-
Previous Article
Bifurcation analysis to Swift-Hohenberg equation with Steklov type boundary conditions
- DCDS Home
- This Issue
-
Next Article
Periodic traveling waves of a mean curvature flow in heterogeneous media
Upper bounds for coarsening for the degenerate Cahn-Hilliard equation
| 1. | Department of Mathematics, Technion-IIT, Haifa 32000 |
| 2. | Institute of Applied Mathematics and Mechanics, 83114 Donetsk |
$u_t=\nabla \cdot (1-u^2) \nabla \[ \frac{\Theta}{2} \{ \ln(1+u)-\ln(1-u)\} - \alpha u -$ Δu$],$
is characterized by the growth of domains in which $u(x,t) \approx u_{\pm},$ where $u_\pm$ denote the ''equilibrium phases;" this process is known as coarsening. The degree of coarsening can be quantified in terms of a characteristic length scale, $l(t)$, where $l(t)$ is prescribed via a Liapunov functional and a $W^{1, \infty}$ predual norm of $u(x,t).$ In this paper, we prove upper bounds on $l(t)$ for all temperatures $\Theta \in (0, \Theta_c),$ where $\Theta_c$ denotes the ''critical temperature," and for arbitrary mean concentrations, $\bar{u}\in (u_{-}, u_{+}).$ Our results generalize the upper bounds obtained by Kohn & Otto [14]. In particular, we demonstrate that transitions may take place in the nature of the coarsening bounds during the coarsening process.
| [1] |
Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 |
| [2] |
Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 |
| [3] |
Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 |
| [4] |
Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 |
| [5] |
Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020377 |
| [6] |
Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 |
| [7] |
Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246 |
| [8] |
Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020350 |
| [9] |
Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020467 |
| [10] |
Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076 |
| [11] |
Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020045 |
| [12] |
Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 |
| [13] |
Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020345 |
| [14] |
Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020384 |
| [15] |
Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020164 |
| [16] |
Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 |
| [17] |
Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020340 |
| [18] |
Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020103 |
| [19] |
Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011 |
| [20] |
Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]