在最优的初始值条件下考虑如下拟线性抛物方程的柯西问题ut-div a(x, t,u, Du)=b(x,t, u, Du),(x,t)属于 ST=RN×(0,T).令a(x,t,u,Du)={ai(x,t,u,Du), 假设 ai(x,t,u,Du)与b(x,t,u, Du)皆为 Caratheodory 函数, 并且假设它们满足Du的单调性, 关于u,|Du|等一定的增长阶条件下, 得到了解的比较定理,证明了解的存在性, 并得到了相关的Harnack不等式.
Abstract
In the paper, the Cauchy problem for the quasi-linear parabolic equation ut-div a(x, t,u, Du)=b(x,t, u, Du),(x,t), Suppose that ai(x,t,u,Du) and b(x,t,u,Du)are Caratheodory functions, and they satisfy some other restrictions such as the monotone property of Du the increasing order condition of u,|Du| etc., the comparison theorem of the equation is established, the existence of the solution is obtained, and the Harnack inequality is proved.
关键词
初始迹 /
Harnack不等式 /
双非线性抛物方程
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Key words
initial trace /
Harnack inequality /
doubly nonlinear parabolic equation
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参考文献
[1] Aronson D. G., Caffarelli L. A., The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc., 1983, 280: 351-366.
[2] Benilan Ph., Crandall M. G., Pierre M., Solutions of the porous medium equation in RN under optimal conditions on initial values, Indiana Univ. Math. J., 1984, 33: 51-87.
[3] Chen C., Wang R., Global existence and L∞ estimates of solution for doubly degenerate parabolic equation, Acta Mathematica Sinica, Chinese Series, 2001, 44(6): 1089-1098.
[4] Di Benedetto E., Degenerate Parabolic Equations, Springer-Verlag, New York, Berlin, Heideberg, 1993.
[5] Di Benedetto E., Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations, Arch. Rational Mech. Anal., 1988, 100: 129-147.
[6] Di Benedetto E., Gianazza U., Vespri V., Harnack estimates forquasi-linear degenerate parabolic differential equations, Acta Math., 2008, 200(2): 181-208.
[7] Di Benedetto E., Herrero M. A., On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc., 1989, 314: 187-224.
[8] Di Benedetto E., Kwong Y. C., Harnack estimates and extinction profile for weak solutions of certain singular parabolic equations, Trans. Amer. Math. Soc., 1992, 330: 783-811.
[9] Di Benedetto E., Urbano J. M., Vespri V., Current issues on singular and degenerate evolution equations, evolutionary equations, in: Handbook of Differential Equations, North-Holland, Amsterdam, 2004, 1: 169- 286.
[10] Esteban J. R., Vazquez J. L., Homogeneous diffusion in R with power-like nonlinear diffusivity, Arch. Rational Mech. Anal., 1988, 103: 39-88.
[11] Gianazza U., Vespri V., A Harnack inequality for a degenerate parabolic equation, J. Evol. Equ., 2006, 200(6): 247-267.
[12] Gu L., Second Order Parabolic Partial Differential Equations (in Chinese), Xiamen University Press, Xiamen, 2002.
[13] Ivanov A. V., Gradient estimates for doubly nonlinear parabolic equations, J. Math. Sci., 1999, 93(5): 661-688.
[14] Ivanov A. V., Hölder estimates for quasilinear doubly degenerate parabolic equations, J. Soviet Math., 1991, 56(2): 2320-2348.
[15] Ivanov A. V., Regularity for doubly nonlinear parabolic equations, Tatra Mount. Math. Publ., 1994, 4: 117-124.
[16] Ivanov A. V., Hölder estimates for equations of the type of fast diffusion, Algebra Analiz, 1994, 6(4): 101-142.
[17] Ivanov A. V., Hölder estimates for a natural class of equations of the type of fast diffusion, J. Math. Sci., 1998, 89: 1607-1630.
[18] Ivanov A. V., Regularity for doubly nonlinear parabolic equations, J. Math. Sci., 1997, 83(1): 22-37.
[19] Ivanov A. V., Mkrtychan P. Z., Regularity up to the boundary for generalized solutions to the first boundaryvalue problem for doubly degenerate quasilinear parabolic equations, J. Math. Sci., 1994, 70(6): 2112-2122.
[20] Ivanov A. V., Mkrtychan P. Z., Jäger W., Existence and uniqueness of a regular solution of the Cauchy- Dirichlet problem for a class of doubly nonlinear parabolic equations, J. Math. Sci., 1997, 84(1): 845-855.
[21] Ivanov A. V., Mkrtychan P. Z., On the existence of Hölder continuous weak solution of the first boundaryvalue problem for quasilinear doubly nonlinear parabolic equations, Zap. Nauchn. Semin. LOMI, 1990, 182: 5-28.
[22] Kalashnikkov A. S., Some problems of nonlinear parabolic equations of second order, USSR Math., Nauk. T., 1987, 42(2): 135-176.
[23] Ladyzenskaja O. A., Solonniklov V. A, Ural’stzeva N. N., Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Mono., 23, A.M.S., Providence, RI, 1968.
[24] Masayoshi T., On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl., 1988, 132: 187-212.
[25] Ragnedda F., Vernier Piro S., Vespri V., Large time behavior of solutions to a class of non-autonomous, degenerate parabolic equations, Math. Ann., 2010, 348: 779-795.
[26] Verspri V., On the local behavior of solutions of a certain class of doubly nonlinear parabolic equations, Manuscript Math., 1992, 75: 65-80.
[27] Wu Z., Zhao J., Yin J., Li H., Nonlinear Diffusion Equations, Word Scientific Publishing, Singapore, 2001.
[28] Yang S., Harnack estimates for weak solutions of equations of non-Newtonian polytropic filtration, Chin. Ann. of Math., 2001, 22B: 63-74.
[29] Yuan H., Harnack inequality for the equation of non-Newtonian polytropic filtration, Acta Sci. Natu. Univ., Jilinensis, 1994, 4: 19-22.
[30] Zhan H., Solution to nonlinear parabolic equations related to the p-Laplacian, Chinese Ann. of Math., 2012, 33B: 767-782.
[31] Zhan H., Harnack estimates for weak solutions of a singular parabolic equation, Chinese Ann. of Math., 2011, 32B: 397-416.
[32] Zhao J., Xu Z. H., Cauchy problem and initial traces for a doubly nonlinear degenerate parabolic equation, Sci. China, Ser. A, 1996, 39: 73-684.
[33] Zhao J., Yuan H., The Cauchy problem of some doubly nonlinear degenerate parabolic equations (in Chinese), Chinese Ann. Math., 1995, 16(2): 179-194.
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脚注
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基金
福建省自然科学基金资助课题(2012J01011);厦门理工学院科研启动基金资助课题
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