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Convergence to travelling waves for quasilinear Fisher-KPP type equations. (English) Zbl 1234.35055

Summary: We consider the Cauchy problem \[ \begin{cases} u_t = \varphi(u)_{xx} + \psi(u), \qquad & (t,x) \in {\mathbb R}^+ \times {\mathbb R}, \\ u(0,x) = u_0(x), \qquad & x \in {\mathbb R}, \end{cases} \] when the increasing function \(\varphi\) satisfies that \(\varphi (0)=0\) and the equation may degenerate at \(u=0\) (in the case of \(\varphi^{\prime}(0)=0\)). We consider the case of \(u_{0}\in L^{\infty}({\mathbb R})\), \(0\leq u_{0}(x)\leq 1\) a.e. \(x\in {\mathbb R}\) and the special case of \(\psi(u)=u - \varphi (u)\). We prove that the solution approaches the travelling wave solution (with speed \(c=1\)), spreading either to the right or to the left, or to the two travelling waves moving in opposite directions.

MSC:

35C07 Traveling wave solutions
35K59 Quasilinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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