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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.

35K55 Nonlinear parabolic equations
35B35 Stability in context of PDEs
35K40 Second-order parabolic systems
Full Text: DOI
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