August  2013, 33(8): 3707-3718. doi: 10.3934/dcds.2013.33.3707

Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations

1. 

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejon 305-701, South Korea

2. 

Center for Partial Di erential Equations, East China Normal University, Minhang, Shanghai, 200241, China

3. 

Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-38 Ookayama, Meguro-ku, Tokyo 152-8552, Japan

Received  April 2012 Revised  September 2012 Published  January 2013

Assume a single reaction-diffusion equation has zero as an asymptotically stable stationary point. Then we prove that there exist no localized travelling waves with non-zero speed. If $[\liminf_{|x|\to\infty}u(x),\limsup_{|x|\to\infty}u(x)]$ is included in an open interval of zero that does not include other stationary points, then the speed has to be zero or the travelling profile $u$ has to be identically zero.
Citation: Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707
References:
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D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics,, in, 446 (1975), 5.

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5.

[3]

H. Berestycki, P.-L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $R^N$,, Indiana Univ. Math. J., 30 (1981), 141. doi: 10.1512/iumj.1981.30.30012.

[4]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555.

[5]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions,, Arch. Rational Mech. Anal., 82 (1983), 347. doi: 10.1007/BF00250556.

[6]

S.-I. Ei, M. Mimura and M. Nagayama, Interacting spots in reaction diffusion systems,, Discrete Contin. Dyn. Syst., 14 (2006), 31.

[7]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.

[8]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, Mathematical Analysis and Applications, 7 (1981), 369.

[9]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1983).

[10]

S. Kawaguchi and M. Mimura, Collision of travelling waves in a reaction-diffusion system with global coupling effect,, SIAM J. Appl. Math., 59 (1999), 920. doi: 10.1137/S003613999630664X.

[11]

A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem,, Moscow University Bulletin of Mathematics, 1 (1937), 1.

[12]

M.-K. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $R^n$,, Arch. Rational Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502.

[13]

Y. Li and W.-M. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, Comm. Partial Differential Equations, 18 (1993), 1043. doi: 10.1080/03605309308820960.

[14]

Z. Nehari, On a nonlinear differential equation arising in nuclear physics,, Proc. Roy. Irish Acad. Sect. A, 62 (1963), 117.

[15]

Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering of traveling spots in dissipative systems,, Chaos, 15 (2005). doi: 10.1063/1.2087127.

[16]

L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $R^n$,, Arch. Rational Mech. Anal., 81 (1983), 181. doi: 10.1007/BF00250651.

[17]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (1984). doi: 10.1007/978-1-4612-5282-5.

[18]

M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations equations,, Discrete Contin. Dyn. Syst., 32 (2012), 1011. doi: 10.3934/dcds.2012.32.1011.

[19]

E. Yanagida, Uniqueness of positive radial solutions of $\Delta u+f(u,|x|)=0$,, Nonlinear Anal., 19 (1992), 1143. doi: 10.1016/0362-546X(92)90188-K.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics,, in, 446 (1975), 5.

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5.

[3]

H. Berestycki, P.-L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $R^N$,, Indiana Univ. Math. J., 30 (1981), 141. doi: 10.1512/iumj.1981.30.30012.

[4]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555.

[5]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions,, Arch. Rational Mech. Anal., 82 (1983), 347. doi: 10.1007/BF00250556.

[6]

S.-I. Ei, M. Mimura and M. Nagayama, Interacting spots in reaction diffusion systems,, Discrete Contin. Dyn. Syst., 14 (2006), 31.

[7]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.

[8]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, Mathematical Analysis and Applications, 7 (1981), 369.

[9]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1983).

[10]

S. Kawaguchi and M. Mimura, Collision of travelling waves in a reaction-diffusion system with global coupling effect,, SIAM J. Appl. Math., 59 (1999), 920. doi: 10.1137/S003613999630664X.

[11]

A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem,, Moscow University Bulletin of Mathematics, 1 (1937), 1.

[12]

M.-K. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $R^n$,, Arch. Rational Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502.

[13]

Y. Li and W.-M. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, Comm. Partial Differential Equations, 18 (1993), 1043. doi: 10.1080/03605309308820960.

[14]

Z. Nehari, On a nonlinear differential equation arising in nuclear physics,, Proc. Roy. Irish Acad. Sect. A, 62 (1963), 117.

[15]

Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering of traveling spots in dissipative systems,, Chaos, 15 (2005). doi: 10.1063/1.2087127.

[16]

L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $R^n$,, Arch. Rational Mech. Anal., 81 (1983), 181. doi: 10.1007/BF00250651.

[17]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (1984). doi: 10.1007/978-1-4612-5282-5.

[18]

M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations equations,, Discrete Contin. Dyn. Syst., 32 (2012), 1011. doi: 10.3934/dcds.2012.32.1011.

[19]

E. Yanagida, Uniqueness of positive radial solutions of $\Delta u+f(u,|x|)=0$,, Nonlinear Anal., 19 (1992), 1143. doi: 10.1016/0362-546X(92)90188-K.

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