半线性高阶抛物型方程组具慢衰减初值问题整体解的存在与不存在性
Existence and Nonexistence of Global Solutions of Higher-Order Parobolic System with Slow Decay Initial Data

作者: 孙福芹 , 胡素娟 :;

关键词: Higher-Order Parobolic System Comparison Principle Slow Decay Initial Data Global Solutions高阶抛物型方程组比较原理慢衰减初值整体解

摘要: 本文研究一类半线性高阶抛物型方程组的Cauchy问题。通过建立高阶抛物型方程组的所谓“比较原理”,利用Schauder不动点理论和试验函数等方法,证明了该问题在慢衰减初值条件下解的整体性存在性与不存在性。

Abstract: This paper concerns with the Cauchy problem of a higher-order parobolic system. By constructing a so called “comparison principle”of the higher-order parobolic system and utilizing Schauder fixed point theorem and the test function method, we prove the existence and nonexistence of global solutions to such a problem with slow decay initial data.

文章引用: 孙福芹 , 胡素娟 (2011) 半线性高阶抛物型方程组具慢衰减初值问题整体解的存在与不存在性。 理论数学, 1, 208-214. doi: 10.12677/pm.2011.13040

参考文献

[1] L. A. Peletier, W. C. Troy. Spacial patterns: Higher order models in physics and mechanics. Boston-Berlin: Birkh Auser, 2001.

[2] C. J. Budd, V. A. Galaktionov and J. F. Williams. Self-similar blow-up in higher-order semilinear parabolic equations. SIAM Journal on Ma-thematics, 2004, 64(5): 1775-1809.

[3] G. Caristi, E. Mitidieri. Existence and nonexistence of global solutions of higher-order parobolic problems with slow decay initial data. Journal of Mathematics Analysis and Application, 2003, 279(2): 710-722.

[4] S. B. Cui. Local and global existence of solutions to semilinear parabolic initial value problems. Nonlinear Analysis, 2001, 43(3): 293-323.

[5] F. Gazzola, H-C. Grunau. Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay. Calculus of Variations and Partial Differential Equations, 2007, 30(3): 389-415.

[6] V. A. Galaktionov, P. J. Harwin. Non-uniqueness and global Similarity solutions for a higher-order semilinear parabolic equation. Nonlinearity, 2005, 18: 717-746.

[7] V. A. Galaktionov, S. I. Pohozaev. Existence and blow-up for higher-order semilinear parabolic equations: Majorizing order-preserving operators. Indiana University Mathematics Journal, 2002, 51: 1321-1338.

[8] V. A. Galaktionov. On a spectrum of blow-up patterns for a higher-order semilinear parabolic equations. Proceedings: Royal Society of London, A., 2001, 457: 1-21.

[9] V. A. Galaktionov, J. F. Williams. On very singular sililarity solutions of a higher-order semilinear parabolic equation. Nonlinearity, 2004, 17: 1075-1099.

[10] P. Y. H. Pang, F. Q. Sun and M. X. Wang. Existence and Non-existence of global solutions for a higher-order semilinear paraboic system. Indiana University Mathematics Journal, 2006, 55(3): 1113-1134.

[11] F. Q. Sun. Life span of blow-up solutions for a higher-order semilinear parabolic equation. Electronic Journal of Differential Equations, 2010, 17: 1-9.

[12] F.Q. Sun, F. Li and X. Q. Jia. Asymptotically self-similar global solutions for a higher-order semilinear parabolic system. Journal of Partial Differential Equations, 2009, 22(3): 282-298.

[13] 孙福芹, 王明新. 高阶半线性抛物型方程组的生命跨度[J]. 数学年刊, 2006, 27(1): 27-38.

[14] H. Fujita. On the blowing up of solutions of the Cauchy problem for . Journal of the Faculty of Science, University of Tokyo, 1966, 13(2): 105-113.

[15] M. E. Taylor. Partial differential Equations III: Nonlinear equations. New York: Springer-Verlag, 1996: 272-276.

[16] Q. S. Zhang. Global existence and local continuity of solutions for semilinear parabolic equations. Communications in Partial Differential Equations, 1997, 22: 1529-1557.

分享
Top