##
**Travelling wave solutions of a fourth-order semilinear diffusion equation.**
*(English)*
Zbl 0932.35017

We are interested in traveling wave solutions \(u= u(x,t)\) of the fourth-order equation
\[
{\partial u\over\partial t}= -\gamma{\partial^4u\over\partial x^4}+ {\partial^2u\over\partial x^2}+ f(u),\quad f(u)= (u-a)(1- u^2),\tag{1}
\]
where \(-1<a\leq 0\) and \(\gamma>0\), connecting the two stable states \(u= \pm 1\) of the ordinary differential equation \(u'= (u-a)(1- u^2)\). When \(\gamma=0\), a travelling wave solution is given by
\[
u(x,t)= \text{tanh}\Biggl({x- a\sqrt 2t\over\sqrt 2}\Biggr),
\]
with wave speed \(c_0= a\sqrt 2\), which is negative if \(a<0\). The wave profile is independent of \(a\), and for \(a=0\) this travelling wave solution is a stationary solution of (1) with \(\gamma=0\).

In this paper, we look for travelling wave solutions of equation (1). The resulting travelling wave equation neither has a conserved energy nor a variational structure and also the symmetry is lost. For small \(\gamma\), however, equation (1) can be seen as a perturbation of (1) with \(\gamma=0\), and it is this view that is taken in this paper. With the methods of geometric singular perturbation theory, we prove the following theorem.

Theorem. For \(\gamma>0\) sufficiently small, there exists a \(c= c(\gamma)\) for which there is a travelling wave solution of (1) connecting the steady states \(u=\pm 1\). The rate of change of the wave speed with respect to \(\gamma\) is given by \[ {dc\over d\gamma}\Biggl|_{\gamma=0}= -{1\over 5}\sqrt 2 a(2a^2- 3). \]

In this paper, we look for travelling wave solutions of equation (1). The resulting travelling wave equation neither has a conserved energy nor a variational structure and also the symmetry is lost. For small \(\gamma\), however, equation (1) can be seen as a perturbation of (1) with \(\gamma=0\), and it is this view that is taken in this paper. With the methods of geometric singular perturbation theory, we prove the following theorem.

Theorem. For \(\gamma>0\) sufficiently small, there exists a \(c= c(\gamma)\) for which there is a travelling wave solution of (1) connecting the steady states \(u=\pm 1\). The rate of change of the wave speed with respect to \(\gamma\) is given by \[ {dc\over d\gamma}\Biggl|_{\gamma=0}= -{1\over 5}\sqrt 2 a(2a^2- 3). \]

### MSC:

35B25 | Singular perturbations in context of PDEs |

35K55 | Nonlinear parabolic equations |

35K30 | Initial value problems for higher-order parabolic equations |

PDFBibTeX
XMLCite

\textit{M. E. Akveld} and \textit{J. Hulshof}, Appl. Math. Lett. 11, No. 3, 115--120 (1998; Zbl 0932.35017)

Full Text:
DOI

### References:

[1] | Peletier, L. A.; Troy, W. C., Spatial patterns described by the Extended Fisher-Kolmogorov (EFK) equation: Kinks, Diff. and Int. Equ., 8, 1279-1304 (1995) · Zbl 0826.34056 |

[2] | Aronson, D. G.; Weinberger, H. F., Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, (Goldstein, J. A., Partial Differential Equations and Related Topics. Partial Differential Equations and Related Topics, Lecture Notes in Math. 446 (1975), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0325.35050 |

[3] | Aronson, D. G.; Weinberger, H. F., Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30, 33-76 (1978) · Zbl 0407.92014 |

[4] | Fife, P. C.; McLeod, J. B., The approach of solutions of nonlinear diffusion equations to travelling wave solutions, Arch. Rational Mech. and Anal., 65, 335-362 (1977) · Zbl 0361.35035 |

[5] | Peletier, L. A.; Troy, W. C., Spatial patterns descibed by the Extended Fisher-Kolmogorov (EFK) equation: Periodic solutions, SIAM Journal of Math. Anal., 28, 1317-1353 (1997) · Zbl 0891.34048 |

[6] | Peletier, L. A.; Troy, W. C., Chaotic spatial patterns described by the Extended Fisher-Kolmogorov equation, Journal of Diff. Equ., 129, 458-508 (1996) · Zbl 0862.34012 |

[7] | Peletier, L. A.; Troy, W. C., A topological shooting method and the existence of kinks of the Extended Fisher-Kolmogorov equation, Topological Methods in Nonlinear Analysis, 6, 331-355 (1995) · Zbl 0862.34030 |

[8] | Peletier, L. A.; Troy, W. C.; van der Vorst, R. C.A. M., Stationary solutions of a fourth order non-linear diffusion equation, Diff. Equ., 31, 327-338 (1995), (in Russian) · Zbl 0856.35029 |

[9] | L.A. Peletier, D. Terman and W.C. Troy, Stationary solutions of the Extended Fisher-Kolmogorov equation, (private communication).; L.A. Peletier, D. Terman and W.C. Troy, Stationary solutions of the Extended Fisher-Kolmogorov equation, (private communication). |

[10] | Gardner, R.; Jones, C. R.K. T., Travelling waves of a perturbed diffusion equation arising in a phase field model, Indiana Univ. Math. Journal, 38, 1197-1222 (1989) · Zbl 0799.35106 |

[11] | Fenichel, N., Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. Journal, 21, 193-226 (1971) · Zbl 0246.58015 |

[12] | Jones, C. K.R. T., Geometric singular perturbation theory, (Johnson, R., Dynamic Systems (1995), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0779.35040 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.