# Asymptotic analysis for a nonlinear parabolic equation on $\mathbb{R}$

Commentationes Mathematicae Universitatis Carolinae (1998)

- Volume: 39, Issue: 3, page 525-544
- ISSN: 0010-2628

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topFašangová, 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 -

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