Exponential Stability of the Traveling Fronts for a Pseudo-Para. Pseudo-Parabolic Fisher-KPP Equation
|
|
- Allan Moore
- 5 years ago
- Views:
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!
The stability of travelling fronts for general scalar viscous balance law
J. Math. Anal. Appl. 35 25) 698 711 www.elsevier.com/locate/jmaa The stability of travelling fronts for general scalar viscous balance law Yaping Wu, Xiuxia Xing Department of Mathematics, Capital Normal
More informationThe influence of a line with fast diffusion on Fisher-KPP propagation : integral models
The influence of a line with fast diffusion on Fisher-KPP propagation : integral models Antoine Pauthier Institut de Mathématique de Toulouse PhD supervised by Henri Berestycki (EHESS) and Jean-Michel
More informationGroup Method. December 16, Oberwolfach workshop Dynamics of Patterns
CWI, Amsterdam heijster@cwi.nl December 6, 28 Oberwolfach workshop Dynamics of Patterns Joint work: A. Doelman (CWI/UvA), T.J. Kaper (BU), K. Promislow (MSU) Outline 2 3 4 Interactions of localized structures
More informationVarying the direction of propagation in reaction-diffusion equations in periodic media
Varying the direction of propagation in reaction-diffusion equations in periodic media Matthieu Alfaro 1 and Thomas Giletti 2. Contents 1 Introduction 2 1.1 Main assumptions....................................
More informationSpectral stability of periodic waves in dispersive models
in dispersive models Collaborators: Th. Gallay, E. Lombardi T. Kapitula, A. Scheel in dispersive models One-dimensional nonlinear waves Standing and travelling waves u(x ct) with c = 0 or c 0 periodic
More informationAn Introduction to Stability Theory for Nonlinear PDEs
An Introduction to Stability Theory for Nonlinear PDEs Mathew A. Johnson 1 Abstract These notes were prepared for the 217 Participating School in Analysis of PDE: Stability of Solitons and Periodic Waves
More informationStability Analysis of Stationary Solutions for the Cahn Hilliard Equation
Stability Analysis of Stationary Solutions for the Cahn Hilliard Equation Peter Howard, Texas A&M University University of Louisville, Oct. 19, 2007 References d = 1: Commun. Math. Phys. 269 (2007) 765
More informationGLOBAL STABILITY OF CRITICAL TRAVELING WAVES WITH OSCILLATIONS FOR TIME-DELAYED REACTION-DIFFUSION EQUATIONS
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 16, Number 3, Pages 375 397 c 2019 Institute for Scientific Computing and Information GLOBAL STABILITY OF CRITICAL TRAVELING WAVES WITH OSCILLATIONS
More informationDispersion relations, linearization and linearized dynamics in PDE models
Dispersion relations, linearization and linearized dynamics in PDE models 1 Dispersion relations Suppose that u(x, t) is a function with domain { < x 0}, and it satisfies a linear, constant coefficient
More informationMonostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space
Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space Yoshihisa Morita and Hirokazu Ninomiya Department of Applied Mathematics and Informatics Ryukoku
More informationStochastic Homogenization for Reaction-Diffusion Equations
Stochastic Homogenization for Reaction-Diffusion Equations Jessica Lin McGill University Joint Work with Andrej Zlatoš June 18, 2018 Motivation: Forest Fires ç ç ç ç ç ç ç ç ç ç Motivation: Forest Fires
More informationPeriodic Schrödinger operators with δ -potentials
Networ meeting April 23-28, TU Graz Periodic Schrödinger operators with δ -potentials Andrii Khrabustovsyi Institute for Analysis, Karlsruhe Institute of Technology, Germany CRC 1173 Wave phenomena: analysis
More informationSlow Modulation & Large-Time Dynamics Near Periodic Waves
