May  2021, 41(5): 2411-2445. doi: 10.3934/dcds.2020370

Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations

Teacher Training Courses, Faculty of Education, Kagawa University, Takamatsu, Kagawa, 760-8522, Japan

Received  April 2020 Revised  September 2020 Published  November 2020

Fund Project: The author is supported by JSPS KAKENHI Grant Numbers 19K14580 and the Overseas Research Fellowship Program by National Institute of Technology

This paper is concerned with strong blow-up instability (Definition 1.3) for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. In the single case, namely the nonlinear Klein-Gordon equation with power type nonlinearity, stability and instability for standing wave solutions have been extensively studied. On the other hand, in the case of our system, there are no results concerning the stability and instability as far as we know.

In this paper, we prove strong blow-up instability for the standing wave to our system. The proof is based on the techniques in Ohta and Todorova [27]. It turns out that we need the mass resonance condition in two or three space dimensions whose cases are the mass-subcritical case.

Citation: Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math., 293 (1981), 489-492.   Google Scholar

[3]

H. Brezis and E. H. Lieb, Minimum action solutions of some vector field equations, Comm. Math. Phys., 96 (1984), 97–113. http://projecteuclid.org/euclid.cmp/1103941720. doi: 10.1007/BF01217349.  Google Scholar

[4]

J. ByeonL. Jeanjean and M. Mariș, Symmetry and monotonicity of least energy solutions, Calc. Var. Partial Differential Equations, 36 (2009), 481-492.  doi: 10.1007/s00526-009-0238-1.  Google Scholar

[5]

T. Cazenave, Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60 (1985), 36-55.  doi: 10.1016/0022-1236(85)90057-6.  Google Scholar

[6]

T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[7]

A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math., 56 (2003), 1565-1607.  doi: 10.1002/cpa.10104.  Google Scholar

[8]

V. D. Dinh, Existence, stability of standing waves and the characterization of finite time blow-up solutions for a system NLS with quadratic interaction, Nonlinear Anal., 190 (2020), 111589, 39 pp. doi: 10.1016/j.na.2019.111589.  Google Scholar

[9]

V. D. Dinh, Strong instability of standing waves for a system NLS with quadratic interaction, Acta Math. Sci. Ser. B (Engl. Ed.), 40 (2020), 515-528.  doi: 10.1007/s10473-020-0214-6.  Google Scholar

[10]

D. Garrisi, On the orbital stability of standing-wave solutions to a coupled non-linear Klein-Gordon equation, Adv. Nonlinear Stud., 12 (2012), 639-658.  doi: 10.1515/ans-2012-0311.  Google Scholar

[11]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅰ, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[12]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅱ, J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[13]

M. Hamano, Global dynamics below the ground state for the quadratic Schrödinger system in 5d, preprint, https://arXiv.org/abs/1805.12245. Google Scholar

[14]

N. HayashiM. Ikeda and P. I. Naumkin, Wave operator for the system of the Dirac-Klein-Gordon equations, Math. Methods Appl. Sci., 34 (2011), 896-910.  doi: 10.1002/mma.1409.  Google Scholar

[15]

N. HayashiT. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690.  doi: 10.1016/j.anihpc.2012.10.007.  Google Scholar

[16]

L. Jeanjean and M. Squassina, Existence and symmetry of least energy solutions for a class of quasi-linear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1701-1716.  doi: 10.1016/j.anihpc.2008.11.003.  Google Scholar

[17]

Y. Kawahara and H. Sunagawa, Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance, J. Differential Equations, 251 (2011), 2549-2567.  doi: 10.1016/j.jde.2011.04.001.  Google Scholar

[18]

E. H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448.  doi: 10.1007/BF01394245.  Google Scholar

[19]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[20]

Y. LiuM. Ohta and G. Todorova, Instabilité forte d'ondes solitaires pour des équations de Klein-Gordon non linéaires et des équations généralisées de Boussinesq, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 539-548.  doi: 10.1016/j.anihpc.2006.03.005.  Google Scholar

[21]

M. Maeda, Stability of bound states of Hamiltonian PDEs in the degenerate cases, J. Funct. Anal., 263 (2012), 511-528.  doi: 10.1016/j.jfa.2012.04.006.  Google Scholar

[22]

F. Merle and H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math., 125 (2003), 1147–1164. http://muse.jhu.edu/journals/american_journal_of_mathematics/v125/125.5merle.pdf. doi: 10.1353/ajm.2003.0033.  Google Scholar

