Asymptotic behavior of solutions to a class of coupled semilinear parabolic systems with gradient terms
-
1957
Downloads
-
3484
Views
Authors
Yang Na
- School of Mathematics, Jilin University, Changchun 130012, P. R. China.
Yuanyuan Nie
- School of Mathematics, Jilin University, Changchun 130012, P. R. China.
Xu Zhou
- College of Computer Science and Technology, Jilin University, Changchun 130012, P. R. China.
Abstract
This paper concerns the asymptotic behavior of solutions
to the Cauchy problem of a class of coupled semilinear parabolic systems with gradient terms.
Using the energy comparison method and comparison principle,
the blow-up theorem of Fujita type is established
and the critical Fujita curve is formulated by spacial dimension,
the behavior of the coefficient of the gradient term at infinity.
Share and Cite
ISRP Style
Yang Na, Yuanyuan Nie, Xu Zhou, Asymptotic behavior of solutions to a class of coupled semilinear parabolic systems with gradient terms, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 11, 5813--5824
AMA Style
Na Yang, Nie Yuanyuan, Zhou Xu, Asymptotic behavior of solutions to a class of coupled semilinear parabolic systems with gradient terms. J. Nonlinear Sci. Appl. (2017); 10(11):5813--5824
Chicago/Turabian Style
Na, Yang, Nie, Yuanyuan, Zhou, Xu. "Asymptotic behavior of solutions to a class of coupled semilinear parabolic systems with gradient terms." Journal of Nonlinear Sciences and Applications, 10, no. 11 (2017): 5813--5824
Keywords
- Asymptotic behavior
- critical Fujita curve
- gradient term
MSC
References
-
[1]
K. Deng, H. A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl., 243 (2000), 85–126.
-
[2]
M. Escobedo, M. A. Herrero, Boundedness and blow up for a semilinear reaction–diffusion system, J. Differential Equations, 89 (1991), 176–202.
-
[3]
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 Sect. I, 13 (1966), 109–124.
-
[4]
W. Guo, M. Lei, Critical Fujita curves for a coupled reaction-convection-diffusion system with singular coefficients, J. Jilin Univ. Sci., 54 (2016), 183–188.
-
[5]
W. Guo, X. Wang, M. Zhou, Asymptotic behavior of solutions to a class of semilinear parabolic equations, Bound. Value Probl., 2016 (2016), 9 pages.
-
[6]
K. Hayakawa , On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503–505.
-
[7]
K. Kobayashi, T. Siaro, H. Tanaka, On the blowing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407–424.
-
[8]
H. A. Levine, The role of critical exponents in blow-up theorems , SIAM Rev., 32 (1990), 262–288.
-
[9]
H.-L. Li, X.-Y. Wang, Y.-Y. Nie, H. He, Asymptotic behavior of solutions to a degenerate quasilinear parabolic equation with a gradient term, Electron. J. Differential Equations, 2015 (2015), 12 pages.
-
[10]
P. Meier , On the critical exponent for reaction–diffusion equations, Arch. Rational Mech. Anal., 109 (1990), 63–71.
-
[11]
Y.-W. Qi, The critical exponents of parabolic equations and blow-up in \(R^n\), Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 123–136.
-
[12]
Y.-W. Qi, M.-X. Wang, Critical exponents of quasilinear parabolic equations , J. Math. Anal. Appl., 267 (2002), 264–280.
-
[13]
P. Quittner, P. Souplet , Superlinear parabolic problems: blow-up, global existence and steady states, Birkhäuser, Basel (2007)
-
[14]
R. Suzuki , Existence and nonexistence of global solutions to quasilinear parabolic equations with convection, Hokkaido Math. J., 27 (1998), 147–196.
-
[15]
C.-P. Wang, Asymptotic behavior of solutions to a class of semilinear parabolic equations with boundary degeneracy, Proc. Amer. Math. Soc., 141 (2013), 3125–3140.
-
[16]
C.-P.Wang, S. Zheng, Critical Fujita exponents of degenerate and singular parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 415–430.
-
[17]
C.-P. Wang, S. Zheng , Fujita-type theorems for a class of nonlinear diffusion equations, Differential Integral Equations, 26 (2013), 555–570.
-
[18]
C.-P. Wang, S. Zheng, Z. Wang, Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data, Nonlinearity, 20 (2007), 1343–1359.
-
[19]
F. B. Weissler , Existence and non–existence of global solutions for semilinear equation, Israel J. Math., 38 (1981), 29–40.
-
[20]
S. Zheng, X.-F. Song, Z.-X. Jiang , Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl., 298 (2004), 308–324.
-
[21]
S. Zheng, C.-P. Wang, Large time behaviour of solutions to a class of quasilinear parabolic equations with convection terms, Nonlinearity, 21 (2008), 2179–2200.
-
[22]
M.-J. Zhou, H.-L. Li, W. Guo, X. Zhou , Critical Fujita exponents to a class of non-Newtonian filtration equations with fast diffusion, Bound. Value Probl., 2016 (2016), 16 pages.
-
[23]
Q. Zhou, Y.-Y. Nie, X.-Y. Han, Large time behavior of solutions to semilinear parabolic equations with gradient , J. Dyn. Control Syst., 22 (2016), 191–205.
