Absence theorems for some classes of nonlinear systems (Q1874806)

The author proves existence and establishes the asymptotic behavior, as \(\varepsilon\rightarrow 0\), of stable stationary solutions to the equation \[ u_{t}=\varepsilon\nabla \cdot[d(x)\nabla u]+ (1-u^{2})[u-a(x)], \] for \((t,x)\in \mathbb{R}^{+}\times\Omega\), where \(\Omega\subset\mathbb{R}^{N},\;N\geq 2\), with Neumann boundary condition. The function \(a(x)\in C^{0,v}(\Omega)\) satisfies \(-1<a(x)<1\) and varies on some hypersurfaces.