Slow Modulation & Large-Time Dynamics Near Periodic Waves Miguel Rodrigues IRMAR Université Rennes 1 France SIAG-APDE Prize Lecture Jointly with Mathew Johnson (Kansas), Pascal Noble (INSA Toulouse), Kevin
More informationSpeed Selection in Coupled Fisher Waves
Speed Selection in Coupled Fisher Waves Martin R. Evans SUPA, School of Physics and Astronomy, University of Edinburgh, U.K. September 13, 2013 Collaborators: Juan Venegas-Ortiz, Rosalind Allen Plan: Speed
More informationThe Evans function and the stability of travelling waves
The Evans function and the stability of travelling waves Jitse Niesen (University of Leeds) Collaborators: Veerle Ledoux (Ghent) Simon Malham (Heriot Watt) Vera Thümmler (Bielefeld) PANDA meeting, University
More informationIntroduction LECTURE 1
LECTURE 1 Introduction The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in
More informationSEMIGROUP APPROACH FOR PARTIAL DIFFERENTIAL EQUATIONS OF EVOLUTION
SEMIGROUP APPROACH FOR PARTIAL DIFFERENTIAL EQUATIONS OF EVOLUTION Istanbul Kemerburgaz University Istanbul Analysis Seminars 24 October 2014 Sabanc University Karaköy Communication Center 1 2 3 4 5 u(x,
More informationBistable entire solutions in cylinder-like domains
Bistable entire solutions in cylinder-like domains 3 ème journées de l ANR NonLocal Antoine Pauthier Institut de Mathématique de Toulouse Thèse dirigée par Henri Berestycki (EHESS) et Jean-Michel Roquejoffre
More informationSHARP ESTIMATE OF THE SPREADING SPEED DETERMINED BY NONLINEAR FREE BOUNDARY PROBLEMS
SHARP ESTIMATE OF THE SPREADING SPEED DETERMINED BY NONLINEAR FREE BOUNDARY PROBLEMS YIHONG DU, HIROSHI MATSUZAWA AND MAOLIN ZHOU Abstract. We study nonlinear diffusion problems of the form u t = u xx
More informationSemigroup Generation
Semigroup Generation Yudi Soeharyadi Analysis & Geometry Research Division Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung WIDE-Workshoop in Integral and Differensial Equations 2017
More information1 Existence of Travelling Wave Fronts for a Reaction-Diffusion Equation with Quadratic- Type Kinetics
1 Existence of Travelling Wave Fronts for a Reaction-Diffusion Equation with Quadratic- Type Kinetics Theorem. Consider the equation u t = Du xx + f(u) with f(0) = f(1) = 0, f(u) > 0 on 0 < u < 1, f (0)
More informationEvans function review
Evans function review Part I: History, construction, properties and applications Strathclyde, April 18th 2005 Simon J.A. Malham http://www.ma.hw.ac.uk/ simonm/talks/ Acknowledgments T.J. Bridges, C.K.R.T.
More informationStability of an abstract wave equation with delay and a Kelvin Voigt damping
Stability of an abstract wave equation with delay and a Kelvin Voigt damping University of Monastir/UPSAY/LMV-UVSQ Joint work with Serge Nicaise and Cristina Pignotti Outline 1 Problem The idea Stability
More informationPositive Stabilization of Infinite-Dimensional Linear Systems
Positive Stabilization of Infinite-Dimensional Linear Systems Joseph Winkin Namur Center of Complex Systems (NaXys) and Department of Mathematics, University of Namur, Belgium Joint work with Bouchra Abouzaid
More informationProblem (p.613) Determine all solutions, if any, to the boundary value problem. y + 9y = 0; 0 < x < π, y(0) = 0, y (π) = 6,
Problem 10.2.4 (p.613) Determine all solutions, if any, to the boundary value problem y + 9y = 0; 0 < x < π, y(0) = 0, y (π) = 6, by first finding a general solution to the differential equation. Solution.
More informationNonlinear convective stability of travelling fronts near Turing and Hopf instabilities
Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities Margaret Beck Joint work with Anna Ghazaryan, University of Kansas and Björn Sandstede, Brown University September
More informationFront propagation directed by a line of fast diffusion : existence of travelling waves.
