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CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION WITH VARIABLE COEFFICIENTS
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 Title & Authors
CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION WITH VARIABLE COEFFICIENTS
Li, Zhongping; Mu, Chunlai; Du, Wanjuan;
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 Abstract
In this paper, we consider the positive solution to a Cauchy problem in of the fast diffusive equation: , with nontrivial, nonnegative initial data. Here < < 2, > 1 and 0 < < . We prove that is the critical Fujita exponent. That is, if 1 < , then every positive solution blows up in finite time, but for > , there exist both global and non-global solutions to the problem.
 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  crossref(new windwow)
 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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

10.
K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad. 49 (1973), 503-505. crossref(new window)

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. crossref(new window)

12.
H. A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev. 32 (1990), no. 2, 262-288. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

22.
Y. W. Qi, The global existence and nonuniqueness of a nonlinear degenerate equation, Nonlinear Anal. 31 (1998), no. 1-2, 117-136. crossref(new window)

23.
Y. W. Qi and H. A. Levine, The critical exponent of degenerate parabolic systems, Z. Angew. Math. Phys. 44 (1993), no. 2, 249-265. crossref(new window)

24.
Y. W. Qi and M. X. Wang, Critical exponents of quasilinear parabolic equations, J. Math. Anal. Appl. 267 (2002), no. 1, 264-280. crossref(new window)

25.
A. E. Scheidegger, The Physics of Flow Through Porous Media, third ed., Buffalo, Toronto, 1974.

26.
C. P. Wang and S. N. Zheng, Critical Fujita exponents of degenerate and singular parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), no. 2, 415-430. crossref(new window)

27.
C. P. Wang, S. N. Zheng, and Z. J. Wang, Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data, Nonlinearity 20 (2007), no. 6, 1343-1359. crossref(new window)

28.
Z. J. Wang, J. X. Yin, C. P. Wang, and H. Gao, Large time behavior of solutions to Newtonian filtration equation with nonlinear boundary sources, J. Evol. Equ. 7 (2007), no. 4, 615-648. crossref(new window)

29.
F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), no. 1-2, 29-40. crossref(new window)

30.
Z. Q. Wu, J. N. Zhao, J. X. Yin, and H. L. Li, Nonlinear Diffusion Equations, Word Scientific, Singapore, 2001.

31.
Y. B. Zeldovich and Y. P. Raizer, Physics and shock waves and high-temperature hydrodynamic phenomena, Moscow, vol. 2, Academic Press, New York, 1967.

32.
S. N. Zheng and C. P. Wang, Large time behaviour of solutions to a class of quasilinear parabolic equations with convection terms, Nonlinearity 21 (2008), no. 9, 2179-2200. crossref(new window)