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Global existence and nonexistence for degenerate parabolic systems

Author(s): Yuxiang Li; Weibing Deng; Chunhong Xie
Journal: Proc. Amer. Math. Soc. 130 (2002), 3661-3670.
MSC (2000): Primary 35K50, 35K55, 35K65
Posted: May 14, 2002
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Abstract: The initial-boundary value problems are considered for the strongly coupled degenerate parabolic system

\begin{displaymath}\begin{split} u_t=v^p(\Delta u+au), v_t=u^q(\Delta v+bv) \end{split}\end{displaymath}

in the cylinder $\Omega\times(0,\infty)$, where $\Omega\subset R^N$ is bounded and $p, q, a, b$ are positive constants. We are concerned with the global existence and nonexistence of the positive solutions. Denote by $\lambda_1$ the first Dirichlet eigenvalue for the Laplacian on $\Omega$. We prove that there exists a global solution iff $\lambda_1\geq \min\{a,b\}$.

References:

1.
J. R. Anderson, Local existence and uniqueness of solutions of degenerate parabolic equations, Comm. Partial Differential Equations 16 (1991), 105-143. MR 92d:35163
2.
R. Courant and D. Hilbert, Methods of mathematical physics I, II, Interscience, New York, 1953, 1962. MR 16:426a; MR 25:4216
3.
E. DiBenedetto, Degenerate parabolic equations, Springer-Verlag, New York, 1993. MR 94h:35130
4.
E. DiBenedetto, Continuity of weak solutions to a general porous media equation, Indiana Univ. Math. J. 32 (1983), 83-118. MR 85c:35010
5.
Z. W. Duan and L. Zhou, Global and blow-up solutions for nonlinear degenerate parabolic systems with crosswise-diffusion, J. Math. Anal. Appl. 244 (2000), 263-278. MR 2001b:35181
6.
M. S. Floater, Blow-up at the boundary for degenerate semilinear parabolic equations, Arch. Rational Mech. Anal. 114 (1991), 57-77. MR 92m:35139
7.
A. Friedman and B. McLeod, Blow-up of solutions of nonlinear degenerate parabolic equations, Arch. Rational Mech. Anal. 96 (1987), 55-80. MR 87j:35051
8.
A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, N. J., 1964. MR 31:6062
9.
V. A. Galaktionov, A boundary value problem for the nonlinear parabolic equation $u_t=\Delta u^{\sigma+1}+u^\beta$, Differential Equations 17 (1981), 836-842. (Russian) MR 82m:35074
10.
V. A. Galaktionov, S. P. Kurdyumov and A. Samarskii, A parabolic system of quasi-linear equations I, Differential Equations 19 (1983), 1558-1571.
11.
V. A. Galaktionov, S. P. Kurdyumov and A. Samarskii, A parabolic system of quasi-linear equations II, Differential Equations 21 (1985), 1049-1062.
12.
H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review 32 (1990), 262-288. MR 91j:35135
13.
O. A. Lady$\breve{z}$enskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Amer. Math. Soc., Providence, 1967. MR 39:3159h
14.
C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992. MR 94c:35002
15.
R. Dal Passo and S. Luckhaus, A degenerate diffusion problem not in divergence form, J. Differential Equations 69 (1987), 1-14. MR 88k:35091
16.
M. A. Protter and H. F. Weinberger, Maximum principles in differential equations, Springer-Verlag, 1999.
17.
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), 681-703. MR 88b:35072
18.
P. E. Sacks, Global behavior for a class of nonlinear evolutionary equations, SIAM J. Math. Anal. 16 (1985), 233-250. MR 86f:35031
19.
M. Wiegner, Blow-up for solutions of some degenerate parabolic equations, Differential Integral Equations 7 (1994), 1641-1647. MR 95b:35120
20.
S. Wang, M. X. Wang and C. H. Xie, A nonlinear degenerate diffusion equation not in divergence form, Z. Angew. Math. Phys. 51 (2000), 149-159. MR 2001a:35101
21.
C. H. Xie, T. P. He and G. H. Bai, Existence and invariance for convection, dispersion, reaction systems with pressure, Acta Math. Sci. 12 (1992), 443-456. MR 94i:35100

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Additional Information:

Yuxiang Li
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China
Email: lieyuxiang@yahoo.com.cn

Weibing Deng
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China

Chunhong Xie
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China

DOI: 10.1090/S0002-9939-02-06630-3
PII: S 0002-9939(02)06630-3
Keywords: Global existence, global nonexistence, degenerate parabolic system
Received by editor(s): July 23, 2001
Posted: May 14, 2002
Communicated by: David S. Tartakoff
Copyright of article: Copyright 2002, American Mathematical Society

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