Korman, Philip Steady states and long time behavior of some convective reaction-diffusion equations. (English) Zbl 0887.35076 Funkc. Ekvacioj, Ser. Int. 40, No. 2, 165-183 (1997). The article discusses nonnegative solutions of the convective reaction-diffusion equation \[ u_t = u_{xx}+g(u)_x + \lambda f(u), \qquad 0<x<L, \] with Dirichlet boundary conditions. For \(g(u)=au\) and rather general convex functions \(f(u)\), it is shown that all steady solutions lie on a single closed curve, which passes through \(\lambda=0\), \(u=0\) and has a fold point for some \(\lambda_0>0\). The upper branch tends to infinity as \(\lambda\rightarrow 0\). For a class of convex-concave functions \(f(u)\) a similar behaviour is found. Further, the asymptotic stability of the steady solutions is computed, and the behaviour of radial solutions of reaction-diffusion equations in a circular domain is investigated. Reviewer: A.Steindl (Wien) Cited in 3 Documents MSC: 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B32 Bifurcations in context of PDEs 35K57 Reaction-diffusion equations 35B35 Stability in context of PDEs Keywords:convex-concave nonlinearity; radial solutions; circular domain PDFBibTeX XMLCite \textit{P. Korman}, Funkc. Ekvacioj, Ser. Int. 40, No. 2, 165--183 (1997; Zbl 0887.35076)