SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP
FOR A NON-LOCAL DIFFUSION SYSTEM

M. Bogoya, C.A. Gomez
Departamento de Matemáticas
Universidad Nacional de Colombia
Bogotá, COLOMBIA

Abstract

In this work, the non-local diffusion system with Neumann boundary conditions
\begin{displaymath} \begin{array}{l} u_t(x,t)=\!\displaystyle\int_{\Omega}\! J... ...0(x), \ \ \ v(x,0)=v_0(x), \ \ \ \ \ x\in\Omega, \end{array} \end{displaymath} (1)

is studied, where $(x,t)\in\Omega\times (0,T)$, $\Omega$ is a bounded, connected and smooth domain, $f$ and $g$ are continuous functions and $(u_0, v_0)\in C(\overline{\Omega})\times C(\overline{\Omega})$, nonnegative and real functions. Existence and uniqueness of solutions is proved. For some particular functions $f$, $g$, the simultaneous and non-simultaneous blow-up for solutions is analyzed. Finally, the blow-up rate for the solution is given.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 36
Issue: 2
Year: 2023

DOI: 10.12732/ijam.v36i2.2

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] P. Bates, P. Fife, X. Ren and X. Wang, Travelling waves in a convolution model for phase transitions, Arch. Rat. Mech. Anal., 138 (1997), 105-136.
  2. [2] M. Bogoya, A nonlocal nonlinear diffusion equation in higher space dimensions, J. Math. Anal. Appl., 344 (2008), 601-615.
  3. [3] M. Bogoya, Sobre la explosi´on de una ecuaci´on de difusi´on no local con t´ermino de reacci´on, Bolet´ın de Matem´aticas, 24, No 2 (2017), 117-130.
  4. [4] M. Bogoya, A non-local diffusion coupled system equations in a bounded domain, Boundary Value Problems, 2018 (2018), 38.
  5. [5] M. Bogoya, C.A. G´omez, Blow-up analysis for a non-local discrete diffusion system, Contemporary Engineering Sciences, 11, No 54 (2018), 2679-2690.
  6. [6] C. Cort´azar, M. Elgueta and J.D. Rossi, A non-local diffusion equation whose solutions develop a free boundary. Ann. Henri Poincar´e, 6, No 2 (2005), 269-281.
  7. [7] E. Chasseigne, M. Chaves and J.D. Rossi, Asymptotic behaviour for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.
  8. [8] F. Dicksteina, M. Escobedo, A maximum principle for semilinear parabolic systems and applications, Nonlinear Analysis, 45 (2001), 825-837.
  9. [9] M. Escobedo, M.A. Herrero, A semilinear parabolic system in a bounded domain, Ann. Matematica Pura Appl., 165, No 4 (1993), 315-336.
  10. [10] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, In: Trends in Nonlinear Analysis, Springer, Berlin (2003), 153-191.
  11. [11] M. P´erez Llanos, J.D. Rossi, Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Analysis, 70, No 4 (2009), 1629-1640.
  12. [12] F. Quiros, J.D. Rossi, Non simultaneous blow-up in a semilinear parabolic system, Zeitschrift fur Angewandte Mathematik und Physik, 52, No 2 (2001).
  13. [13] P. Quittner, P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkha¨user Advanced Texts Basler Lehrb¨ucher (2007).
  14. [14] J.D. Rossi, On existence and nonexistence in the large for an N dimensional system of heat equations with nontrivial coupling at the boundary, New Zealand J. Math., 26 (1997), 275-285.
  15. [15] J.D. Rossi, P. Souplet, Coexistence of simultaneous and non-simultaneous blow-up in a semilinear parabolic system, Differential and Integral Equations, 18, No 4 (2005), 405-418.
  16. [16] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, and A.P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin (1995).
  17. [17] P. Souplet, S. Tayachi, Optimal condition for non-simultaneous blow-up in a reaction-difusion system, J. Math. Soc. Japan, 56, No 2 (2004), 571-584.
  18. [18] M. Wang, Blow-up rate estimates for semilinear parabolic systems, J. Math. Anal. Appl., 257 (2001), 46-51.