具测度初值的非线性抛物方程组的Cauchy问题

doi:10.6040/j.issn.1671-9352.0.2015.613

山东大学学报（理学版） ›› 2016, Vol. 51 ›› Issue (10): 41-47.doi: 10.6040/j.issn.1671-9352.0.2015.613

具测度初值的非线性抛物方程组的Cauchy问题

宋萌萌,尚海锋

河南理工大学数学与信息科学学院, 河南焦作 454000

收稿日期:2015-12-17 出版日期:2016-10-20 发布日期:2016-10-17
作者简介:宋萌萌(1991— ),女,硕士研究生,研究方向为偏微分方程. E-mail:mmsonglucky@foxmail.com
基金资助:
国家自然科学基金青年基金资助项目(11201124);河南省高等学校青年骨干教师资助计划(2015GGJS-070);河南理工大学杰出青年基金资助项目(J2014-03)

Cauchy problem for nonlinear parabolic equation systems with initial data measures

SONG Meng-meng, SHANG Hai-feng

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, Henan, China

Received:2015-12-17 Online:2016-10-20 Published:2016-10-17

摘要/Abstract

摘要： 研究了一类具强耦合源的退化抛物方程组的Cauchy问题,其中初值为Radon测度。当指标满足一定范围时,克服了方程退化性与强耦合源同时存在带来的困难, 从而得到了解的存在性。还进一步证明了指标的限制范围对解的存在性来说是最优的。

关键词: 抛物方程组, Cauchy问题, 测度初值, 耦合非线性源

Abstract: The Cauchy problem for a class of degenerate parabolic system with strongly coupling source is studied, where initial data are measured. When the known parameters satisfy some conditions, the difficulties are avoided which come from the interactions between the degeneracy of the principal and the strongly coupling source, and the existence of solutions is obtained. Moreover, it is proved that these conditions are optimal for the existence of solutions.

Key words: parabolic equation system, Cauchy problem, measures as initial data, coupling nonlinear source

中图分类号:

O175.2

宋萌萌,尚海锋. 具测度初值的非线性抛物方程组的Cauchy问题[J]. 山东大学学报（理学版）, 2016, 51(10): 41-47.

SONG Meng-meng, SHANG Hai-feng. Cauchy problem for nonlinear parabolic equation systems with initial data measures[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(10): 41-47.

参考文献

[1] BEBERNESS J, EBERLY D. Mathematical problems from combustion theory[M]. New York: Springer-Verlag, 1989.
[2] GALAKTIONOV V A, KURDYUMOV S P, SAMARSKII A A. A parabolic system of quasi-linear equations I[J]. Differential Equations, 1983, 19: 1558-1571.
[3] GALAKTIONOV V A, KURDYUMOV S P, SAMARSKII A A. A parabolic system of quasi-linear equations II[J]. Differential Equations, 1985, 21: 1049-1062.
[4] QUITTNER P, SOUPLET P. Suplinear parabolic problems:blow-up,global existence and steady states[M]. Germany: Birkhäuser-Verlag, 2007.
[5] 伍卓群, 赵俊宁, 尹景学,等. 非线性扩散方程[M]. 吉林:吉林大学出版社, 2001. WU Zhuoqun, ZHAO Junning, YIN Jingxue, et al. Nonlinear diffusion equation[M]. Jilin: Jilin University Press, 2001.
[6] AMANN H, QUITTNER P. Semilinear parabolic equations involving measures and low regularity data[J]. Transactions of the American Mathematical Society, 2004, 356(3):1045-1119.
[7] BARAS P, PIERRE M. Problèmes paraboliques semi-linéaires avec données mesures[J]. Applicable Analasis, 1984, 18(1-2): 111-149.
[8] BARAS P, PIERRE M. Critère dexistence de solutions positives pour des équations semi-linéaires non nonotones[J]. Annales de linstitut Henri Poincaré(C)Analyse non Linéaire, 1985, 2: 185-212.
[9] SHANG Haifeng. Cauchy problem and initial traces for fast diffusion equation with space-dependent source[J]. Zeitschrift für Angewandte Mathematik und Physik, 2013, 64(3): 785-798.
[10] ANDREUCCI D, DIBENEDETTO E. On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources[J]. Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze, 1991, 18(3):95-105.
[11] ANDREUCCI D. Degenerate parabolic equations with initial data measures[J]. Transactions of the American Mathematical Society, 1997, 349(10): 3911-3923.
[12] ANDREUCCI D, HERRERO M A, VELÁZQUEZ J J L. Liouville theorems and blow up behavior in semilinear reaction-diffusion systems[J]. Annales de linstitut Henri Poincaré(C)Analyse non Linéaire, 1997, 14(1): 1-53.
[13] ESCOBEDO M, HERRER0 M A. Boundedness and blow up for a semilinear reaction-diffusion system[J]. Journal of Differential Equations, 1991, 89: 176-202.
[14] ESCOBEDO M, LEVINE H A. Explosion et existence globale pour un système faiblement couplé dèquations de réaction diffusion[J]. Comptes Rendus de lAcadémie des Sciences-Série I-Mathématics, 1992, 314: 735-739.
[15] ESCOBEDO M, LEVINE H A. Critical Blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations[J]. Archive for Rational Mechanics and Analysis, 1995, 129: 47-100.
[16] ANDREUCCI D. New results on the Cauchy problem for parabolic systems and equations with strongly nonlinear sources[J]. Manuscripta Mathematica, 1992, 77: 127-159.
[17] LADYZHENSKAJA O A, SOLONNIKOV V A, URALCEVA N N. Linear and quasilinear equations of parabolic type[M]. Providence, RI:American Mathematical Society, 1968.
[18] CHEN Yazhe, DI BENEDETTO E. H(¨overo)lder estimates of solutions of singular parabolic equations with measurable coefficients[J]. Archive for Rational Mechanics and Analysis, 1992, 118: 257-271.
[19] DI BENEDETTO E. Continuity of weak solutions to a general porous medium equation[J]. Indiana University Mathematics Journal, 1983, 32: 83-118.

多维度评价

Viewed

Full text

Abstract

Cited

Shared

Discussed

具测度初值的非线性抛物方程组的Cauchy问题

Cauchy problem for nonlinear parabolic equation systems with initial data measures

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 2

多维度评价

本文评价

推荐阅读 0

[1]	成军祥,尚海锋. 具时间依赖源的双非线性抛物方程的Cauchy问题[J]. J4, 2013, 48(8): 50-55.
[2]	邓利华,尚海锋. 具强非线性源的多孔介质方程的Cauchy 问题[J]. J4, 2012, 47(8): 50-54.