Asymptotic analysis for a nonlinear parabolic equation on $ℝ$

Fašangová, Eva. "Asymptotic analysis for a nonlinear parabolic equation on $\mathbb {R}$." Commentationes Mathematicae Universitatis Carolinae 39.3 (1998): 525-544. <http://eudml.org/doc/248283>.

@article{Fašangová1998,
abstract = {We show that nonnegative solutions of \[ \begin\{aligned\} & u\_\{t\}-u\_\{xx\}+f(u)=0,\quad x\in \mathbb \{R\},\quad t>0, \\ & u=\alpha \bar\{u\},\quad x\in \mathbb \{R\},\quad t=0, \quad \operatorname\{supp\}\bar\{u\} \hbox\{ compact \} \end\{aligned\} \] either converge to zero, blow up in $\operatorname\{L\}^\{2\}$-norm, or converge to the ground state when $t\rightarrow \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.},
author = {Fašangová, Eva},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {parabolic equation; stationary solution; convergence; parabolic equation; stationary solution; asymptotic behaviour},
language = {eng},
number = {3},
pages = {525-544},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Asymptotic analysis for a nonlinear parabolic equation on $\mathbb \{R\}$},
url = {http://eudml.org/doc/248283},
volume = {39},
year = {1998},
}

TY - JOUR
AU - Fašangová, Eva
TI - Asymptotic analysis for a nonlinear parabolic equation on $\mathbb {R}$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 3
SP - 525
EP - 544
AB - We show that nonnegative solutions of \[ \begin{aligned} & u_{t}-u_{xx}+f(u)=0,\quad x\in \mathbb {R},\quad t>0, \\ & u=\alpha \bar{u},\quad x\in \mathbb {R},\quad t=0, \quad \operatorname{supp}\bar{u} \hbox{ compact } \end{aligned} \] either converge to zero, blow up in $\operatorname{L}^{2}$-norm, or converge to the ground state when $t\rightarrow \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.
LA - eng
KW - parabolic equation; stationary solution; convergence; parabolic equation; stationary solution; asymptotic behaviour
UR - http://eudml.org/doc/248283
ER -