## Perturbation analysis of a semilinear parabolic problem with nonlinear boundary conditions.(English)Zbl 0858.35066

The objective of the paper is the study of the nonlinear, parabolic, initial-boundary value problem: $u_t=u_{xx}-\lambda u^p, \qquad x\in(0,1), \quad t>0,$
$u_x(0,t)=0, \;\;u_x(1,t)=u^q(1,t), \quad t>0, \qquad u(x,0)=u_0(x),$ where $$\lambda$$, $$p$$, $$q$$ are constants, $$\lambda>0$$, $$p>1$$, $$q>1$$ and $$u_0>0$$. A formal asymptotic description of the steady states is presented and the stability of these steady states is studied in the case when $$p$$ and $$q$$ exceed unity. In particular, a perturbation analysis is made taking $$p=1+\varepsilon \alpha$$ and $$q=1+\varepsilon\beta$$ where $$\varepsilon$$ is a small positive parameter. The analysis emphasizes the role of $$\alpha/\beta$$ especially when this parameter is near one or a half. The authors expect that their response diagrams will give a reliable guide to the size of stationary solutions for larger values of $$\varepsilon$$. They indicate also how the steady states can be used to give information on the solution of the unsteady problem.
Reviewer: S.Totaro (Firenze)

### MSC:

 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35C20 Asymptotic expansions of solutions to PDEs

### Keywords:

perturbation analysis
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### References:

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