Front propagation directed by a line of fast diffusion : existence of travelling waves. Laurent Dietrich Ph.D supervised by H. Berestycki and J.-M. Roquejoffre Institut de mathématiques de Toulouse Workshop
More informationAsymptotic stability of the critical Fisher-KPP front using pointwise estimates
Asymptotic stability of the critical Fisher-KPP front using pointwise estimates Grégory Faye and Matt Holzer 2 CNS, UM 529, Institut de Mathématiques de Toulouse, 3062 Toulouse Cede, France 2 Department
More informationTRAVELLING WAVES. Morteza Fotouhi Sharif Univ. of Technology
TRAVELLING WAVES Morteza Fotohi Sharif Univ. of Technology Mini Math NeroScience Mini Math NeroScience Agst 28 REACTION DIFFUSION EQUATIONS U = DU + f ( U ) t xx x t > U n D d 1 = d j > d n 2 Travelling
More informationTraveling Waves and Steady States of S-K-T Competition Model
Traveling Waves and Steady States of S-K-T Competition Model with Cross diffusion Capital Normal University, Beijing, China (joint work with Wei-Ming Ni, Qian Xu, Xuefeng Wang and Yanxia Wu) 2015 KAIST
More informationASYMPTOTIC SPEED OF SPREAD AND TRAVELING WAVES FOR A NONLOCAL EPIDEMIC MODEL. Dashun Xu and Xiao-Qiang Zhao. (Communicated by Hal Smith)
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS SERIES B Volume 5, Number 4, November 25 pp. 143 156 ASYMPTOTIC SPEED OF SPREAD AND TRAVELING WAVES FOR A NONLOCAL EPIDEMIC MODEL
More informationMath 337, Summer 2010 Assignment 5
Math 337, Summer Assignment 5 Dr. T Hillen, University of Alberta Exercise.. Consider Laplace s equation r r r u + u r r θ = in a semi-circular disk of radius a centered at the origin with boundary conditions
More informationDiagonalization of the Coupled-Mode System.
Diagonalization of the Coupled-Mode System. Marina Chugunova joint work with Dmitry Pelinovsky Department of Mathematics, McMaster University, Canada Collaborators: Mason A. Porter, California Institute
More informationQualitative properties of monostable pulsating fronts : exponential decay and monotonicity
Qualitative properties of monostable pulsating fronts : exponential decay and monotonicity François Hamel Université Aix-Marseille III, LATP, Faculté des Sciences et Techniques Avenue Escadrille Normandie-Niemen,
More informationKPP Pulsating Traveling Fronts within Large Drift
KPP Pulsating Traveling Fronts within Large Drift Mohammad El Smaily Joint work with Stéphane Kirsch University of British olumbia & Pacific Institute for the Mathematical Sciences September 17, 2009 PIMS
More informationGENERALIZED FRONTS FOR ONE-DIMENSIONAL REACTION-DIFFUSION EQUATIONS
GENERALIZED FRONTS FOR ONE-DIMENSIONAL REACTION-DIFFUSION EQUATIONS ANTOINE MELLET, JEAN-MICHEL ROQUEJOFFRE, AND YANNICK SIRE Abstract. For a class of one-dimensional reaction-diffusion equations, we establish
More informationAuthor(s) Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 41(1)
Title On the stability of contact Navier-Stokes equations with discont free b Authors Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 4 Issue 4-3 Date Text Version publisher URL
More informationTravelling waves. Chapter 8. 1 Introduction
Chapter 8 Travelling waves 1 Introduction One of the cornerstones in the study of both linear and nonlinear PDEs is the wave propagation. A wave is a recognizable signal which is transferred from one part
More informationTwo examples of reaction-diffusion front propagation in heterogeneous media
Two examples of reaction-diffusion front propagation in heterogeneous media Soutenance de thèse Antoine Pauthier Institut de Mathématique de Toulouse Thèse dirigée par Henri Berestycki (EHESS) et Jean-Michel
More informationThe circular law. Lewis Memorial Lecture / DIMACS minicourse March 19, Terence Tao (UCLA)
The circular law Lewis Memorial Lecture / DIMACS minicourse March 19, 2008 Terence Tao (UCLA) 1 Eigenvalue distributions Let M = (a ij ) 1 i n;1 j n be a square matrix. Then one has n (generalised) eigenvalues
More informationUPPER AND LOWER SOLUTIONS FOR A HOMOGENEOUS DIRICHLET PROBLEM WITH NONLINEAR DIFFUSION AND THE PRINCIPLE OF LINEARIZED STABILITY
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 30, Number 4, Winter 2000 UPPER AND LOWER SOLUTIONS FOR A HOMOGENEOUS DIRICHLET PROBLEM WITH NONLINEAR DIFFUSION AND THE PRINCIPLE OF LINEARIZED STABILITY ROBERT
More informationTraveling waves of a kinetic transport model for the KPP-Fisher equation
Traveling waves of a kinetic transport model for the KPP-Fisher equation Christian Schmeiser Universität Wien and RICAM homepage.univie.ac.at/christian.schmeiser/ Joint work with C. Cuesta (Bilbao), S.
