为了描述物理学中多孔介质力学、流体力学、气体流量等问题3种介质的反应扩散问题,研究了一类具有3个变量耦合且同时具有加权非局部边界和非线性内部源的多孔介质抛物型方程组解的渐近性态,打破常用的第一特征值等构造上下解的方法,而采用常微分方程方法构造了该方程组的上、下解,引用比较定理,证明得到了由幂函数和指数函数完全耦合的一类抛物型方程组解的存在及爆破问题,在推广了已有的结果的基础上,为多孔介质及流体力学等问题提供理论支持.
In order to describe the reaction diffusion problems of three media in physics, such as mechanics of porous media, fluid mechanics, gas flow and so on, the asymptotic behavior of a class of parabolic equations of porous media,with 3 variables coupled with a weighted nonlocal boundary and a nonlinear internal source is studied.The upper and lower solutions of the first eigenvalue are broken, and the upper and lower solutions of the equations are constructed by using the ordinary differential equation method, and the comparison theorem is quoted. It is proved that the whole existence of the solution of the parabolic equations of porous media with the power function and the exponential function fully coupled and the sufficient conditions for the solution in the finite time blasting are obtained.On the basis of the existing results, provide more theoretical support for porous media and fluid mechanics are achieved.
[1] 杨婕,刘丙辰,张长城.具有非局部边界的退化抛物方程组的爆破解[J].数学杂志,2017,37(6): 1275-1286.DOI:10.13548/j.sxzz.20161108.002.
[2] 钟光胜,田立新.具有非线性范数型源的反应扩散方程组解的爆破性质[J].数学杂志,2017(1):129-137.DOI:10.13548/j.sxzz.20160322.002.
[3] LIU BC, ZHANG CC. Critical exponents in porous media equations with weighted nonlocal boundary conditions[J]. Journal of Mathematics,2015,35(1):51-62.DOI:10.13548/j.sxzz.2015.01.006.
[4] 王文海.具非局部源和非局部边界条件抛物方程组解的性质[J].中北大学学报(自然科学版), 2012, 33(4): 372-375.DOI:10.3969/j.issn.1673-3193.2012.04.005.
[5] MU C L, SU Y. Global existence and blow up for a quasilinear degenerateparabolic system in a ylinder[J]. Appi Math Letters,2001,14(1):715-723. DOI:10.1016/S0893-9659(01)80032-X.
[6] PAO C V.Nonlinear parabolic and elliptic equations[M].London:Plenum Press,1992.
[7] WANG S,WANG M X,XIE C H.A nonlinear degenerate diffusion equation not in divergence form[J].Zeitschrift Für Angewandte Mathematik Und Physik Zamp,2000,51(1): 149-159.DOI:10.1007/PL00001503.
[8] LI Y X,DENG W B,XIE C H.Global existence and nonexistence for degenerate parabolic systems[J].Nonlinear Analysis,2002,55(3):233-244.DOI:10.1090/S0002-9939-02-06630-3.
[9] DENG W B,LI Y X,XIE C H.Existence and nonexistence of global solution of some nonlocal degenerate parabolic system[J].Proceedings of the American Mathematical Society,2003,131(5):1573-1582.DOI: 10.1016 /s0893-9659( 03)80118-0.
[10] ZHANG Y,SONG X J.Global existence and blow-up for a degenerate parabolic systems with nonlocal source [J].Journal of Southwest University(Natural Science Edition),2008,30(9):80-84.DOI:10.13718/j.cnki.xdzk.2008.09.013.
[11] BUDDAL C J,DOLD J W,GALAKTIONOV V A.Global blow-up for a semilinear heat equation on a subspace[J].Proceedings of the Royal Society of Edinburgh:Section A Mathematics,2015(5):893-923. DOI: 10.1017 /s0308210515000256.
[12] ZHANG H,KONG L H,ZHENG S N. Propagations of singularities in a parabolic system with coupling nonlocal sources[J].Sci in China: Ser A,2009,52(1):181-194.DOI:10.3969 /j.issn.1006-8341.2009. 02.009.
[13] 叶其孝,李正元,王明新,等.反应扩散方程引论[M] 第2版.北京:科学出版社,2011.
