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- W2013200330 abstract "Let u be a solution of a second order semilinear parabolic equation which satisfies either Dirichlet or mixed boundary conditions. Let v be a stationary solution of this problem. Then by substituting $u = v + varepsilon w$, and omitting terms of higher order in $varepsilon $ than $O( varepsilon )$, we may formally derive a linear problem. In this paper we discuss the asymptotic behavior of solutions u of the nonlinear problem in relation to such behavior of solutions w of the linear problem. We show that if, for some $lambda _0 > 0$, $w = O( {e^{ - lambda _0 t} } )$ as $t to infty $ (in the supremum norm), then $u - v = O( {e^{ - lambda _0 t} } )$ as $t to infty $ for suitably restricted initial data, provided the nonlinear term in the original problem is sufficiently smooth in a neighborhood of v. An example is given which demonstrates that without this smoothness condition, the convergence of u towards v may take place at a slower rate than is the case for the linear problem." @default.
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- W2013200330 date "1975-01-01" @default.
- W2013200330 modified "2023-09-23" @default.
- W2013200330 title "On the Asymptotic Behavior of Solutions of Semilinear Parabolic Equations" @default.
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- W2013200330 doi "https://doi.org/10.1137/0128005" @default.
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