Differential and Integral Equations

Large time behaviour of solutions of a generalized Haraux-Weissler equation

Claus Dohmen

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Abstract

We characterize the possible large time behaviors of solutions to $$ (|U|^{m-1} U)'' + {{N-1}\over\eta } (|U|^{m-1} U)' + \beta\eta U' + \alpha U + \gamma |U|^{p-1} U \, =\, 0 \quad \text{ in }\ \mathbb{R}^+ $$ with $m>0$, $p>1$, $\alpha, \beta >0$, $\alpha (m-1)+2\beta >0$ and $\gamma \in \{ -1,0,1\}$. It turns out that if $U$ is bounded and non-constant, $L:= \lim_{\eta\rightarrow\infty } \eta^{\alpha /\beta } U(\eta )$ is always finite and that in the case in which $L=0$, the solutions have compact support ($m>1$), decay exponentially ($m=1$) or decay like $\eta ^{-{2\over{1-m}}}$ ($m<1$), respectively. We want to stress that we impose no sign restriction on the solution.

Article information

Source
Differential Integral Equations, Volume 8, Number 8 (1995), 2065-2078.

Dates
First available in Project Euclid: 20 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1369056140

Mathematical Reviews number (MathSciNet)
MR1348965

Zentralblatt MATH identifier
0839.35069

Subjects
Primary: 35Q99: None of the above, but in this section
Secondary: 34C99: None of the above, but in this section 35B40: Asymptotic behavior of solutions 35K99: None of the above, but in this section

Citation

Dohmen, Claus. Large time behaviour of solutions of a generalized Haraux-Weissler equation. Differential Integral Equations 8 (1995), no. 8, 2065--2078. https://projecteuclid.org/euclid.die/1369056140


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