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Existence and stability of multidimensional travelling waves in the monostable case. (English) Zbl 0929.35064
The paper deals with the coupled system $\frac{\partial u}{\partial t}=a(x')\Delta u+\sum_{j=1}^m b_j(x') \frac{\partial u}{\partial x_j}+F(u, x')$ in a cylinder $$\Omega=\Omega'\times \mathbb{R}$$. Here $u=u(x,t)=(u_1(x,t), \dots, u_n(x,t)), \quad x=(x_1, \dots, x_m), \quad x'=(x_2, \dots, x_m),$ $$x_1$$ is the variable along the axis of the cylinder, $$\Omega'$$ is a bounded domain. In addition $$F=(F_j(x))$$ is a smooth vector-valued function defined on $$\mathbb{R}^n\times \Omega'$$; $$a, b_j$$ are diagonal matrices. The travelling wave solution $$w$$ is a solution of the form $$u(x,t)=w(x_1-ct, x_2, \dots,x_m)$$ with a constant $$c$$. Under consideration, existence and stability conditions for travelling waves are derived provided the condition $\frac{\partial F_i}{\partial u_j}\geq 0, \quad i\neq j; \quad i, j=1, \dots, n,$ holds.

##### MSC:
 35K55 Nonlinear parabolic equations 35B35 Stability in context of PDEs 35K40 Second-order parabolic systems
##### Keywords:
quasilinear parabolic systems; stability conditions
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##### References:
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