Topological Methods in Nonlinear Analysis

Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$

Perikles G. Papadopoulos and Nikos M. Stavrakakis

Full-text: Open access

Abstract

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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 17, Number 1 (2001), 91-109.

Dates
First available in Project Euclid: 22 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1471875799

Mathematical Reviews number (MathSciNet)
MR1846980

Zentralblatt MATH identifier
0989.35091

Citation

Papadopoulos, Perikles G.; Stavrakakis, Nikos M. Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$. Topol. Methods Nonlinear Anal. 17 (2001), no. 1, 91--109. https://projecteuclid.org/euclid.tmna/1471875799


Export citation

References

  • A. Arosio and S. Garavaldi, On the mildly degenerate Kirchhoff string , Methods Appl. Sci., 14 (1991), 177–195 \ref\key2
  • C. Bandle and N. Stavrakakis, Global existence and stability results for a semilinear parabolic equation on $\Bbb R^N$ , in progress \ref\key3
  • K. J. Brown and N. M. Stavrakakis, Global bifurcation results for a semilinear elliptic equation on all of $\Bbb R^N$ , Duke Math. J., 85 (1996), 77–94 \ref\key4
  • H. R. Crippa, On local solutions of some mildly degenerate hyperbolic equations , Nonlinear Anal., 21 (1993), 565–574 \ref\key5
  • P. D'Ancona and Y. Shibata, On global solvability for the degenerate Kirchhoff equation in the analytic category , Math. Methods Appl. Sci., 17 (1994), 477–489 \ref\key6 ––––, Global solvability for the degenerate Kirchhoff equation with real analytic data , Invent. Math., 108 (1992), 247–262 \ref\key7
  • P. D'Ancona and S. Spagnolo, Nonlinear perturbations of the Kirchhoff equation , Comm. Pure Appl. Math., 47 (1994), 1005–1029 \ref\key8
  • M. Hosoya and Y. Yamada, On some nonlinear wave equations II: global existence and energy decay of solutions , J. Fac. Sci. Univ. Tokyo Sect. IA Math., 38 (1991), 239–250 \ref\key9
  • R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms , Nonlinear Anal., 27 (1996), 1165–1175 \ref\key10
  • N. I. Karahalios and N. M. Stavrakakis, Existence of global attractors for semilinear dissipative wave equations on $\Bbb R^N$ , J. Differential Equations, 157 (1999), 183–205 \ref\key11 ––––, Functional invariant sets for semilinear dissipative wave equations on $\Bbb R^N$ , to appear \ref\key12 ––––, Global existence and blow-up results for some nonlinear wave equations on $\Bbb R^N$, Adv. Differental Equations, 6 (2001), 155–174 \ref\key13
  • G. Kirchhoff, Vorlesungen Über Mechanik, Teubner, Leipzig (1883) \ref\key14
  • T. Kobayashi, H. Pecher and Y. Shibata, On a global in time existence theorem of smooth solutions to a nonlinear wave equations with viscosity , Math. Ann., 296 (1993), 215–234 \ref\key15
  • H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form ${\Cal P}u_{tt}= -{\Cal A}u + {\Cal F}(u)$ , Trans. Math. Soc., 192 (1974), 1–21 \ref\key16 ––––, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations , SIAM J. Math. Anal., 5 (1974), 138–146 \ref\key17
  • M. P. Matos and D. C. Pereira, On a hyperbolic equation with strong damping , Funkcial. Ekvac., 34 (1991), 303–311 \ref\key18
  • M. Nakao, Decay of solutions of some nonlinear evolution equations , J. Math. Anal. Appl., 60 (1977), 542–549 \ref\key19 ––––, A difference inequality and its application to nonlinear evolution equation , J. Math. Soc. Japan, 30 (1978), 747–762 \ref\key20 ––––, Energy decay for the quasilinear wave equation with viscosity , Math. Z., 219 (1995), 289–299 \ref\key21
  • M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equation , Math. Z., 214 (1993), 325–342 \ref\key22
  • K. Nishihara, Degenerate quasilinear hyperbolic equation with strong damping , Funkcial. Ekvac., 27 (1984), 125–145 \ref\key23 ––––, Decay properties of solutions of some quasilinear hyperbolic equations with strong damping , Nonlinear Anal., 21 (1993), 17–21 \ref\key24
  • K. Nishihara and K. Ono, Asymptotic behaviour of solutions of some nonlinear oscillation equations with strong damping , Adv. Math. Sci. Appl., 4 (1994), 285–295 \ref\key25
  • K. Nishihara and Y. Yamada, On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms , Funkcial. Ekvac., 33 (1990), 151–159 \ref\key26
  • K. Ono, On global existence, asymptotic stability and blowing-up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation , Math. Methods Appl. Sci., 20 (1997), 151–177 \ref\key27 ––––, Global existence and decay properties of solutions for some mildly degenerate nonlinear dissipative Kirchhoff strings , Funkcial. Ekvac., 40 (1997), 255–270 \ref\key28 ––––, Global existence, decay, and blow-up of solutions for some mildly degenerate nonlinear Kirchhoff strings , J. Differential Equations, 137 (1997), 273–301 \ref\key29
  • K. Ono and K. Nishihara, On a nonlinear degenerate integro-differential equation of hyperbolic type with a strong dissipation , Adv. Math. Sci. Appl., 5 (1995), 457–476 \ref\key30
  • L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations , Israel J. Math., 22 (1975), 273–303 \ref\key31
  • S. I. Pohozaev, On a class of quasilinear hyperbolic equations , Math. USSR Sb., 25 (1975), 145–158 \ref\key32
  • G. Todorova, The Cauchy problem for nonlinear wave equations with nonlinear damping and source terms , Nonlinear Anal., 41 (2000), 891–905 \ref\key33
  • T. Yamazaki, On local solutions of some quasilinear degenerate hyperbolic equations , Funkcial. Ekvac., 31 (1988), 439–457 \ref\key34
  • E. Zeidler, Nonlinear Functional Analysis and its Applications, Monotone Operators, Springer-Verlag, II (1990)