Acta mathematica scientia, Series B >
OPTIMAL CONDITIONS OF GLOBAL EXISTENCE AND BLOW-UP FOR A NONLINEAR PARABOLIC EQUATION
Received date: 2015-02-06
Revised date: 2015-11-22
Online published: 2016-06-25
Supported by
The first author was supported by National Natural Science Foundation of China (11126336 and 11201324), New Teachers' Fund for Doctor Stations, Ministry of Education (20115134120001), Fok Ying Tuny Education Foundation (141114), Youth Fund of Sichuan Province (2013JQ0027)
According to the variational analysis and the potential well argument, we get the optimal conditions of global existence and blow-up for a type of nonlinear parabolic equations. Furthermore, we give its application in the instability of the steady states.
Yi JIANG , Chuan LUO . OPTIMAL CONDITIONS OF GLOBAL EXISTENCE AND BLOW-UP FOR A NONLINEAR PARABOLIC EQUATION[J]. Acta mathematica scientia, Series B, 2016 , 36(3) : 872 -880 . DOI: 10.1016/S0252-9602(16)30046-7
[1] Wang M X, Ding X Q. Global existence, asymptotic behavior, and blow-up problems for a semilinear heat equation. Science in China (Series A), 1993, 36(4): 420-430
[2] Gan Z H, Zhang J. Sharp conditions of global existence for the generalized Davey-Stewartson system in three dimensional space. Acta Mathematica Scientia, 2006, 26A(1): 087-092
[3] Todorova G. Stable and unstable sets for the Cauchy ptoblem for a nonlinear wave equation with nonlinear damping and soutce terms. J Math Anal Appl, 1999, 239: 213-226
[4] Ma L. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Commun Pure Appl Anal, 2013, 12(2): 1103-1110
[5] Zhang J. Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations. Nolinear Analysis TMA, 2002, 48: 191-207
[6] Kobayashi K, Sirao T, et al. On the growing up problem for semilinear heat equations. J Math Soc Japan, 1977, 29: 407-424
[7] Tsutsumi M. Existence & non-existence of global solutions for nonlinear parabolic equations. Publ RIMS, 1972, 8(73): 211-299
[8] Weissler F B. Existence & non-existence of global solutions for a semilinear heat equations. Isreal J Math, 1981, 38: 29-40
[9] Lou B D. Positive equilibrium solutions of semilinear parabolic equations. Acta Math Sci, 2006, 26B(4): 670-678
[10] Adams R. Sobolev Spaces. New York: Academic Press, 1975
[11] Zhang J. Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials. Z Angew Math Phys, 2000, 51: 498-503
/
| 〈 |
|
〉 |