[23]

H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity, J. Math. Soc. Japan, 46 (1994), 557-586.  doi: 10.2969/jmsj/04640557.  Google Scholar

[24]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[25]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111 (1991), 487-496.  doi: 10.2307/2048340.  Google Scholar

[26]

M. Ohta and G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 12 (2005), 315-322.  doi: 10.3934/dcds.2005.12.315.  Google Scholar

[27]

M. Ohta and G. Todorova, Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system, SIAM J. Math. Anal., 38 (2007), 1912-1931.  doi: 10.1137/050643015.  Google Scholar

[28]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[29]

H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270.  doi: 10.1007/BF01181697.  Google Scholar

[30]

J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys., 91 (1983), 313–327, http://projecteuclid.org/euclid.cmp/1103940612. doi: 10.1007/BF01208779.  Google Scholar

[31]

J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710.  doi: 10.1090/S0002-9947-1985-0792821-7.  Google Scholar

[32]

J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173–190, http://projecteuclid.org/euclid.cmp/1103943442. doi: 10.1007/BF01212446.  Google Scholar

[33]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149–162, http://projecteuclid.org/euclid.cmp/1103900983. doi: 10.1007/BF01626517.  Google Scholar

[34]

H. Sunagawa, On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension, J. Differential Equations, 192 (2003), 308-325.  doi: 10.1016/S0022-0396(03)00125-6.  Google Scholar

[35]

Y. Tsutsumi, Stability of constant equilibrium for the Maxwell-Higgs equations, Funkcial. Ekvac., 46 (2003), 41-62.  doi: 10.1619/fesi.46.41.  Google Scholar

[36]

B. Wang, On existence and scattering for critical and subcritical nonlinear Klein-Gordon equations in $H^s$, Nonlinear Anal., 31 (1998), 573-587.  doi: 10.1016/S0362-546X(97)00424-0.  Google Scholar

[37]

Y. Wu, Instability of the standing waves for the nonlinear Klein-Gordon equations in one dimension, preprint, https://arXiv.org/abs/1705.04216. Google Scholar

[38]

J. ZhangZ.-h. Gan and B.-l. Guo, Stability of the standing waves for a class of coupled nonlinear Klein-Gordon equations, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 427-442.  doi: 10.1007/s10255-010-0008-z.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math., 293 (1981), 489-492.   Google Scholar

[3]

H. Brezis and E. H. Lieb, Minimum action solutions of some vector field equations, Comm. Math. Phys., 96 (1984), 97–113. http://projecteuclid.org/euclid.cmp/1103941720. doi: 10.1007/BF01217349.  Google Scholar

[4]

J. ByeonL. Jeanjean and M. Mariș, Symmetry and monotonicity of least energy solutions, Calc. Var. Partial Differential Equations, 36 (2009), 481-492.  doi: 10.1007/s00526-009-0238-1.  Google Scholar

[5]

T. Cazenave, Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60 (1985), 36-55.  doi: 10.1016/0022-1236(85)90057-6.  Google Scholar

[6]

T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[7]

A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math., 56 (2003), 1565-1607.  doi: 10.1002/cpa.10104.  Google Scholar

[8]

V. D. Dinh, Existence, stability of standing waves and the characterization of finite time blow-up solutions for a system NLS with quadratic interaction, Nonlinear Anal., 190 (2020), 111589, 39 pp. doi: 10.1016/j.na.2019.111589.  Google Scholar

[9]

V. D. Dinh, Strong instability of standing waves for a system NLS with quadratic interaction, Acta Math. Sci. Ser. B (Engl. Ed.), 40 (2020), 515-528.  doi: 10.1007/s10473-020-0214-6.  Google Scholar

[10]

D. Garrisi, On the orbital stability of standing-wave solutions to a coupled non-linear Klein-Gordon equation, Adv. Nonlinear Stud., 12 (2012), 639-658.  doi: 10.1515/ans-2012-0311.  Google Scholar

[11]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅰ, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[12]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅱ, J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[13]

M. Hamano, Global dynamics below the ground state for the quadratic Schrödinger system in 5d, preprint, https://arXiv.org/abs/1805.12245. Google Scholar

[14]

N. HayashiM. Ikeda and P. I. Naumkin, Wave operator for the system of the Dirac-Klein-Gordon equations, Math. Methods Appl. Sci., 34 (2011), 896-910.  doi: 10.1002/mma.1409.  Google Scholar

