Browse > Article
http://dx.doi.org/10.4134/JKMS.j200369

THE STABILITY OF WEAK SOLUTIONS TO AN ANISOTROPIC POLYTROPIC INFILTRATION EQUATION  

Zhan, Huashui (School of Applied Mathematics Xiamen University of Technology)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.5, 2021 , pp. 1109-1129 More about this Journal
Abstract
This paper considers an anisotropic polytropic infiltration equation with a source term $$u_t={\sum\limits_{i=1}^{N}}{\frac{{\partial}}{{\partial}x_i}}\(a_1(x){\mid}u{\mid}^{{\alpha}_i}{\mid}u_{x_i}{\mid}^{p_i-2}u_{x_i}\)+f(x,t,u)$$, where pi > 1, αi > 0, ai(x) ≥ 0. The existence of weak solution is proved by parabolically regularized method. Based on local integrability $u_{x_i}{\in}W_{loc}^{1,p_i}(\Omega)$, the stability of weak solutions is proved without boundary value condition by the weak characteristic function method. One of the essential characteristics of an anisotropic equation different from an isotropic equation is found originally.
Keywords
The anisotropic polytropic infiltration equation; the weak characteristic function method; stability; boundary value condition;
Citations & Related Records
연도 인용수 순위
  • Reference
1 J. Weickert, Anisotropic diffusion in image processing, European Consortium for Mathematics in Industry, B. G. Teubner, Stuttgart, 1998.
2 H. J. Fan, Cauchy problem of some doubly degenerate parabolic equations with initial datum a measure, Acta Math. Sin. (Engl. Ser.) 20 (2004), no. 4, 663-682. https://doi.org/10.1007/s10114-004-0375-6   DOI
3 R. Gianni, A. Tedeev, and V. Vespri, Asymptotic expansion of solutions to the Cauchy problem for doubly degenerate parabolic equations with measurable coefficients, Nonlinear Anal. 138 (2016), 111-126. https://doi.org/10.1016/j.na.2015.09.006   DOI
4 Q. Li, Weak Harnack estimates for supersolutions to doubly degenerate parabolic equations, Nonlinear Anal. 170 (2018), 88-122. https://doi.org/10.1016/j.na.2017.12.017   DOI
5 M. Mihailescu, P. Pucci, and V. Radulescu, Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C. R. Math. Acad. Sci. Paris 345 (2007), no. 10, 561-566. https://doi.org/10.1016/j.crma.2007.10.012   DOI
6 F. Otto, L1-contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differential Equations 131 (1996), no. 1, 20-38. https://doi.org/10.1006/jdeq.1996.0155   DOI
7 H. Shang and J. Cheng, Cauchy problem for doubly degenerate parabolic equation with gradient source, Nonlinear Anal. 113 (2015), 323-338. https://doi.org/10.1016/j.na.2014.10.006   DOI
8 J. Sun, J. Yin, and Y. Wang, Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation, Nonlinear Anal. 74 (2011), no. 6, 2415-2424. https://doi.org/10.1016/j.na.2010.11.044   DOI
9 A. F. Tedeev, The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations, Appl. Anal. 86 (2007), no. 6, 755-782. https://doi.org/10.1080/00036810701435711   DOI
10 M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl. 132 (1988), no. 1, 187-212. https://doi.org/10.1016/0022-247X(88)90053-4   DOI
11 I. Fragala, F. Gazzola, and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincare Anal. Non Lineaire 21 (2004), no. 5, 715-734. https://doi.org/10.1016/j.anihpc.2003.12.001   DOI
12 H. Zhan, The stability of the solutions of an anisotropic diffusion equation, Lett. Math. Phys. 109 (2019), no. 5, 1145-1166. https://doi.org/10.1007/s11005-018-1135-3   DOI
13 H. Zhan, The weak solutions of an evolutionary p(x)-Laplacian equation are controlled by the initial value, Comput. Math. Appl. 76 (2018), no. 9, 2272-2285. https://doi.org/10.1016/j.camwa.2018.08.026   DOI
14 Z. Wu, J. Zhao, J. Yin, and H. Li, Nonlinear Diffusion Equations, World Scientic Publishing, Singapore, 2001.
15 H. J. Yuan, S. Z. Lian, C. L. Cao, W. J. Gao, and X. J. Xu, Extinction and positivity for a doubly nonlinear degenerate parabolic equation, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 10, 1751-1756. https://doi.org/10.1007/s10114-007-0944-6   DOI
16 H. Zhan, Solutions to polytropic filtration equations with a convection term, Electron. J. Differential Equations 2017 (2017), Paper No. 207, 10 pp.
