Browse > Article
http://dx.doi.org/10.4134/BKMS.2012.49.6.1179

DOUBLY NONLINEAR PARABOLIC EQUATIONS INVOLVING p-LAPLACIAN OPERATORS VIA TIME-DISCRETIZATION METHOD  

Shin, Kiyeon (Department of Mathematics Pusan National University)
Kang, Sujin (Department of Nanomaterials Engineering Pusan National University)
Publication Information
Bulletin of the Korean Mathematical Society / v.49, no.6, 2012 , pp. 1179-1192 More about this Journal
Abstract
In this paper, we consider a doubly nonlinear parabolic partial differential equation $\frac{{\partial}{\beta}(u)}{{\partial}t}-{\Delta}_pu+f(x,t,u)=0$ in ${\Omega}{\times}[0,T]$, with Dirichlet boundary condition and initial data given. We prove the existence of a discrete approximate solution by means of the Rothe discretization in time method under some conditions on ${\beta}$, $f$ and $p$.
Keywords
doubly nonlinear; p-Laplacian; Rothe method;
Citations & Related Records
연도 인용수 순위
  • Reference
1 R. Adams, Sobolev Spaces, Academic Press, New York, 1975.
2 V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noord-hoff Internat. Publ., Leyden, 1976.
3 A. Bensoussan, L. Boccardo, and F. Murat, On a nonlinear partial differential equation having natural growth terms and unbounded solution, Ann. Inst. H. Poincare Anal. Non Lineaire 5 (1988), no. 4, 347-364.   DOI
4 J. I. Diaz and J. F. Padial, Uniqueness and existence of solutions in the BVt(Q) space to a doubly nonlinear parabolic problem, Differential Integral Equations 40 (1996), no. 2, 527-560.
5 A. Eden, B. Michaux, and J. M. Rakotoson, Semi-discretized nonlinear evolution equa- tions as discrete dynamical systems and error analysis, Indiana Univ. Math. J. 39 (1990), no. 3, 737-783.   DOI
6 A. Eden, B. Michaux, and J. M. Rakotoson, Doubly nonlinear parabolic type equations as dynamical systems, J. Dynam. Differential Equations 3 (1991), no. 1, 87-131.   DOI
7 M. Efendiev and A. Miranville, New models of Cahn-Hilliard-Gurtin equations, Contin. Mech. Thermodyn. 16 (2004), no. 5, 441-451.   DOI
8 A. El Hachimi and H. El Ouardi, Existence and regularity of a global attractor for doubly nonlinear parabolic equations, Electron. J. Differential Equations 2002 (2002), no. 45, 1-15.
9 M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D. 92 (1996), no. 3-4, 178-192.   DOI   ScienceOn
10 A. G. Kartasatos and M. E. Parrott, The weak solution of a functional-differential equation in a general Banach space, J. Differential Equations 75 (1988), no. 2, 290-302.   DOI
11 V. Le and K. Schmit, Global Bifurcation in Variational Inequalities, Springer-Verlag, New York, 1997.
12 N. Merazga and A. Bouziani, On a time-discretization method for a semilinear heat equation with purely integral conditions in a nonclassical function space, Nonlinear Analysis 66 (2007), no. 3, 604-623.   DOI   ScienceOn
13 A. Miranville and G. Schimperna, Global solution to a phase transition model based on a microforce balance, J. Evol. Equ. 5 (2005), no. 2, 253-276.   DOI
14 F. Otto, $L^1$-contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differential Equations 131 (1996), no. 1, 20-38.   DOI   ScienceOn
15 A. Rougirel, Convergence to steady state and attractors for doubly nonlinear equations, J. Math. Anal. Appl. 339 (2008), no. 1, 281-294.   DOI   ScienceOn
16 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs 49, AMS, 1997.
17 J. M. Rakotoson, On some degenerate and nondegenerate quasilinear elliptic systems with nonhomogeneous Dirichlet boundary condition, Nonlinear Analysis 13 (1989), no. 2, 165-183.   DOI   ScienceOn
18 M. Schatzman, Stationary solutions and asymptotic behavior of a quasilinear degenerate parabolic equation, Indiana Univ. Math. J. 33 (1984), no. 1, 1-29.   DOI