More informationLecture 15: Biological Waves
Lecture 15: Biological Waves Jonathan A. Sherratt Contents 1 Wave Fronts I: Modelling Epidermal Wound Healing 2 1.1 Epidermal Wound Healing....................... 2 1.2 A Mathematical Model.........................
More informationDynamics of weakly interacting front and back waves in three-component systems
Toyama Math. J. Vol. 3(27), 1-34 Dynamics of weakly interacting front and back waves in three-component systems Hideo Ikeda Abstract. Weak interaction of stable travelling front and back solutions is considered
More informationNonlinear Modulational Instability of Dispersive PDE Models
Nonlinear Modulational Instability of Dispersive PDE Models Jiayin Jin, Shasha Liao, and Zhiwu Lin Georgia Tech ICERM workshop on water waves, 4/28/2017 Jiayin Jin, Shasha Liao, and Zhiwu Lin Georgia Tech
More informationarxiv: v1 [math.ap] 5 Feb 2015
Spectra and stability of spatially periodic pulse patterns: Evans function factorization via Riccati transformation Björn de Rijk brijk@math.leidenuniv.nl Jens Rademacher rademach@math.uni-bremen.de Arjen
More informationANDREJ ZLATOŠ. u 0 and u dx = 0. (1.3)
SHARP ASYMPOICS FOR KPP PULSAING FRON SPEED-UP AND DIFFUSION ENHANCEMEN BY FLOWS ANDREJ ZLAOŠ Abstract. We study KPP pulsating front speed-up and effective diffusivity enhancement by general periodic incompressible
More informationOn Solutions of Evolution Equations with Proportional Time Delay
On Solutions of Evolution Equations with Proportional Time Delay Weijiu Liu and John C. Clements Department of Mathematics and Statistics Dalhousie University Halifax, Nova Scotia, B3H 3J5, Canada Fax:
More informationarxiv: v1 [nlin.ps] 18 Sep 2008
Asymptotic two-soliton solutions solutions in the Fermi-Pasta-Ulam model arxiv:0809.3231v1 [nlin.ps] 18 Sep 2008 Aaron Hoffman and C.E. Wayne Boston University Department of Mathematics and Statistics
More informationMemoirs on Differential Equations and Mathematical Physics
Memoirs on Differential Equations and Mathematical Physics Volume 51, 010, 93 108 Said Kouachi and Belgacem Rebiai INVARIANT REGIONS AND THE GLOBAL EXISTENCE FOR REACTION-DIFFUSION SYSTEMS WITH A TRIDIAGONAL
More informationFast propagation for KPP equations with slowly decaying initial conditions
Fast propagation for KPP equations with slowly decaying initial conditions arxiv:0906.3164v1 [math.ap] 17 Jun 2009 François Hamel a and Lionel Roques b a Aix-Marseille Université, LATP, Faculté des Sciences
More informationAPPROXIMATING TRAVELLING WAVES BY EQUILIBRIA OF NON LOCAL EQUATIONS
APPROXIMATING TRAVELLING WAVES BY EQUILIBRIA OF NON LOCAL EQUATIONS JOSE M. ARRIETA, MARÍA LÓPEZ-FERNÁNDEZ, AND ENRIQUE ZUAZUA Abstract. We consider an evolution equation of parabolic type in R having
More informationsystem CWI, Amsterdam May 21, 2008 Dynamic Analysis Seminar Vrije Universiteit
CWI, Amsterdam heijster@cwi.nl May 21, 2008 Dynamic Analysis Seminar Vrije Universiteit Joint work: A. Doelman (CWI/UvA), T.J. Kaper (BU), K. Promislow (MSU) Outline 1 2 3 4 Outline 1 2 3 4 Paradigm U
More informationOn the Standard Linear Viscoelastic model
On the Standard Linear Viscoelastic model M. Pellicer (Universitat de Girona) Work in collaboration with: J. Solà-Morales (Universitat Politècnica de Catalunya) (bounded problem) B. Said-Houari (Alhosn
More informationMS: Nonlinear Wave Propagation in Singular Perturbed Systems
MS: Nonlinear Wave Propagation in Singular Perturbed Systems P. van Heijster: Existence & stability of 2D localized structures in a 3-component model. Y. Nishiura: Rotational motion of traveling spots
More informationThe KPP minimal speed within large drift in two dimensions
The KPP minimal speed within large drift in two dimensions Mohammad El Smaily Joint work with Stéphane Kirsch University of British Columbia & Pacific Institute for the Mathematical Sciences Banff, March-2010
More information7 Hyperbolic Differential Equations
Numerical Analysis of Differential Equations 243 7 Hyperbolic Differential Equations While parabolic equations model diffusion processes, hyperbolic equations model wave propagation and transport phenomena.
More informationLIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 21(21), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LIFE SPAN OF BLOW-UP
More informationSpatial decay of rotating waves in parabolic systems
Spatial decay of rotating waves in parabolic systems Nonlinear Waves, CRC 701, Bielefeld, June 19, 2013 Denny Otten Department of Mathematics Bielefeld University Germany June 19, 2013 CRC 701 Denny Otten
More informationContinuous dependence estimates for the ergodic problem with an application to homogenization
Continuous dependence estimates for the ergodic problem with an application to homogenization Claudio Marchi Bayreuth, September 12 th, 2013 C. Marchi (Università di Padova) Continuous dependence Bayreuth,
More informationFORCED OSCILLATIONS OF A CLASS OF NONLINEAR DISPERSIVE WAVE EQUATIONS AND THEIR STABILITY
Jrl Syst Sci & Complexity (2007) 20: 284 292 FORCED OSCILLATIONS OF A CLASS OF NONLINEAR DISPERSIVE WAVE EQUATIONS AND THEIR STABILITY Muhammad USMAN Bingyu ZHANG Received: 14 January 2007 Abstract It
More informationNonlinear stability of semidiscrete shocks for two-sided schemes
Nonlinear stability of semidiscrete shocks for two-sided schemes Margaret Beck Boston University Joint work with Hermen Jan Hupkes, Björn Sandstede, and Kevin Zumbrun Setting: semi-discrete conservation
More informationHardy inequalities, heat kernels and wave propagation
Outline Hardy inequalities, heat kernels and wave propagation Basque Center for Applied Mathematics (BCAM) Bilbao, Basque Country, Spain zuazua@bcamath.org http://www.bcamath.org/zuazua/ Third Brazilian
More informationOn Pressure Stabilization Method and Projection Method for Unsteady Navier-Stokes Equations 1
On Pressure Stabilization Method and Projection Method for Unsteady Navier-Stokes Equations 1 Jie Shen Department of Mathematics, Penn State University University Park, PA 1682 Abstract. We present some
More informationFronts for Periodic KPP Equations
Fronts for Periodic KPP Equations Steffen Heinze Bioquant University of Heidelberg September 15th 2010 ontent Fronts and Asymptotic Spreading Variational Formulation for the Speed Qualitative onsequences
More informationRational Chebyshev pseudospectral method for long-short wave equations
Journal of Physics: Conference Series PAPER OPE ACCESS Rational Chebyshev pseudospectral method for long-short wave equations To cite this article: Zeting Liu and Shujuan Lv 07 J. Phys.: Conf. Ser. 84
More informationDeforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary
Deforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary Weimin Sheng (Joint with Li-Xia Yuan) Zhejiang University IMS, NUS, 8-12 Dec 2014 1 / 50 Outline 1 Prescribing
More informationStability and instability of nonlinear waves:
Stability and instability of nonlinear waves: Introduction 1. Nonlinear Waves 2. Stability problems 3. Stability of pulses and fronts 4. Stability of periodic waves 1 Nonlinear waves particular solutions
More informationKAM for quasi-linear KdV
KAM for quasi-linear KdV Massimiliano Berti ST Etienne de Tinée, 06-02-2014 KdV t u + u xxx 3 x u 2 + N 4 (x, u, u x, u xx, u xxx ) = 0, x T Quasi-linear Hamiltonian perturbation N 4 := x {( u f )(x, u,
More informationParameter Dependent Quasi-Linear Parabolic Equations
CADERNOS DE MATEMÁTICA 4, 39 33 October (23) ARTIGO NÚMERO SMA#79 Parameter Dependent Quasi-Linear Parabolic Equations Cláudia Buttarello Gentile Departamento de Matemática, Universidade Federal de São
More informationAvailable online at J. Math. Comput. Sci. 2 (2012), No. 1, ISSN:
Available online at http://scik.org J. Math. Comput. Sci. 2 (2012), No. 1, 15-22 ISSN: 1927-5307 BRIGHT AND DARK SOLITON SOLUTIONS TO THE OSTROVSKY-BENJAMIN-BONA-MAHONY (OS-BBM) EQUATION MARWAN ALQURAN
More informationEntire solutions of the Fisher-KPP equation in time periodic media
Dynamics of PDE, Vol.9, No.2, 133-145, 2012 Entire solutions of the Fisher-KPP equation in time periodic media Wei-Jie Sheng and Mei-Ling Cao Communicated by Y. Charles Li, received September 11, 2011.
