ 0, then the solution u(z, t) to the system (12) exists globally and satisfies u(, t) U ε c ( ) H 2 α (R) C u 0 (z) U ε c (z) H 2 α (R)e βt, t 0, with Hα(R) 2 = {u u(z)w α (z) H 2 (R)}, w α (z) = 1 + e αz.

21 1 Introduction of problem and known results 2

22 In the moving coordinate z = x ct, equation (2) can be written as u t = εu zzt εcu zzz + u zz + cu z + u(1 u), (12) u(z, 0) = u 0 (z).

23 Consider the linearized system of (12) around U ε c (z) u t = εu zzt εcu zzz + u zz + cu z + (1 2Uc ε (z))u ( ) 1 = I ε 2 z [ εc c 2 z 3 z 2 z + (1 2Uε c (z)]u L ε u (13) For each fixed small ε > 0, the linear operator L ε only generates a strongly continuous semigroup (which is also called C 0 -semigroup) on H 2 (R).

24 Based on C 0 -semigroup theories, to prove the linear exponential stability of the waves, we need to verify L ε satisfies the following spectral estimates (spectral stability) sup Re{σ(L ε ) \ {0}} δ < 0 and zero is a simple eigenvalue of L ε or zero is not an eigenvalue of L ε. (uniform bound on resolvent) sup (λi L ε ) 1 X X M α. Reλ 0

25 Based on C 0 -semigroup theories, to prove the linear exponential stability of the waves, we need to verify L ε satisfies the following spectral estimates (spectral stability) sup Re{σ(L ε ) \ {0}} δ < 0 and zero is a simple eigenvalue of L ε or zero is not an eigenvalue of L ε. (uniform bound on resolvent) sup (λi L ε ) 1 X X M α. Reλ 0

26 σ(l ε ) = σ ess (L ε ) + σ n (L ε ) where σ n (L ε ) is the set of all the isolated eigenvalues of L ε with finite algebraic multiplicity.

27 Imλ no eigenvalue (λi L ε α ) 1 Xα Xα < M α,2 no eigenvalue σ ess(l ε α ) no eigenvalue (λi L ε α ) 1 Xα Xα < M α,3 (λi L ε α ) 1 Xα Xα < M α,1 O Reλ no eigenvalue (λi L ε α ) 1 Xα Xα < M α,2

28 Main difficulties: the uniform boundedness of isolated unstable eigenvalues and the uniform boundedness on resolvent of the operator L ε α for small ε > 0 the existence and convergence of Evans function D ε (λ) for ε 0 and λ in bounded region

29 Main difficulties: the uniform boundedness of isolated unstable eigenvalues and the uniform boundedness on resolvent of the operator L ε α for small ε > 0 the existence and convergence of Evans function D ε (λ) for ε 0 and λ in bounded region

30 Step 3: nonexistence of eigenvalue in bounded region Consider the eigenvalue problem L ε φ = i.e. ) 1 (I ε 2 z 2 [ εcφ zzz + φ zz + cφ z + (1 2Uc ε (z))φ] = λφ, ε(cφ zzz λφ zz ) φ zz cφ z + (λ 1 + 2U ε c (z))φ = 0. (14) Notice that the equation with small ε > 0 is a singular perturbation of the equation with ε = 0, so the classical spectral perturbation theories or the classical Evans function methods can t be applied directly to get the location and estimates of isolated eigenvalues of L ε α.

31 Step 3: nonexistence of eigenvalue in bounded region By choosing appropriate Evans function( dual Evans function) and applying more detailed geometric singular perturbation estimates on Evans function, we can prove the continuity of the dual Evans function D(λ, ε) in ε for λ in bounded region, i.e. D(λ, ε) D(λ, 0) as ε 0 +, then using Evans function methods and the available spectral results for ε = 0, we can prove the nonexistence of unstable eigenvalues in bounded region, which with step 1 and step 2 guarantees the linear exponential stability of the wave.

