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Blow-up set of type I blowing up solutions for nonlinear parabolic systems

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Abstract

We consider the blow-up problem for systems of nonlinear parabolic inequalities and establish a criterion for the location of the blow-up set. Our criterion enables us to obtain sufficient conditions for the boundedness of the blow-up set and no boundary blow-up for type I blowing up solutions.

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References

  1. Andreucci, D., Herrero, M.A., Velázquez, J.J.L.: Liouville theorems and blow up behaviour in semilinear reaction diffusion systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 14, 1–53 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Caristi, G., Mitidieri, E.: Blow-up estimates of positive solutions of a parabolic system. J. Differ. Equ. 113, 265–271 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, X.-Y., Matano, H.: Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations. J. Differ. Equ. 78, 160–190 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chlebík, M., Fila, M.: From critical exponents to blow-up rates for parabolic problems. Rend. Mat. Appl. 19, 449–470 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Deng, K.: Blow-up rates for parabolic systems. Z. Angew. Math. Phys. 47, 132–143 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dickstein, F.: Blowup stability of solutions of the nonlinear heat equation with a large life span. J. Differ. Equ. 223, 303–328 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Escobedo, M., Herrero, M.A.: A uniqueness result for a semilinear reaction-diffusion system. Proc. Am. Math. Soc. 112, 175–185 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Escobedo, M., Herrero, M.A.: Boundedness and blow up for a semilinear reaction-diffusion system. J. Differ. Equ. 89, 176–202 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Escobedo, M., Herrero, M.A.: A semilinear parabolic system in a bounded domain. Ann. Mat. Pura Appl. 165, 315–336 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fila, M., Souplet, P.: The blow-up rate for semilinear parabolic problems on general domains. NoDEA Nonlinear Differ. Equ. Appl. 8, 473–480 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Filippas, S., Merle, F.: Modulation theory for the blowup of vector-valued nonlinear heat equations. J. Differ. Equ. 116, 119–148 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Földes, J.: Liouville theorems, a priori estimates, and blow-up rates for solutions of indefinite superlinear parabolic problems. Czech. Math. J. 61, 169–198 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Friedman, A., Giga, Y.: A single point blow-up for solutions of semilinear parabolic systems. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34, 65–79 (1987)

  14. Friedman, A., McLeod, B.: Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 34, 425–447 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fujishima, Y.: Location of the blow-up set for a superlinear heat equation with small diffusion. Differ. Integr. Equ. 25, 759–786 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Fujishima, Y.: Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete Contin. Dyn. Syst. 34, 4617–4645 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fujishima, Y., Ishige, K.: Blow-up set for a semilinear heat equation with small diffusion. J. Differ. Equ. 249, 1056–1077 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fujishima, Y., Ishige, K.: Blow-up for a semilinear parabolic equation with large diffusion on \({ R}^N\). J. Differ. Equ. 250, 2508–2543 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fujishima, Y., Ishige, K.: Blow-up for a semilinear parabolic equation with large diffusion on \({ R}^N\). II. J. Differ. Equ. 252, 1835–1861 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fujishima, Y., Ishige, K.: Blow-up set for a semilinear heat equation and pointedness of the initial data. Indiana Univ. Math. J. 61, 627–663 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fujishima, Y., Ishige, K.: Blow-up set for type I blowing up solutions for a semilinear heat equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 231–247 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Giga, Y.: A bound for global solutions of semilinear heat equations. Commun. Math. Phys. 103, 415–421 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. Giga, Y., Kohn, R.V.: Asymptotically self-similar blow-up of semilinear heat equations. Commun. Pure Appl. Math. 38, 297–319 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Giga, Y., Kohn, R.V.: Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36, 1–40 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Giga, Y., Kohn, R.V.: Nondegeneracy of blowup for semilinear heat equations. Commun. Pure Appl. Math. 42, 845–884 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ishige, K.: Blow-up time and blow-up set of the solutions for semilinear heat equations with large diffusion. Adv. Differ. Equ. 7, 1003–1024 (2002)

    MathSciNet  MATH  Google Scholar 

  27. Ishige, K., Kawakami, T., Sierżȩga, M.: Supersolutions of parabolic systems with power nonlinearities. J. Differ. Equ. 260, 6084–6107 (2016)

