Blow-Up, Exponential Grouth of Solution for a Nonlinear Parabolic Equation with p(x) - Laplacian

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Amar Ouaoua, Messaoud Maouni, Blow-Up, Exponential Grouth of Solution for a Nonlinear Parabolic Equation with p(x) - Laplacian, Int. J. Anal. Appl., 17 (4) (2019), 620-629.

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Amar Ouaoua, Messaoud Maouni

Abstract

In this paper, we consider the following equation

$u_{t}-\func{div}\left( \left\vert \nabla u\right\vert ^{p\left( x\right)-2}\nabla u\right) +\omega \left\vert u\right\vert ^{m\left( x\right)-2}u_{t}=b\left\vert u\right\vert ^{r\left( x\right) -2}u.$

We prove a finite time blowup result for the solutions in the case $\omega =0$ and exponential growth in the case $\omega >0$, with the negative initial energy in the both case.

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References

  1. G. Akagi, Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian, J. Differential Equations 241 (2007), 359-385.
  2. G. Akagi and M. Otani, Evolutions inclusions governed by subdifferentials in reflexive Banach spaces, J. Evol. Equ. 4 ˆ (2004), 519-541.
  3. G. Akagi and M. Otani, Evolutions inclusions governed by the difference of two subdifferentials in reflexive Banach spaces, ˆ J. Differential Equations 209 (2005), 392-415.
  4. S.N. Antontsev and V. Zhikov, Higher integrability for parabolic equations of p(x, t)-Laplacian type. Adv. Differ. Equ. 10 (2005), 1053-1080.
  5. Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functions in image restoration. SIAM J. Appl. Math. 66 (2006), 1383-1406.
  6. D. Edmunds and J. Rakosnik, Sobolev embeddings with variable exponent, Stud. Math. 143 (3) (2000), 267-293.
  7. D. Edmunds and J. Rakosnik Sobolev embeddings with variable exponent. II, Math. Nachr. 246 (1) (2002), 53-67.
  8. X. Fan and D. Zhao, On the spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2) (2001), 424-446.
  9. H. Fujita, On the blowing up solutions of the Cauchy problem for ut = ∆u+ u 1+α, J. Fac. Sci. Univ. Tokyo Sect. A.Math. 16 (1966), 105-113.
  10. Y. Gao, B. Guo and W.Gao, Weak solutions for a high-order pseudo-parabolic equation with variable exponents. Appl. Anal. 93 (2) (2014), 322-338.
  11. Z. Jiang, S. Zheng, and X. Song, Blow-up analysis for a nonlinear diffusion equation with nonlinear boundary conditions, Appl. Math. Lett. 17 (2) (2004), 193-199.
  12. A.M. Kbiri, S.A. Messaoudi. and H.B. Khenous, A blow-up result for nonlinear generalized heat equation, Comput. Math. Appl. 68 (12) (2014), 1723-1732.
  13. D. Lars, P. Harjulehto, P. Hasto and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, in: Lecture Notes in Mathematics, Springer, 2011.
  14. J. Leray and J.L.Lions, Quelques r ´esultats de Visick sur les probl ´emes elliptiques non lin ´eaires pour les m ´ethodes de Minty-Browder, Bull. Soc. Math. France 93 (1965), 97-107.
  15. S.Z. Lian, W.J. Gao, CL. Cao and HJ. Yuan, Study of the solutions to a model porous medium equation with variable exponents of nonlinearity. J. Math. Anal. Appl. 342 (2008), 27-38.
  16. S.A. Messaoudi and A. Talahmeh, Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Math. Methods Appl. Sci. 40 (18) (2017), 6976-6986.
  17. M. Otani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy ˆ problems, J. Differential Equations 46 (1982), 268-299.
  18. N. Polat, Blow up of solution for a nonlinear reaction diffusion equation with multiple nonlinearities, Int. J. Sci. Technol. 2 (2) (2007), 123-128.