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Travelling wavefronts of reaction-diffusion equations in cylindrical domains. (English) Zbl 0816.35058
From the introduction: This paper is concerned with the existence and uniqueness (up to translation) of some bounded solutions of \[ \begin{aligned} \Delta &u+ cu_ x+ f(u)=0 \quad \text{in } \mathbb{R}\times \Omega,\tag{1}\\ &u=0 \quad \text{at } \mathbb{R}\times \partial \Omega,\tag{2}\\ &u (x,\cdot) \to v_ \pm \quad \text{pointwise as }x\to \pm\infty, \tag{3}\end{aligned} \] that may be viewed (after a suitable change of variables) as travelling wave solutions of the parabolic problem \(\partial u/\partial t= \Delta u+ f(u)\) in \(\mathbb{R}\times \Omega\), with the boundary condition (2) and an appropriate initial condition. Here, \(\Omega\subset \mathbb{R}^{n-1}\) \((n\geq 2)\) is a bounded domain, with a \(C^{2,\alpha}\) boundary, for some \(\alpha>0\) if \(n\leq 2\), and the spatial coordinates are written as \((x,y)\), where \(x=x_ 1\) and \(y= (x_ 2,\dots, x_ n)\). The constant \(c\) is the wave velocity that is to be determined, \(f: \mathbb{R}\to \mathbb{R}\) is a \(C^ 2\)-function, and \(v_ -\) and \(v_ +\) are solutions of \(\Delta v+ f(v)= 0\) in \(\Omega\), \(v=0\) at \(\partial \Omega\). We assume that \(v_ - (y)< v_ + (y)\) for all \(y\in \Omega\), and we consider only the solution of (1)–(3) such that \(v_ - (y)\leq u(x,y)\leq v_ + (y)\) for all \((x,y)\in \mathbb{R}\times \Omega\).

MSC:
35K57 Reaction-diffusion equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J65 Nonlinear boundary value problems for linear elliptic equations
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