More informationBreakdown of Pattern Formation in Activator-Inhibitor Systems and Unfolding of a Singular Equilibrium
Breakdown of Pattern Formation in Activator-Inhibitor Systems and Unfolding of a Singular Equilibrium Izumi Takagi (Mathematical Institute, Tohoku University) joint work with Kanako Suzuki (Institute for
More informationInstability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs
Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs Zhiwu Lin and Chongchun Zeng School of Mathematics Georgia Institute of Technology Atlanta, GA 30332, USA Abstract Consider
More informationSuperlinear Parabolic Problems
Birkhäuser Advanced Texts Basler Lehrbücher Superlinear Parabolic Problems Blow-up, Global Existence and Steady States Bearbeitet von Pavol Quittner, Philippe Souplet 1. Auflage 2007. Buch. xii, 584 S.
More informationMotivation Power curvature flow Large exponent limit Analogues & applications. Qing Liu. Fukuoka University. Joint work with Prof.
On Large Exponent Behavior of Power Curvature Flow Arising in Image Processing Qing Liu Fukuoka University Joint work with Prof. Naoki Yamada Mathematics and Phenomena in Miyazaki 2017 University of Miyazaki
More informationA review of stability and dynamical behaviors of differential equations:
A review of stability and dynamical behaviors of differential equations: scalar ODE: u t = f(u), system of ODEs: u t = f(u, v), v t = g(u, v), reaction-diffusion equation: u t = D u + f(u), x Ω, with boundary
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 1. Linear systems Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Linear systems 1.1 Differential Equations 1.2 Linear flows 1.3 Linear maps
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
More informationResearch Article On Existence, Uniform Decay Rates, and Blow-Up for Solutions of a Nonlinear Wave Equation with Dissipative and Source
Abstract and Applied Analysis Volume, Article ID 65345, 7 pages doi:.55//65345 Research Article On Existence, Uniform Decay Rates, and Blow-Up for Solutions of a Nonlinear Wave Equation with Dissipative
More informationThe comparison of optimal homotopy asymptotic method and homotopy perturbation method to solve Fisher equation
Computational Methods for Differential Equations http://cmdetabrizuacir Vol 4, No, 206, pp 43-53 The comparison of optimal homotopy asymptotic method and homotopy perturbation method to solve Fisher equation
More informationExistence of Positive Solutions of Fisher-KPP Equations in Locally Spatially Variational Habitat with Hybrid Dispersal
Journal of Mathematics Research; Vol. 9, No. 1; February 2017 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science Education Existence of Positive Solutions of Fisher-KPP Equations in
More informationRapid travelling waves in the nonlocal Fisher equation connect two unstable states
apid travelling waves in the nonlocal Fisher equation connect two unstable states Matthieu Alfaro a, Jérôme Coville b a I3M, Université Montpellier 2, CC51, Place Eugène Bataillon, 3495 Montpellier Cedex
More informationLecture 18: Bistable Fronts PHYS 221A, Spring 2017
Lecture 18: Bistable Fronts PHYS 221A, Spring 2017 Lectures: P. H. Diamond Notes: Xiang Fan June 15, 2017 1 Introduction In the previous lectures, we learned about Turing Patterns. Turing Instability is
More informationCoherent structures near the boundary between excitable and oscillatory media
Coherent structures near the boundary between excitable and oscillatory media Jeremy Bellay University of Minnesota Department of Computer Science 200 Union St. S.E. Minneapolis, MN 55455, USA Arnd Scheel
More informationNODAL PROPERTIES FOR p-laplacian SYSTEMS
Electronic Journal of Differential Equations, Vol. 217 (217), No. 87, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NODAL PROPERTIES FOR p-laplacian SYSTEMS YAN-HSIOU