17 H. Zhan, Infiltration equation with degeneracy on the boundary, Acta Appl. Math. 153 (2018), 147-161. https://doi.org/10.1007/s10440-017-0124-3   DOI
18 H. Zhan, The nonnegative weak solution of a degenerate parabolic equation with variable exponent growth order, Bound. Value Probl. 2020 (2020), Paper No. 69, 20 pp. https://doi.org/10.1186/s13661-020-01364-x   DOI
19 H. Zhan and Z. Feng, The stability theorems of non-Newtonian fluid and electrorheological fluid mixed type equation, preprint, 2020.
20 H. Zhan and Z. Feng, The definite condition for the evolutionary p(x)-Laplacian equation, preprint, 2020.
21 S. N. Antontsev and S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal. 60 (2005), no. 3, 515-545. https://doi.org/10.1016/j.na.2004.09.026   DOI
22 J. N. Zhao and H. J. Yuan, The Cauchy problem for a class of nonlinear doubly degenerate parabolic equations, Chinese Ann. Math. Ser. A 16 (1995), no. 2, 181-196.
23 H. Zhan and Z. Feng, Well-posedness problem of an anisotropic parabolic equation, J. Differential Equations 268 (2020), no. 2, 389-413. https://doi.org/10.1016/j.jde.2019.08.014   DOI
24 V. V. Zhikov, On the density of smooth functions in Sobolev-Orlicz spaces, J. Math. Sci. (N.Y.) 132 (2006), no. 3, 285-294; translated from Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34], 67-81, 226. https://doi.org/10.1007/s10958-005-0497-0   DOI
25 S. N. Antontsev and S. I. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear Anal. 65 (2006), no. 4, 728-761. https://doi.org/10.1016/j.na.2005.09.035   DOI
26 W. Zou and J. Li, Existence and uniqueness of solutions for a class of doubly degenerate parabolic equations, J. Math. Anal. Appl. 446 (2017), no. 2, 1833-1862. https://doi.org/10.1016/j.jmaa.2016.10.002   DOI
27 Z. Zhou, Z. Guo, and B. Wu, A doubly degenerate diffusion equation in multiplicative noise removal models, J. Math. Anal. Appl. 458 (2018), no. 1, 58-70. https://doi.org/10.1016/j.jmaa.2017.08.049   DOI
28 M. Mihailescu, P. Pucci, and V. Radulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008), no. 1, 687-698. https://doi.org/10.1016/j.jmaa.2007.09.015   DOI
29 D. Andreucci, G. R. Cirmi, S. Leonardi, and A. F. Tedeev, Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary, J. Differential Equations 174 (2001), no. 2, 253-288. https://doi.org/10.1006/jdeq.2000.3948   DOI
30 A. El Hamidi and J.-M. Rakotoson, On a perturbed anisotropic equation with a critical exponent, Ric. Mat. 55 (2006), no. 1, 55-69. https://doi.org/10.1007/s11587-006-0004-z   DOI
31 M. Bendahmane, M. Langlais, and M. Saad, On some anisotropic reaction-diffusion systems with L1-data modeling the propagation of an epidemic disease, Nonlinear Anal. 54 (2003), no. 4, 617-636. https://doi.org/10.1016/S0362-546X(03)00090-7   DOI
32 C. S. Chen and R. Y. Wang, Global existence of and L∞ estimates for solutions for a doubly degenerate parabolic equation, Acta Math. Sinica (Chin. Ser.) 44 (2001), no. 6, 1089-1098.   DOI
33 J. Droniou, R. Eymard, and K. S. Talbot, Convergence in C([0, T]; L2(Ω)) of weak solutions to perturbed doubly degenerate parabolic equations, J. Differential Equations 260 (2016), 7821-7860.   DOI
34 E. Eisenriegler, Anisotropic colloidal particles in critical fluids, J. Chem. Phys. 121 (2004), 32-99.   DOI
35 E. Eisenriegler, Anisotropic colloidal particles interacting with polymers in a good solvent, J. Chem. Phys. 124 (2006), 144-912.   DOI
36 A. El Hamidi and J.-M. Rakotoson, Extremal functions for the anisotropic Sobolev inequalities, Ann. Inst. H. Poincare Anal. Non Lineaire 24 (2007), no. 5, 741-756. https://doi.org/10.1016/j.anihpc.2006.06.003   DOI
37 S. N. Antontsev, J. I. Diaz, and S. Shmarev, Energy methods for free boundary problems, Progress in Nonlinear Differential Equations and their Applications, 48, Birkhauser Boston, Inc., Boston, MA, 2002. https://doi.org/10.1007/978-1-4612-0091-8   DOI