Global stability of traveling fronts and convergence towards stacked families of waves in monotone parabolic systems. (English) Zbl 0861.35013

The purpose of this paper is to study the long-time behaviour of the solutions of the Cauchy problem for a class of semilinear parabolic systems of the form \[ u_t- Du_{xx}= F(u),\quad u(0,x)= u_0(x).\tag{1} \] Let \(s(t)u_0\) be the solution of (1), \(U\subset C(\mathbb{R},\mathbb{R}^n)\) be the set of all bounded, uniformly continuous functions from \(\mathbb{R}\) to \(\mathbb{R}^n\). \(\omega_0\in\mathbb{R}^n\) is a stable rest point of \(F\), if \(F(\omega_0)=0\) and if the eigenvalues of the matrix \(F'(\omega_0)\) have negative real parts. The authors obtain the following main result.
Assume that \(\omega'\leq\omega^+\) are two stable rest points connected by a monotone front \(\omega\) with speed \(c\). Let \(u_0\in U\) satisfy \(\omega^-\leq u_0(x)\leq \omega^+\), and set \[ \varepsilon(u_0)= \sup\Biggl(\limsup_{x\to-\infty}|u_0(x)-\omega^-|,\limsup_{x\to+\infty}|u_0(x)-\omega^+|). \] Then, if \(\varepsilon(u_0)\) is small enough, the solution \(S(t)u_0\) is defined for all times, and there exist \(x_0\in\mathbb{R}\) and \(\omega>0\) such that \[ |S(t)u_0-\omega(\cdot+x_0+ct)|_\infty=O(e^{-\omega t}). \]


35B40 Asymptotic behavior of solutions to PDEs
35K45 Initial value problems for second-order parabolic systems
35K55 Nonlinear parabolic equations
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