CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION WITH VARIABLE COEFFICIENTS

Title & Authors
CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION WITH VARIABLE COEFFICIENTS
Li, Zhongping; Mu, Chunlai; Du, Wanjuan;

Abstract
In this paper, we consider the positive solution to a Cauchy problem in $\small{\mathbb{B}^N}$ of the fast diffusive equation: ${\mid}x{\mid}^mu_t Keywords critical Fujita exponent;fast diffusive equation;variable coefficients; Language English Cited by 1. Existence and nonexistence of global solutions for a semilinear reaction–diffusion system, Journal of Mathematical Analysis and Applications, 2017, 445, 1, 97 References 1. K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. 243 (2000), no. 1, 85-126. 2. E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. 3. A. Friedman and J. B.McLeod, Blow-up of positive solutions of semilinear heat equation, Indiana Univ. Math. J. 34 (1985), no. 2, 425-447. 4. H. Fujita, On the blowing up of solutions of the Cauchy problem for${u_t} = {\Delta}u + {u^{1+{\alpha}}}$, J. Fac. Sci. Univ. Tokyo Sec. I 13 (1966), 109-124. 5. V. A. Galaktionov, Conditions for global nonexistence and localization for a class of nonlinear parabolic equations, Comput. Math. Math. Phys. 23 (1983), 35-44. 6. V. A. Galaktionov, Blow-up for quasilinear heat equations with critical Fujita's exponents, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 3, 517-525. 7. V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov, and A. A. Samarskii, On unbounded solutions of the Cauchy problem for the parabolic equation${u_t} = {\nabla}({u^{\sigma}}{\nabla}u)+{u^B}$, Soviet Phys. Dokl. 25 (1980), 458-459. 8. V. A. Galaktionov, Blowup in Quasilinear Parabolic Equations, De Gruyter Expositions in Mathematics, Springer, Berlin, 1995. 9. V. A. Galaktionov and H. A. Levine, A general approach to critical Fujita exponents and systems, Nonlinear Anal. 34 (1998), no. 7, 1005-1027. 10. K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad. 49 (1973), 503-505. 11. Q. Huang, K. Mochizuki, and K. Mukai, Life span and asymptotic behavior for a semilinear parabolic system with slowly decaying initial values, Hokkaido Math. J. 27 (1998), no. 2, 393-407. 12. H. A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev. 32 (1990), no. 2, 262-288. 13. Z. P. Li and C. L. Mu, Critical exponents for a fast diffusive polytropic filtration equation with nonlinear boundary flux, J. Math. Anal. Appl. 346 (2008), no. 1, 55-64. 14. Z. P. Li, C. L. Mu, and L. Xie, Critical curves for a degenerate parabolic equation with multiple nonlinearities, J. Math. Anal. Appl. 359 (2009), no. 1, 39-47. 15. Z. P. Li, C. L. Mu, and Z. J. Cui, Critical curves for a fast diffusive polytropic filtration system coupled via nonlinear boundary flux, Z. Angew. Math. Phys. 60 (2008), no. 2, 284-298. 16. A. V. Martynenko and A. F. Tedeev, The Cauchy problem for a quasilinear parabolic equation with a source and nonhomogeneous density, Comput. Math. Math. Phys. 47 (2007), no. 2, 238-48. 17. A. V. Martynenko and A. F. Tedeev, On the behavior of solutions to the Cauchy problem for a degenerate parabolic equation with inhomogeneous density and a source, Comput. Math. Math. Phys. 48 (2008), no. 7, 1145-1160. 18. K. Mochizuki and K. Mukai, Existence and nonexistence of global solutions to fast diffusions with source, Methods Appl. Anal. 2 (1995), no. 1, 92-102. 19. K. Mochizuki and R. Suzuki, Critical exponent and critical blow-up for quasilinear parabolic equations, Israel J. Math. 98 (1997), no. 1, 141-156. 20. Y. W. Qi, Critical exponents of degenerate parabolic equations, Sci. China Ser. A 38 (1995), no. 10, 1153-1162. 21. Y. W. Qi, The critical exponents of parabolic equations and blow-up in${\mathbb{R}^N}\$, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 1, 123-136.

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