This paper deals with the existence and non-xistence of the solutions to the Cauchy problem of a class of quasilinear parabolic equations with a gradient term. We establish Fujita-type blowup theorems and determine the critical Fujita exponent in terms of spatial dimension, the asymptotic behavior of the coefficients of the gradient term at infinity, the exponents of spatial positions in the coefficients of the time-derivative term, and the source term. In particular, we classify the critical case as the blowup case.
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J. Aguirre and M. Escobedo, On the blow-up of solutions of a convective reaction diffusion equation, Proc. R. Soc. Edinb., Sect. A, Math., 123(3):433–460, 1993.
D. Andreucci, G. Cirmi, S. Leonardi, and A. Tedeev, Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary, J. Differ. Equations, 174(2):253–288, 2001.
K. Deng and H. Levine, The role of critical exponents in blow-up theorems: The sequel, J. Math. Anal. Appl., 243(1): 85–126, 2000.
M. Fira and B. Kawohl, Large time behavior of solutions to a quasilinear parabolic equation with a nonlinear boundary condition, Adv.Math. Sci. Appl., 11(1):113–126, 2001.
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964.
H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = Δu + u1+α, J. Fac. Sci., Univ. Tokyo, Sect. I, 13:109–124, 1966.
K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49: 503–525, 1973.
K. Kobayashi, T. Siaro, and H. Tanaka, On the blowing up problem for semilinear heat equations, J. Math. Soc. Japan, 29:407–424, 1977.
H. Levine, The role of critical exponents in blow-up theorems, SIAM Rev., 32(2):262–288, 1990.
H. Levine and Q. Zhang, The critical Fujita number for a semilinear heat equation in exterior domains with homogeneous Neumann boundary values, Proc. R. Soc. Edinb., Sect. A, Math., 130(3):591–602, 2000.
H. Li, X. Wang, Y. Nie, and H. He, Asymptotic behavior of solutions to a degenerate quasilinear parabolic equation with a gradient term, Electron. J. Differ. Equ., 2015:298, 2015.
P. Meier, On the critical exponent for reaction-diffusion equations, Arch. Ration. Mech. Anal., 109(1):63–71, 1990.
Y. Na, M. Zhou, X. Zhou, and G. Gai, Blow-up theorems of Fujita type for a semilinear parabolic equation with a gradient term, Adv. Difference Equ., 2018:128, 2018.
Y. Qi, The critical exponents of parabolic equations and blow-up in Rn, Proc. R. Soc. Edinb., Sect. A, Math., 128(1): 123–136, 1998.
Y. Qi and M. Wang, Critical exponents of quasilinear parabolic equations, J. Math. Anal. Appl., 267(1):264–280, 2002.
R. Suzuki, Existence and nonexistence of global solutions to quasilinear parabolic equations with convection, Hokkaido Math. J., 27(1):147–196, 1998.
C.Wang and S. Zheng, Critical Fujita exponents of degenerate and singular parabolic equations, Proc. R. Soc. Edinb., Sect. A, Math., 136(2):415–430, 2006.
C. Wang and S. Zheng, Fujita-type theorems for a class of nonlinear diffusion equations, Differ. Integral Equ., 26(5–6):555–570, 2013.
C. Wang, S. Zheng, and Z. Wang, Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data, Nonlinearity, 20(6):1343–1359, 2007.
Z.Wang, J. Yin, and C. Wang, Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition, Appl. Math. Lett., 20(2):142–147, 2007.
M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25(11):911–925, 2002.
Z. Wu, J. Zhao, J. Yin, and H. Li, Nonlinear Diffusion Equations, World Scientific, Singapore, 2001.
Q. Zhang, A general blow-up result on nonlinear boundary-value problems on exterior domains, Proc. R. Soc. Edinb., Sect. A, Math., 131(2):451–475, 2001.
S. Zheng, X. Song, and Z. Jiang, Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl., 298(1):308–324, 2004.
S. Zheng and C.Wang, Large time behaviour of solutions to a class of quasilinear parabolic equations with convection terms, Nonlinearity, 21(9):2179–2200, 2008.
Q. Zhou, Y. Nie, and X. Han, Large time behavior of solutions to semilinear parabolic equations with gradient, J. Dyn. Control Syst., 22(1):191–205, 2016.
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Zhou, M., Leng, Y. Existence and Nonexistence of the Solutions to the Cauchy Problem of Quasilinear Parabolic Equation with a Gradient Term. Lith Math J (2021). https://doi.org/10.1007/s10986-021-09511-2
- critical Fujita exponent
- quasilinear parabolic equation
- gradient term