Description

• Title:
  Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$
• Creator:
  Papadopoulos, Perikles G.;
  Stavrakakis, Nikos M.
• Description:
  We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff Type $$ u_{tt}-\phi (x)\Vert\nabla u(t)\Vert^{2}\Delta u+\delta u_{t} = |u|^{a}u,\quad x\in {\mathbb R}^N,\ t\geq 0,$$ with initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N \geq 3$, $\delta \geq 0$ and $(\phi (x))^{-1} =g (x)$ is a positive function lying in $L^{N/2}(\mathbb R^N)\cap L^{\infty}(\mathbb R^N )$. When the initial energy $ E(u_{0},u_{1})$, which corresponds to the problem, is non-negative and small, there exists a unique global solution in time. When the initial energy $E(u_{0},u_{1})$ is negative, the solution blows-up in finite time. A combination of the modified potential well method and the concavity method is widely used.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2001.03.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 17, No 1 (March 2001); 91 - 110;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2001.006/9957
• Subject:
  Quasilinear hyperbolic equations;
  global solution;
  blow-up;
  dissipation;
  potential well;
  concavity method;
  unbounded domains;
  Kirchhoff strings;
  generalised Sobolev spaces;
  weighted $L^p$ spaces;
  35A07;
  35B30;
  35B40;
  35B45;
  35L15;
  35L70;
  35L80;
  47F05;
  47H20
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2001.006
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other