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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 \[ {dc\over d\gamma}\Biggl|_{\gamma=0}= -{1\over 5}\sqrt 2 a(2a^2- 3). \]

MSC:

35B25 Singular perturbations in context of PDEs
35K55 Nonlinear parabolic equations
35K30 Initial value problems for higher-order parabolic equations
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