# American Institute of Mathematical Sciences

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, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707
##### References:
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##### References:
 [1] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, in "Partial Differential Equations and Related Topics" (ed. J. A. Goldstein), Lecture Notes in Mathematics, 446 (1975), 5-49.  Google Scholar [2] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar [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), 141-157. doi: 10.1512/iumj.1981.30.30012.  Google Scholar [4] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar [5] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556.  Google Scholar [6] S.-I. Ei, M. Mimura and M. Nagayama, Interacting spots in reaction diffusion systems, Discrete Contin. Dyn. Syst., 14 (2006), 31-62.  Google Scholar [7] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar [8] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7 (1981), 369-402, Academic Press, New York.  Google Scholar [9] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.  Google Scholar [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), 920-941. doi: 10.1137/S003613999630664X.  Google Scholar [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), 1-25. Google Scholar [12] M.-K. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.  Google Scholar [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), 1043-1054. doi: 10.1080/03605309308820960.  Google Scholar [14] Z. Nehari, On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish Acad. Sect. A, 62 (1963), 117-135.  Google Scholar [15] Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering of traveling spots in dissipative systems, Chaos, 15 (2005), 047509, 10 pp. doi: 10.1063/1.2087127.  Google Scholar [16] L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $R^n$, Arch. Rational Mech. Anal., 81 (1983), 181-197. doi: 10.1007/BF00250651.  Google Scholar [17] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, Berlin, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar [18] M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations equations, Discrete Contin. Dyn. Syst., 32 (2012), 1011-1046. doi: 10.3934/dcds.2012.32.1011.  Google Scholar [19] E. Yanagida, Uniqueness of positive radial solutions of $\Delta u+f(u,|x|)=0$, Nonlinear Anal., 19 (1992), 1143-1154. doi: 10.1016/0362-546X(92)90188-K.  Google Scholar
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