Taiwanese Journal of Mathematics

Blow-up for Parabolic Equations and Systems with Nonnegative Potential

Yung-Jen Lin Guo and Masahiko Shimojo

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Abstract

We study the blow-up behaviors of two parabolic problems on a bounded domain. One is the heat equation with nonlinear memory and the other is a parabolic system with power nonlinearity in which the coefficients of the reaction terms (potentials) are nonnegative and spatially inhomogeneous. Our aim is to show that any zero of the potential, where there is no reaction, is not a blow-up point, if the solution is monotone in time. We also give sufficient conditions for the time monotonicity of solutions.

Article information

Source
Taiwanese J. Math., Volume 15, Number 3 (2011), 995-1005.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406280

Digital Object Identifier
doi:10.11650/twjm/1500406280

Mathematical Reviews number (MathSciNet)
MR2829893

Zentralblatt MATH identifier
1235.35049

Subjects
Primary: 35K05: Heat equation 35K15: Initial value problems for second-order parabolic equations 35K55: Nonlinear parabolic equations 35K61: Nonlinear initial-boundary value problems for nonlinear parabolic equations

Keywords
blow-up parabolic equation parabolic system

Citation

Lin Guo, Yung-Jen; Shimojo, Masahiko. Blow-up for Parabolic Equations and Systems with Nonnegative Potential. Taiwanese J. Math. 15 (2011), no. 3, 995--1005. doi:10.11650/twjm/1500406280. https://projecteuclid.org/euclid.twjm/1500406280


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References

  • H. Bellout, Blow-up of solutions of parabolic equations with nonlinear memory, J. Differ. Equations, 70 (1987), 42-68.
  • K. Deng, Blow-up rates for parabolic systems, Z. Angew. Math. Phys., 47 (1996), 132-143.
  • A. Friedman, J. B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.
  • J.-S. Guo, S. Sasayama and C.-J. Wang, Blowup rate estimate for a system of semilinear parabolic equations, Comm. Pure Appl, Anal., 8 (2009), 711-718.
  • J.-S. Guo and M. Shimojo, Blowing up at zero points of potential for an initial boundary value problem, preprint.
  • Y. Li and C. Xie, Blow-up for semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys., 55 (2004) 15-27.
  • F. Quirós and J. D. Rossi, Non-simultaneous blow-up in a semilinear parabolic system, Z. angew. Math. Phys., 52 (2001), 342-346.
  • P. Quittner and Ph. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basler Lehrbücher, 2007.
  • Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 28 (1998), 1301-1334.
  • Ph. Souplet, Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys., 55 (2004), 28-31.
  • M. Wang, Blowup estimates for a semilinear reaction diffusion system, J. Math. Anal. Appl., 257 (2001), 46-51.
  • M. Wang, Blow-up rate estimates for semilinear parabolic systems, J. Diff. Equations, 170 (2001), 317-324.
  • M. Wang, Blow-up rate for a semilinear reaction diffusion system, Computers Math. Appl., 44 (2002), 573-585.