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Travelling front solution for a class of competition-diffusion system with high-order singular point. (English) Zbl 0931.35086

Summary: The existence of travelling front solutions for a class of competition-diffusion system with high-order singular point \[ w_{it}= d_iw_{ixx} -w_i^{\alpha_i} f_i(w),\;x\in \mathbb{R},\;t>0,\;i=1,2 \tag{I} \] is studied, where \(d_i\), \(\alpha_i>0\) \((i=1,2)\) and \(w=(w_1(x,t), w_2(x,t))\). Under the certain assumptions on \(f\), it is shown that if \(\alpha_i<1\) for some \(i\), then (I) has no travelling front solution. If \(\alpha_i\geq 1\) for \(i=1,2\), then there exist. \(c_0\geq c^*>0\), where \(c^*\) is called the minimal wave speed of (I), such that, if \(c\geq c_0\) or \(c=c^*\), then (I) has a travelling front solution, if \(c<c^*\), then (I) has no travelling front solution. We use the shooting method in combination with a compactness argument.

MSC:

35K65 Degenerate parabolic equations
35K57 Reaction-diffusion equations
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K55 Nonlinear parabolic equations
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References:

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