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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
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References:

[1] M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions , Acta Math. Univ. Comenian. 60 (1991), 35-103. · Zbl 0743.35038
[2] J. López-Gómez, V. Márquez and N. Wolanski, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition , J. Differential Equations 92 (1991), 384-401. · Zbl 0735.35016
[3] ——–, Global behaviour of positive solutions to a semilinear equation with a nonlinear flux condition , IMA preprint series #810, 1991.
[4] H.A. Levine and L.E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time , J. Differential Equations 16 (1974), 319-334. · Zbl 0285.35035
[5] M. Fila, Boundedness of global solutions for the heat equation with nonlinear boundary conditions , Comment. Math. Univ. Carolin. 30 (1989), 479-484. · Zbl 0702.35141
[6] P. Quittner, On global existence and stationary solutions for two classes of semilinear parabolic problems , Comment. Math. Univ. Carolin. 34 (1993), 105-124. · Zbl 0794.35089
[7] W. Walter, On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition , SIAM J. Math. Anal. 6 (1975), 85-90. · Zbl 0268.35052
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