一类半线性反应扩散方程组自由边界问题的爆破性
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摘要
提出了一类半线性反应扩散方程组的自由边界模型.首先分析了该模型在爆破时正解之间的关系,通过重新刻画自变量,得到解同时爆破的结论.其次,构造辅助函数和利用内部Schauder估计,证明了模型的解在某些区域上关于空间变量是单调递减的结论.最后,利用反证法,通过构造辅助函数和利用最大值原理,得到了爆破集为初始区域的紧子集,也得到了此时自由边界有界性的结论.
In this paper, the free boundary model of a class of semi-linear reaction diffusion equations is presented.Firstly, the relationship between the positive solutions of the model when the blow-up case occurs is analyzed, and the results of simultaneous blow-up are obtained by rescaling the independent variables. Secondly, it is proved that the solution of the model is monotonically decreasing in some regions, by constructing auxiliary functions and using internal Schauder estimation. Finally, by using the contradiction method, the result that the blow-up set is the compact subset of the initial area is obtained by constructing the auxiliary function and using the maximum principle,while in this case, the free boundary is bounded.
In this paper, the free boundary model of a class of semi-linear reaction diffusion equations is presented.Firstly, the relationship between the positive solutions of the model when the blow-up case occurs is analyzed, and the results of simultaneous blow-up are obtained by rescaling the independent variables. Secondly, it is proved that the solution of the model is monotonically decreasing in some regions, by constructing auxiliary functions and using internal Schauder estimation. Finally, by using the contradiction method, the result that the blow-up set is the compact subset of the initial area is obtained by constructing the auxiliary function and using the maximum principle,while in this case, the free boundary is bounded.
引文
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[5] WANG M X, ZHAO Y G. A semilinear parabolic system with a free boundary[J]. Zeitschrift für Angewandte Mathematik und Physik, 2015, 66(6):3309-3332.
[6] CAFFARELLI L, SALSA S. A geometric approach to free boundary problems[M]. Providence, Rhode Island:American Mathematical Society, 2005.
[7] LIN Z G, LING Z, PEDERSEN M. A free boundary problem for a reaction-diffusion system with nonlinear memory[J]. Zeitschrift für Angewandte Mathematik und Physik,2014, 65(3):521-530.
[8] SUN N K. Blow-up and asymptotic behavior of solutions for reaction-diffusion equations with free boundaries[J]. Journal of Mathematical Analysis and Applications, 2015, 428(2):838-854.
[9] DU Y H, LIN Z G. Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary[J]. SIAM Journal on Mathematical Analysis, 2010, 42(1):377-405.
[10] WANG M X. A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment[J]. Journal of Functional Analysis, 2016,270(2):483-508.
[11] XIANG Z Y, CHEN Q, MU C L. Blow-up rate estimates for a system of reaction-diffusion equations with absorption[J]. Journal of the Korean Mathematical Society, 2007, 44(4):779-786.
[2] ZHOU P, BAO J, LIN Z G. Global existence and blowup of a localized problem with free boundary[J]. Nonlinear Analysis:Theory, Methods&Applications, 2011, 74(7):2523-2533.
[3] LI F C, XIE C H. Global existence and blow-up for a nonlinear porous medium equation[J]. Applied Mathematics Letters, 2003, 16(2):185-192.
[4] GHIDOUCHE H. SOUPLET P, TARZIA D, Decay of global solutions, stability and blowup for a reaction-diffusion problem with free boundary[J]. Proceedings of the American Mathematical Society, 2001, 129(3):781-792.
[5] WANG M X, ZHAO Y G. A semilinear parabolic system with a free boundary[J]. Zeitschrift für Angewandte Mathematik und Physik, 2015, 66(6):3309-3332.
[6] CAFFARELLI L, SALSA S. A geometric approach to free boundary problems[M]. Providence, Rhode Island:American Mathematical Society, 2005.
[7] LIN Z G, LING Z, PEDERSEN M. A free boundary problem for a reaction-diffusion system with nonlinear memory[J]. Zeitschrift für Angewandte Mathematik und Physik,2014, 65(3):521-530.
[8] SUN N K. Blow-up and asymptotic behavior of solutions for reaction-diffusion equations with free boundaries[J]. Journal of Mathematical Analysis and Applications, 2015, 428(2):838-854.
[9] DU Y H, LIN Z G. Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary[J]. SIAM Journal on Mathematical Analysis, 2010, 42(1):377-405.
[10] WANG M X. A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment[J]. Journal of Functional Analysis, 2016,270(2):483-508.
[11] XIANG Z Y, CHEN Q, MU C L. Blow-up rate estimates for a system of reaction-diffusion equations with absorption[J]. Journal of the Korean Mathematical Society, 2007, 44(4):779-786.