    Article  MATH  Google Scholar 

  28. Ishige, K., Mizoguchi, N.: Blow-up behavior for semilinear heat equations with boundary conditions. Differ. Integr. Equ. 16, 663–690 (2003)

    MathSciNet  MATH  Google Scholar 

  29. Ishige, K., Yagisita, H.: Blow-up problems for a semilinear heat equation with large diffusion. J. Differ. Equ. 212, 114–128 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mahmoudi, N., Souplet, P., Tayachi, S.: Improved conditions for single-point blow-up in reaction-diffusion systems. J. Differ. Equ. 259, 1898–1932 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Merle, F.: Solution of a nonlinear heat equation with arbitrarily given blow-up points. Commun. Pure Appl. Math. 45, 263–300 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Merle, F., Zaag, H.: A Liouville theorem for vector-valued nonlinear heat equations and applications. Math. Ann. 316, 103–137 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Poláčik, P., Quittner, P., Souplet, P.: Singularity and decay estimates in superlinear problems via Liouville-type theorems. II. Parabolic equations. Indiana Univ. Math. J. 56, 879–908 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Quittner, P.: Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. Math. Ann. 364, 269–292 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Quittner, P.: Liouville theorems, universal estimates and periodic solutions for cooperative parabolic Lotka–Volterra systems. J. Differ. Equ. 260, 3524–3537 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Quittner, P., Souplet, P.: Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel (2007)

    MATH  Google Scholar 

  37. Souplet, P.: Single-point blow-up for a semilinear parabolic system. J. Eur. Math. Soc. 11, 169–188 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Souplet, P., Tayachi, S.: Single-point blow-up for parabolic systems with exponential nonlinearities and unequal diffusivities. Nonlinear Anal. 138, 428–447 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Velázquez, J.J.L.: Estimates on the \((n-1)\)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation. Indiana Univ. Math. J. 42, 445–476 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, M.: Blowup estimates for a semilinear reaction diffusion system. J. Math. Anal. Appl. 257, 46–51 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Weissler, F.B.: Single point blow-up for a semilinear initial value problem. J. Differ. Equ. 55, 204–224 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yagisita, H.: Blow-up profile of a solution for a nonlinear heat equation with small diffusion. J. Math. Soc. Japan 56, 993–1005 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zaag, H.: A Liouville theorem and blowup behavior for a vector-valued nonlinear heat equation with no gradient structure. Commun. Pure Appl. Math. 54, 107–133 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zaag, H.: On the regularity of the blow-up set for semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 19, 505–542 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zaag, H.: Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation. Duke Math. J. 133, 499–525 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zheng, S.: Nonexistence of positive solutions to a semilinear elliptic system and blow-up estimates for a reaction-diffusion system. J. Math. Anal. Appl. 232, 293–311 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zou, H.: Blow-up rates for semi-linear reaction-diffusion systems. J. Differ. Equ. 257, 843–867 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their useful comments. The second author wishes to thank Professor Pavol Quittner and Professor Philippe Souplet for stimulating discussions and useful suggestions about parabolic Liouville type theorems. He was supported partially by the Grant-in-Aid for Scientific Research (A) (No. 15H02058) from Japan Society for the Promotion of Science.

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Authors and Affiliations

  1. Department of Mathematical and Systems Engineering, Faculty of Engineering, Shizuoka University, 3-5-1, Johoku, Hamamatsu, 432-8561, Japan

    Yohei Fujishima

  2. Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan

    Kazuhiro Ishige & Hiroki Maekawa

Authors
  1. Yohei Fujishima
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  2. Kazuhiro Ishige
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  3. Hiroki Maekawa
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Correspondence to Kazuhiro Ishige.

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Communicated by Y. Giga.

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Fujishima, Y., Ishige, K. & Maekawa, H. Blow-up set of type I blowing up solutions for nonlinear parabolic systems. Math. Ann. 369, 1491–1525 (2017). https://doi.org/10.1007/s00208-016-1498-7

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  • DOI: https://doi.org/10.1007/s00208-016-1498-7

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