JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2016, Vol. 51 ›› Issue (10): 41-47.doi: 10.6040/j.issn.1671-9352.0.2015.613

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Cauchy problem for nonlinear parabolic equation systems with initial data measures

SONG Meng-meng, SHANG Hai-feng   

  1. 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

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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

CLC Number: 

  • O175.2
[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.
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