Journal of Integral Equations and Applications

Blow-up behavior of Hammerstein-type Volterra integral equations

H. Brunner and Z.W. Yang

Full-text: Open access

Article information

Source
J. Integral Equations Applications, Volume 24, Number 4 (2012), 487-512.

Dates
First available in Project Euclid: 7 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1362669519

Digital Object Identifier
doi:10.1216/JIE-2012-24-4-487

Mathematical Reviews number (MathSciNet)
MR3041508

Zentralblatt MATH identifier
1261.45003

Subjects
Primary: 45D05: Volterra integral equations [See also 34A12] 45G10: Other nonlinear integral equations

Keywords
Volterra integral equations Volterra integro-differential equations blow-up critical exponent

Citation

Brunner, H.; Yang, Z.W. Blow-up behavior of Hammerstein-type Volterra integral equations. J. Integral Equations Applications 24 (2012), no. 4, 487--512. doi:10.1216/JIE-2012-24-4-487. https://projecteuclid.org/euclid.jiea/1362669519


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References

  • C. Bandle and H. Brunner, Blow-up in diffusion equations: A survey, J. Comput. Appl. Math. 97 (1998), 3-22.
  • H. Brunner, Collocation methods for Volterra and related functional differential equations, Camb. Mono. Appl. Comp. Math. 15, Cambridge University Press, Cambridge, 2004.
  • H. Brunner and L.Z. Li, Finite-time blow-up in semilinear parabolic integro-differential equations on unbounded domains, to appear.
  • F. Calabrò and G. Capobianco, Blowing up behavior for a class of nonlinear VIEs connected with parabolic PDEs, J. Comput. Appl. Math. 228 (2009), 580-588.
  • A.Z. Fino, Critical exponent for damped wave equations with nonlinear memory, Elsevier, submitted.
  • H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\triangle u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Math. 13 (1966), 109-124.
  • A. Goriely and C. Hyde, Finite-time blow-up in dynamical systems, Phys. Lett. A 250 (1998), 311-318.
  • C.M. Kirk and C.A. Roberts, A review of quenching results in the context of nonlinear Volterra integral equations, Dynam. Cont. Discr. Impulse Syst. Math. Anal. 10 (2003), 343-356.
  • H.A. Levine, The role of critical exponents on blowup theorems, SIAM Rev. 32 (1990), 262-288.
  • D. Liu and C. Mu, Blow-up analysis for a semilinear parabolic equation with nonlinear memory and nonlocal nonlinear boundary condition, Electron. J. Qual. Theory Diff. Equat. 51 (2010), 1-17.
  • T. Ma\lolepszy, Nonlinear Volterra integral equations and the Schröder functional equation, Nonlinear Anal. 74 (2011), 424-432.
  • T. Ma\lolepszy and W. Okrasiński, Conditions for blow-up of solutions of some nonlinear Volterra integral equations, J. Comput. Appl. Math. 205 (2007), 744-750.
  • –––, Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity, Appl. Math. Lett. 21 (2008), 307-312.
  • –––, Blow-up time for solutions to some nonlinear Volterra integral equations, J. Math. Anal. Appl. 366 (2010), 372-384.
  • P. Meier, Blow up of solutions of semilinear parabolic equations, Z. Angew. Math. Phys. 39 (1988), 135-149.
  • R.K. Miller, Nonlinear Volterra integral equations, W.A. Benjamin, Menlo Park, CA, 1971.
  • W. Mydlarczyk, A condition for finite blow-up time for a Volterra integral equation, J. Math. Appl. Anal. 181 (1994), 248-253.
  • –––, The blow-up solutions of integral equations, Colloq. Math. 79 (1999), 147-156.
  • W. Mydlarczyk and W. Okrasiński, On a generalization of the Osgood condition, J. Inequal. Appl. 5 (2000), 497-504.
  • –––, Nonlinear Volterra integral equations with convolution kernels, Bull. London Math. Soc. 35 (2003), 484-490.
  • W. Okrasiński, Nontrivial solutions for a class of nonlinear Volterra equations with convolution kernel, J. Integral Equations Appl. 3 (1991), 399-409.
  • –––, Nontrivial solutions to nonlinear Volterra integral equations, SIAM J. Math. Anal. 11 (1991), 1007-1015.
  • –––, Note on kernels to some nonlinear Volterra integral equations, Bull. London Math. Soc. 24 (1992), 373-376.
  • W.E. Olmstead, Ignition of a combustible half space, SIAM J. Appl. Math. 43 (1983), 1-15.
  • W.E. Olmstead and C.A. Roberts, Quenching for the heat equation with a nonlocal nonlinearity, Nonlinear Prob. Appl. Math., SIAM, Philadelphia, PA, 1996.
  • W.E. Olmstead and C.A. Roberts, Thermal blow-up in a subdiffusive medium, SIAM J. Appl. Math. 69 (2008), 514-523.
  • C.A. Roberts, Characterizing the blow-up solutions for nonlinear Volterra integral equations, Nonlinear Anal. Theory Methods Appl. 30 (1997), 923-933.
  • –––, Analysis of explosion for nonlinear Volterra equations, J. Comput. Appl. Math. 97 (1998), 153-166.
  • –––, Recent results on blow-up and quenching for nonlinear Volterra equations, J. Comput. Appl. Math. 205 (2007), 736-743.
  • C.A. Roberts, D.G. Lasseigne and W.E. Olmstead, Volterra equations which model explosion in a diffusive medium, J. Integral Equations Appl. 5 (1993), 531-546.
  • C.A. Roberts and W.E. Olmstead, Growth rates for blow-up solutions of nonlinear Volterra equations, Quart. Appl. Math. 54 (1996), 153-159. \noindentstyle