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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