A reaction-diffusion equation on a thin \(L\)-shaped domain. (English) Zbl 0828.35055

Hale, Jack K.; Raugel, Geneviève

Proc. R. Soc. Edinb., Sect. A 125, No. 2, 283-327 (1995).

The authors study the long-time behaviour of solutions to the equation \(u_t - \Delta u + f(u) = G\), where the spatial variable belongs to a thin \(L\)-shaped domain in \(R^2\). A typical example of such a domain is the set \[ \biggl\{ (x_1, x_2) \bigl |0 < x_1 < 1,\;0 < x_2 < \varepsilon \biggr\} \bigcup \biggl\{ (x_1, x_2) \bigr |0 < x_1 < \varepsilon,\;0 < x_2 < 1 \biggr\} \] with \(\varepsilon > 0\) small. The dynamics is compared with that of the corresponding limit problem for \(\varepsilon = 0\). The limit equation for \(\varepsilon \to 0\) is determined and the upper semicontinuity of the attractors is shown. Moreover, the lower semicontinuity of the attractors is proved provided the equilibrium points of the limit problem are hyperbolic. If the limit equation is one-dimensional, any orbit converges to a singleton provided \(\varepsilon\) is sufficiently close to zero.

Reviewer: E.Feireisl (Praha)

MSC:

35K55	Nonlinear parabolic equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

semilinear heat equation; continuity properties of the attractor; thin \(L\)-shaped domain

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