More informationThe speed of propagation for KPP type problems. II - General domains
The speed of propagation for KPP type problems. II - General domains Henri Berestycki a, François Hamel b and Nikolai Nadirashvili c a EHESS, CAMS, 54 Boulevard Raspail, F-75006 Paris, France b Université
More informationA quantitative Fattorini-Hautus test: the minimal null control time problem in the parabolic setting
A quantitative Fattorini-Hautus test: the minimal null control time problem in the parabolic setting Morgan MORANCEY I2M, Aix-Marseille Université August 2017 "Controllability of parabolic equations :
More informationAn unfortunate misprint
An unfortunate misprint Robert L. Jerrard Department of Mathematics University of Toronto March 22, 21, BIRS Robert L. Jerrard (Toronto ) An unfortunate misprint March 22, 21, BIRS 1 / 12 Consider the
More informationLecture17: Generalized Solitary Waves
Lecture17: Generalized Solitary Waves Lecturer: Roger Grimshaw. Write-up: Andrew Stewart and Yiping Ma June 24, 2009 We have seen that solitary waves, either with a pulse -like profile or as the envelope
More informationA REMARK ON THE GLOBAL DYNAMICS OF COMPETITIVE SYSTEMS ON ORDERED BANACH SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 A REMARK ON THE GLOBAL DYNAMICS OF COMPETITIVE SYSTEMS ON ORDERED BANACH SPACES KING-YEUNG LAM
More informationExact controllability of the superlinear heat equation
Exact controllability of the superlinear heat equation Youjun Xu 1,2, Zhenhai Liu 1 1 School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410075, P R China
More informationReaction-Diffusion Equations In Narrow Tubes and Wave Front P
Outlines Reaction-Diffusion Equations In Narrow Tubes and Wave Front Propagation University of Maryland, College Park USA Outline of Part I Outlines Real Life Examples Description of the Problem and Main
More information1 Sectorial operators
1 1 Sectorial operators Definition 1.1 Let X and A : D(A) X X be a Banach space and a linear closed operator, respectively. If the relationships i) ρ(a) Σ φ = {λ C : arg λ < φ}, where φ (π/2, π); ii) R(λ,
More informationMath 46, Applied Math (Spring 2009): Final
Math 46, Applied Math (Spring 2009): Final 3 hours, 80 points total, 9 questions worth varying numbers of points 1. [8 points] Find an approximate solution to the following initial-value problem which
More informationModeling II Linear Stability Analysis and Wave Equa9ons
Modeling II Linear Stability Analysis and Wave Equa9ons Nondimensional Equa9ons From previous lecture, we have a system of nondimensional PDEs: (21.1) (21.2) (21.3) where here the * sign has been dropped
More informationWave propagation in optical waveguides
Wave propagation in optical waveguides Giulio Ciraolo November, 005 Abstract We present a mathematical framework for studying the problem of electromagnetic wave propagation in a -D or 3-D optical waveguide
More informationA new integrable system: The interacting soliton of the BO
Phys. Lett., A 204, p.336-342, 1995 A new integrable system: The interacting soliton of the BO Benno Fuchssteiner and Thorsten Schulze Automath Institute University of Paderborn Paderborn & Germany Abstract
More informationSome recent results on controllability of coupled parabolic systems: Towards a Kalman condition
Some recent results on controllability of coupled parabolic systems: Towards a Kalman condition F. Ammar Khodja Clermont-Ferrand, June 2011 GOAL: 1 Show the important differences between scalar and non
More information1 Introduction We will consider traveling waves for reaction-diusion equations (R-D) u t = nx i;j=1 (a ij (x)u xi ) xj + f(u) uj t=0 = u 0 (x) (1.1) w
Reaction-Diusion Fronts in Periodically Layered Media George Papanicolaou and Xue Xin Courant Institute of Mathematical Sciences 251 Mercer Street, New York, N.Y. 10012 Abstract We compute the eective
More information