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Steady states and long time behavior of some convective reaction-diffusion equations. (English) Zbl 0887.35076

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)

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
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