[15]

N. HayashiT. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690.  doi: 10.1016/j.anihpc.2012.10.007.  Google Scholar

[16]

L. Jeanjean and M. Squassina, Existence and symmetry of least energy solutions for a class of quasi-linear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1701-1716.  doi: 10.1016/j.anihpc.2008.11.003.  Google Scholar

[17]

Y. Kawahara and H. Sunagawa, Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance, J. Differential Equations, 251 (2011), 2549-2567.  doi: 10.1016/j.jde.2011.04.001.  Google Scholar

[18]

E. H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448.  doi: 10.1007/BF01394245.  Google Scholar

[19]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[20]

Y. LiuM. Ohta and G. Todorova, Instabilité forte d'ondes solitaires pour des équations de Klein-Gordon non linéaires et des équations généralisées de Boussinesq, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 539-548.  doi: 10.1016/j.anihpc.2006.03.005.  Google Scholar

[21]

M. Maeda, Stability of bound states of Hamiltonian PDEs in the degenerate cases, J. Funct. Anal., 263 (2012), 511-528.  doi: 10.1016/j.jfa.2012.04.006.  Google Scholar

[22]

F. Merle and H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math., 125 (2003), 1147–1164. http://muse.jhu.edu/journals/american_journal_of_mathematics/v125/125.5merle.pdf. doi: 10.1353/ajm.2003.0033.  Google Scholar

[23]

H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity, J. Math. Soc. Japan, 46 (1994), 557-586.  doi: 10.2969/jmsj/04640557.  Google Scholar

[24]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[25]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111 (1991), 487-496.  doi: 10.2307/2048340.  Google Scholar

[26]

M. Ohta and G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 12 (2005), 315-322.  doi: 10.3934/dcds.2005.12.315.  Google Scholar

[27]

M. Ohta and G. Todorova, Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system, SIAM J. Math. Anal., 38 (2007), 1912-1931.  doi: 10.1137/050643015.  Google Scholar

[28]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[29]

H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270.  doi: 10.1007/BF01181697.  Google Scholar

[30]

J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys., 91 (1983), 313–327, http://projecteuclid.org/euclid.cmp/1103940612. doi: 10.1007/BF01208779.  Google Scholar

[31]

J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710.  doi: 10.1090/S0002-9947-1985-0792821-7.  Google Scholar

[32]

J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173–190, http://projecteuclid.org/euclid.cmp/1103943442. doi: 10.1007/BF01212446.  Google Scholar

[33]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149–162, http://projecteuclid.org/euclid.cmp/1103900983. doi: 10.1007/BF01626517.  Google Scholar

[34]

H. Sunagawa, On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension, J. Differential Equations, 192 (2003), 308-325.  doi: 10.1016/S0022-0396(03)00125-6.  Google Scholar

[35]

Y. Tsutsumi, Stability of constant equilibrium for the Maxwell-Higgs equations, Funkcial. Ekvac., 46 (2003), 41-62.  doi: 10.1619/fesi.46.41.  Google Scholar

[36]

B. Wang, On existence and scattering for critical and subcritical nonlinear Klein-Gordon equations in $H^s$, Nonlinear Anal., 31 (1998), 573-587.  doi: 10.1016/S0362-546X(97)00424-0.  Google Scholar

[37]

Y. Wu, Instability of the standing waves for the nonlinear Klein-Gordon equations in one dimension, preprint, https://arXiv.org/abs/1705.04216. Google Scholar

[38]

J. ZhangZ.-h. Gan and B.-l. Guo, Stability of the standing waves for a class of coupled nonlinear Klein-Gordon equations, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 427-442.  doi: 10.1007/s10255-010-0008-z.  Google Scholar

[1]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448

[2]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[3]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166

[4]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

[5]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

[6]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[7]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[8]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427

[9]

Dmitry Treschev. Travelling waves in FPU lattices. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 867-880. doi: 10.3934/dcds.2004.11.867

[10]

Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565

[11]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233

[12]

Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065

[13]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035

[14]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009

[15]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[16]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

[17]

Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051

[18]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[19]

Pascal Noble, Sebastien Travadel. Non-persistence of roll-waves under viscous perturbations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 61-70. doi: 10.3934/dcdsb.2001.1.61

[20]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (72)
  • HTML views (140)
  • Cited by (0)

Other articles
by authors

[Back to Top]