32 Step 3: nonexistence of eigenvalue in bounded region Let y 1 = φ, y 2 = φ z, y 3 = cφ zz λφ z + (1 2Uc 0 λ)y 1 + cy 2, rewrite the eigenvalue problem (14) as the following system y y 2 = λ 1 + 2Uc 0 (z) c 1 εy 3 εb 31 (z, ε, λ) ε(1 2Uc 0 2λ c 2 1 ) c + λ c ε + cε (15) There exists a unique solution Y1 ε (z, λ) = (y11 ε ε (z, λ), y12 (z, λ),ε 13 (z, λ))t of (15), which is analytic in λ for Re λ δ α and satisfies Y1 ε (z, λ)e σε 1 (λ)z ( 1, σ1 ε (λ), εa 1(λ) ) T, as z. (16)

33 Step 3: nonexistence of eigenvalue in bounded region Consider the adjoint equation of (15), γ 0 1 λ 2U 1 0 c (z) b 31 (z, ε, λ) γ 2 = 1 c 1 + 2Uc 0 (z) + 2λ + c 2 εγ 3 0 ε 1 c λ c ε cε (17) There exists a unique solution Υ ε+ 1 (z, λ) = (γε+ 11 (z, λ), γε+ 12 (z, λ), γε+ 13 (z, λ))t of (17), which is analytic in λ for Re λ δ α and satisfies γ 1 γ 2 γ 3 Υ ε+ 1 (z, λ)eσε+ 1 (λ)z (a 2 (λ), 1, εa 3 (λ)) T, as z +. (18)

34 Step 3: nonexistence of eigenvalue in bounded region We define the dual Evans function of system (15) (the eigenvalue problem) with ε > 0 by D ε (λ) = Υ ε+ ε 1 (z, λ) Y1 (z, λ) = (y11 ε 11 + y 12 ε 12 + ỹ ε 13 γε+ 13 )(z, λ). (19) Similarly, we define the dual Evans function of system (15) with ε = 0 by D 0 (λ) = Υ + 1 (z, λ) Y 1 (z, λ) = (y 11 γ y 12 γ+ 12 )(z, λ). (20) (i) D 0 (λ)(d ε (λ)) is independent of z and analytic in λ. (ii) λ is an eigenvalue of L 0 α(l ε α), if and only if D 0 (λ)(d ε (λ)) = 0, for Re λ δ α.

35 Step 3: nonexistence of eigenvalue in bounded region We define the dual Evans function of system (15) (the eigenvalue problem) with ε > 0 by D ε (λ) = Υ ε+ ε 1 (z, λ) Y1 (z, λ) = (y11 ε 11 + y 12 ε 12 + ỹ ε 13 γε+ 13 )(z, λ). (19) Similarly, we define the dual Evans function of system (15) with ε = 0 by D 0 (λ) = Υ + 1 (z, λ) Y 1 (z, λ) = (y 11 γ y 12 γ+ 12 )(z, λ). (20) (i) D 0 (λ)(d ε (λ)) is independent of z and analytic in λ. (ii) λ is an eigenvalue of L 0 α(l ε α), if and only if D 0 (λ)(d ε (λ)) = 0, for Re λ δ α.

36 Step 3: nonexistence of eigenvalue in bounded region D ε (λ) = (y ε 11 γε y ε 12 γε ỹ ε 13 γε+ 13 D 0 (λ) = (y11 γ y 12 γ+ 12 )(0, λ). )(0, λ), By more detailed geometric singular perturbation estimates, we can prove the continuation of Evans function in ε, i.e. the dual Evans function D ε (λ) D 0 (λ) uniformly as ε 0 for λ K 0 K.

37 Step 3: nonexistence of eigenvalue in bounded region The nonexistence of eigenvalue of L 0 α for λ K 0 K guarantees D 0 (λ) 0, for λ K 0 K, (21) then by Rouché Theorem for small ε > 0, we have D ε (λ) 0, for λ K 0 K, (22) which implies there is no isolated eigenvalue of L ε α in bounded region.

38 Imλ no eigenvalue (λi L ε α ) 1 Xα Xα < M α,2 no eigenvalue σ ess(l ε α ) no eigenvalue (λi L ε α ) 1 Xα Xα < M α,3 (λi L ε α ) 1 Xα Xα < M α,1 O Reλ no eigenvalue (λi L ε α ) 1 Xα Xα < M α,2

39 Theorem For ε > 0 small enough and each fixed c < 2 and α > 0 satisfying c c 2 4 < α < c + c 2 4, 2 2 there exists small βα > 0 such that Re {σ(l ε α)} < β α. and sup (λi L ε α) 1 Xα Xα < +. Reλ 0 This implies the non-linear exponential stability of traveling fronts.

40 Thank you for your attention!