Geodesic
Asymptotic analysis for a nonlinear parabolic equation on $\Bbb R$
Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 3, pp. 525-544 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We show that nonnegative solutions of $$ \begin{aligned} u_{t}-u_{xx}+f(u)=0,\quad x\in \Bbb R,\quad t>0, \\ u=\alpha \bar u,\quad x\in \Bbb R,\quad t=0, \quad \operatorname{supp}\bar u \hbox{ compact } \end{aligned} $$ either converge to zero, blow up in $\operatorname{L}^{2}$-norm, or converge to the ground state when $t\to\infty$, where the latter case is a threshold phenomenon when $\alpha>0$ varies. The proof is based on the fact that any bounded trajectory converges to a stationary solution. The function $f$ is typically nonlinear but has a sublinear growth at infinity. We also show that for superlinear $f$ it can happen that solutions converge to zero for any $\alpha>0$, provided $\operatorname{supp}\bar u$ is sufficiently small.
We show that nonnegative solutions of $$ \begin{aligned} u_{t}-u_{xx}+f(u)=0,\quad x\in \Bbb R,\quad t>0, \\ u=\alpha \bar u,\quad x\in \Bbb R,\quad t=0, \quad \operatorname{supp}\bar u \hbox{ compact } \end{aligned} $$ either converge to zero, blow up in $\operatorname{L}^{2}$-norm, or converge to the ground state when $t\to\infty$, where the latter case is a threshold phenomenon when $\alpha>0$ varies. The proof is based on the fact that any bounded trajectory converges to a stationary solution. The function $f$ is typically nonlinear but has a sublinear growth at infinity. We also show that for superlinear $f$ it can happen that solutions converge to zero for any $\alpha>0$, provided $\operatorname{supp}\bar u$ is sufficiently small.
Classification : 35B05, 35B40, 35K55
Keywords: parabolic equation; stationary solution; convergence 
@article{CMUC_1998_39_3_a8,
     author = {Fa\v{s}angov\'a, Eva},
     title = {Asymptotic analysis for a nonlinear parabolic equation on $\Bbb R$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {525--544},
     year = {1998},
     volume = {39},
     number = {3},
     mrnumber = {1666798},
     zbl = {0963.35080},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1998_39_3_a8/}
}
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Fašangová, Eva. Asymptotic analysis for a nonlinear parabolic equation on $\Bbb R$. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 3, pp. 525-544. http://geodesic.mathdoc.fr/item/CMUC_1998_39_3_a8/