Results of nonexistence of solutions for some nonlinear evolution problems

Medjahed Djilali; Ali Hakem

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 2, page 269-284
  • ISSN: 0010-2628

Abstract

top
In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), u t t + f ( x ) u t + ( - Δ ) α / 2 ( u m ) = h ( t , x ) | u | p , posed in ( 0 , T ) × ℝ N , where ( - Δ ) α / 2 , 0 < α ≤ 2 is α / 2 -fractional power of - Δ . Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a 2 × 2 system of the same type.

How to cite

top

Djilali, Medjahed, and Hakem, Ali. "Results of nonexistence of solutions for some nonlinear evolution problems." Commentationes Mathematicae Universitatis Carolinae 60.2 (2019): 269-284. <http://eudml.org/doc/294828>.

@article{Djilali2019,
abstract = {In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), \[ u\_\{tt\} +f(x)u\_t +(-\Delta )^\{\alpha /2\}(u^m)= h(t,x) |u|^\{p\}, \] posed in $(0,T)\times \mathbb \{R\}^\{N\},$ where $(-\Delta )^\{\{\alpha \}/\{2\}\}, 0<\alpha \le 2$ is $\{\alpha \}/\{2\}$-fractional power of $\,-\Delta .$ Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a $2\times 2$ system of the same type.},
author = {Djilali, Medjahed, Hakem, Ali},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nonexistence; test functions; global weak solution; fractional Laplacian; critical exponent},
language = {eng},
number = {2},
pages = {269-284},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Results of nonexistence of solutions for some nonlinear evolution problems},
url = {http://eudml.org/doc/294828},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Djilali, Medjahed
AU - Hakem, Ali
TI - Results of nonexistence of solutions for some nonlinear evolution problems
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 2
SP - 269
EP - 284
AB - In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), \[ u_{tt} +f(x)u_t +(-\Delta )^{\alpha /2}(u^m)= h(t,x) |u|^{p}, \] posed in $(0,T)\times \mathbb {R}^{N},$ where $(-\Delta )^{{\alpha }/{2}}, 0<\alpha \le 2$ is ${\alpha }/{2}$-fractional power of $\,-\Delta .$ Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a $2\times 2$ system of the same type.
LA - eng
KW - nonexistence; test functions; global weak solution; fractional Laplacian; critical exponent
UR - http://eudml.org/doc/294828
ER -

References

top
  1. Cazenave T., Haraux A., Introduction aux problèmes d'évolution semi-linéaires, Mathématiques & Applications, 1. Ellipses, Paris, 1990 (French). MR1299976
  2. Fino A. Z., Ibrahim H., Wehbe A., 10.1016/j.camwa.2017.03.030, Comput. Math. Appl. 73 (2017), no. 11, 2415–2420. MR3648022DOI10.1016/j.camwa.2017.03.030
  3. Fujita H., On the blowing up of solutions of the problem for u t = Δ u + u 1 + α , J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124. MR0214914
  4. Guedda M., Kirane M., 10.36045/bbms/1103055577, Bull. Belg. Math. Soc. Simon Stevin 6 (1999), no. 4, 491–497. MR1732885DOI10.36045/bbms/1103055577
  5. Guedda M., Kirane M., Local and global nonexistence of solutions to semilinear evolution equations, Proc. of the 2002 Fez Conf. on Partial Differential Equations, Electron. J. Differ. Equ. Conf. 09 (2002), 149–160. MR1976692
  6. Hakem A., 10.36045/bbms/1113318131, Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 1, 73–82. MR2134858DOI10.36045/bbms/1113318131
  7. Hakem A., Berbiche M., On the blow-up behavior of solutions to semi-linear wave models with fractional damping, IAENG Int. J. Appl. Math. 41 (2011), no. 3, 206–212. MR2954229
  8. Ju N., 10.1007/s00220-004-1256-7, Comm. Math. Phys. 255 (2005), 161–181. MR2123380DOI10.1007/s00220-004-1256-7
  9. Ogawa T., Takeda H., 10.1016/j.na.2008.07.025, Nonlinear Anal. 70 (2009), no. 10, 3696–3701. MR2504456DOI10.1016/j.na.2008.07.025
  10. Pozrikidis C., The Fractional Laplacian, CRC Press, Boca Raton, 2016. MR3470013
  11. Sun F., Wang M., 10.1016/j.na.2006.04.012, Nonlinear Anal. 66 (2007), no. 12, 2889–2910. MR2311644DOI10.1016/j.na.2006.04.012
  12. Takeda H., 10.1016/j.jmaa.2009.06.072, J. Math. Anal. Appl. 360 (2009), no. 2, 631–650. MR2561260DOI10.1016/j.jmaa.2009.06.072
  13. Todorova G., Yordanov B., 10.1016/S0764-4442(00)00228-7, C. R. Acad. Sci. Paris Sèrie I Math. 330 (2000), no. 7, 557–562. MR1760438DOI10.1016/S0764-4442(00)00228-7
  14. Zhang Q. S., 10.1016/S0764-4442(01)01999-1, C. R. Acad. Sci. Paris Sèrie I Math. 333 (2001), no. 2, 109–114. MR1847355DOI10.1016/S0764-4442(01)01999-1
  15. Zheng S., Nonlinear Evolution Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 113, Chapman & Hall/CRC Press, Boca Raton, 2004. MR2088362

NotesEmbed ?

top

You must be logged in to post comments.