Acta mathematica scientia, Series B >
ON THE CAUCHY PROBLEM FOR A REACTION-DIFFUSION SYSTEM WITH SINGULAR NONLINEARITY
Received date: 2012-05-22
Revised date: 2012-09-26
Online published: 2013-07-20
Supported by
This research was supported by NSFC (11201380), the Fundamental Research Funds for the Central Universities (XDJK2012B007), Doctor Fund of Southwest University (SWU111021) and Educational Fund of Southwest University (2010JY053).
We consider the growth rate and quenching rate of the following problem with singular nonlinearity
ut = △u − v−λ, vt = △v − u−μ, (x, t) ∈ Rn × (0,∞),
u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈Rn
for any n ≥ 1, where λ, μ > 0 are constants. More precisely, for any u0(x), v0(x) satisfying A11(1+|x|2) α11 ≤ u0 ≤ A12(1+|x|2) α12 , A21(1+|x|2) α21 ≤ v0 ≤ A22(1+|x|2) α22 for some constants α12 ≥ α11, α22 ≥α 21, A12 ≥ A11, A22 ≥ A21, the global solution (u, v) exists and satisfies A11(1+|x|2+b1t) α11 ≤ u ≤ A12(1+|x|2+b2t) α12 , A21(1+|x|2+b1t) α21 ≤ v ≤ A22(1+|x|2+b2t) α22 for some positive constants b1, b2 (see Theorem 3.3 for the parameters Aij , αij , bi, i, j = 1, 2). When (1 − λ)(1 − λμ) > 0, (1 − λ)(1 − λμ) > 0 and 0 < u0 ≤A1(b1T +|x|2)1−λ/1−λμ , 0 < v0 ≤ A2(b2T +|x|2)1−μ/1−λμ in Rn for some constants Ai, bi (i = 1, 2)satisfying A−λ2 > 2nA11−λ/1−λμ , A−μ1 > 2nA21−μ/1−λμ and 0 < b1 ≤ (1−λμ)A−λ2−(1−λ)2nA1/(1−λ)A1, 0 < b2 ≤ (1−λμ)A−μ1−(1−μ)2nA2/(1−μ)A2, we prove that u(x, t) ≤ A1(b1(T −t)+|x|2)1−λ/1−λμ , v(x, t) ≤A2(b2(T − t) + |x|2)1−μ/1−λμ in Rn × (0, T). Hence, the solution (u, v) quenches at the origin x = 0 at the same time T (see Theorem 4.3). We also find various other conditions for the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time.
Key words： Cauchy problems; singular nonlinearity; growth rate; quenching rate
ZHOU Jun . ON THE CAUCHY PROBLEM FOR A REACTION-DIFFUSION SYSTEM WITH SINGULAR NONLINEARITY[J]. Acta mathematica scientia, Series B, 2013 , 33(4) : 1031 -1048 . DOI: 10.1016/S0252-9602(13)60061-2
[1] Chen H. Global existence and blow-up for a nonlinear reaction-diffusion system. J Math Anal Appl, 1997, 212: 481–492
[2] Chen S. Global existence and nonexistence for some degenerate and quasilinear parabolic systems. J Differ Equ, 2008, 245: 1112–1136
[3] Escobedo M, Herrero M A. Boundedness and blow up for a semilinear reaction-diffusion system. J Differ Equ, 1991, 89: 176–202
[4] Esposito P, Ghoussoub N, Guo Y. Mathematical analysis of partial differential equations modeling elec-trostatic MEMS//Volume 20 of Courant Lecture Notes in Mathematics. New York: Courant Institute of Mathematical Sciences, 2010
[5] Fila M, Hulshof J. A note on the quenching rate. Proc Amer Math Soc, 1991, 112: 473–477
[6] Fila M, Levine H A, V´azquez J L. Stabilization of solutions of weakly singular quenching problems. Proc Amer Math Soc, 1993, 119: 555–559
[7] Flores G, Mercado G A, Pelesko J A. Dynamics and touchdown in electrostatic mems//MEMS, NANO and Smart Systems, 2003. Proceedings International Conference on IEEE, 2003: 182–187
[8] Friedman A. Partial Differential Equations of Parabolic Type. Englewood Cliffs, N J: Prentice-Hall Inc, 1964
[9] Ghoussoub N, Guo Y. On the partial differential equations of electrostatic MEMS devices: stationary case. SIAM J Math Anal, 2006, 38: 1423–1449
[10] Ghoussoub N, Guo Y. On the partial differential equations of electrostatic MEMS devices. II. Dynamic case. NoDEA Nonlinear Differ Equ Appl, 2008, 15: 115–145
[11] Guo Y. On the partial differential equations of electrostatic MEMS devices. III. Refined touchdown behavior. J Differ Equ, 2008, 244: 2277–2309
[12] Guo Y, Pan Z, Ward M J. Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties. SIAM J Appl Math, 2005 66: 309–338
[13] Guo Z, Wei J. On the Cauchy problem for a reaction-diffusion equation with a singular nonlinearity. J Differ Equ, 2007, 240: 279–323
[14] Hui K M. A Fatou theorem for the solution of the heat equation at the corner points of a cylinder. Trans Amer Math Soc, 1992, 333: 607–642
[15] Ladyˇzenskaja O A, Solonnikov V A, Ural′ceva N N. Linear and Quasilinear Equations of Parabolic Type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol 23. Providence, RI: American Mathematical Society, 1967
[16] Mu C, Hu X, Li Y, Cui Z. Blow-up and global existence for a coupled system of degenerate parabolic equations in a bounded domain. Acta Math Sci, 2007, 27B(1): 92–106
[17] Pao C V. Nonlinear Parabolic and Elliptic Equations. New York: Plenum Press, 1992
[18] Pelesko J A, Bernstein D H. Modeling MEMS and NEMS. Boca Raton, FL: Chapman & Hall/CRC, 2003
[19] Souplet P. Decay of heat semigroups in L∞ and applications to nonlinear parabolic problems in unbounded domains. J Funct Anal, 2000, 173: 343–360
[20] Winkler M. Nonuniqueness in the quenching problem. Math Ann, 2007, 339: 559–597
[21] Zheng S, Wang W. Non-simultaneous versus simultaneous quenching in a coupled nonlinear parabolic system. Nonlinear Anal, 2008, 69: 2274–2285
[22] Zhou J, He Y, Mu C. Incomplete quenching of heat equations with absorption. Appl Anal, 2008, 87: 523–529
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