## 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 $|S(t)u_0-\omega(\cdot+x_0+ct)|_\infty=O(e^{-\omega t}).$

### MSC:

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

### Keywords:

semilinear parabolic systems
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