Title: Blow-Up, Exponential Grouth of Solution for a Nonlinear Parabolic Equation with p(x) - Laplacian
Author(s): Amar Ouaoua, Messaoud Maouni
Pages: 620-629
Cite as:
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.

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.


Full Text: PDF

 

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. Google Scholar

  2. G. Akagi and M. Otani, Evolutions inclusions governed by subdifferentials in reflexive Banach spaces, J. Evol. Equ. 4 ˆ (2004), 519–541. Google Scholar

  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. Google Scholar

  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. Google Scholar

  5. Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functions in image restoration. SIAM J. Appl. Math. 66 (2006), 1383-1406. Google Scholar

  6. D. Edmunds and J. Rakosnik, Sobolev embeddings with variable exponent, Stud. Math. 143 (3) (2000), 267–293. Google Scholar

  7. D. Edmunds and J. Rakosnik Sobolev embeddings with variable exponent. II, Math. Nachr. 246 (1) (2002), 53–67. Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  13. D. Lars, P. Harjulehto, P. Hasto and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, in: Lecture Notes in Mathematics, Springer, 2011. Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  17. M. Otani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy ˆ problems, J. Differential Equations 46 (1982), 268–299. Google Scholar

  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. Google Scholar