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Existence of multidimensional travelling wave solutions of an initial boundary value problem. (English) Zbl 0549.35066
Travelling wave solutions of the bistable diffusion equation \[ (1)\quad u_ t=u_{xx}+u_{yy}+u(1-u)(u-\alpha),\quad u|_{\partial\Omega }=0, \] are obtained, where \(\Omega\) is a planar channel \(x\in {\mathbb{R}}^ 1\), \(o<y<L\). The solution takes form \(u(\xi,y)=u(x-\theta t,y);\) it tends to limits \(u_{\pm}(y)\) as \(\xi\) approaches \(\pm\infty \). The asymptotic states are governed by the two point b.v.p. \((2)\quad 0=u_{yy}+u(1- u)(u-\alpha),\quad u(0)=u(L)=0.\) It is assumed that \(\alpha\in (0,1/2)\) so that for large enough L, (2) admits exactly three solutions, \(0=u_ 0<u_{\alpha}(y)<u_ 1(y)\). In this parameter range it is proved that (1) admits a travelling wave \(u(\xi\),y), \(\theta\) asymptotic to \(u_ 0\) (resp. \(u_ 1(y))\) at \(\xi =-\infty\) (resp. \(\xi =+\infty)\). The method of proof is to apply Conley’s index to an approximating system of differential-difference equations obtained by replacing the finite variable y with a discrete net of points \(\{y_ i\}\). The index is computed by deforming the boundary conditions from the Dirichlet problem to the Neumann problem. For the latter problem, most of the components of the equations decouple and in the index can easily be shown to be nontrivial. The constructions supply enough compactness so that solutions of the continuous problem can be obtained in the limit as the mesh tends to zero. - Finally, it is shown that the wave speed \(\theta\) depends on the channel width L (in contrast to the Neumann problem). For large L we have that \(\theta <0\) as in the case of one dimensional waves. However for smaller L it can be shown that \(\theta >0\).

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI
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