
Previous Article
Bifurcation analysis to SwiftHohenberg 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 CahnHilliard equation
1.  Department of Mathematics, TechnionIIT, Haifa 32000 
2.  Institute of Applied Mathematics and Mechanics, 83114 Donetsk 
$u_t=\nabla \cdot (1u^2) \nabla \[ \frac{\Theta}{2} \{ \ln(1+u)\ln(1u)\}  \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 multidimensional space. Discrete & Continuous Dynamical Systems  A, 2021, 41 (1) : 395412. doi: 10.3934/dcds.2020364 
[3] 
Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixednorm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 54875508. 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) : 55815590. 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) : 54375473. 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) : 54135436. 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 MullinsSekerka/Stokes system for twophase flows. Discrete & Continuous Dynamical Systems  S, 2020 doi: 10.3934/dcdss.2020467 
[10] 
Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast nonconvex lowrank 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) : 413438. 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. SecondOrder characterizations for setvalued 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 infinitetime blowup in a haptotaxis system with nonlinear zeroorder interaction. Discrete & Continuous Dynamical Systems  A, 2021, 41 (1) : 439454. 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 statedependent delay in Banach spaces. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020103 
[19] 
Xuefeng Zhang, Yingbo Zhang. Faulttolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 112. doi: 10.3934/naco.2020011 
[20] 
Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems  A, 2021, 41 (1) : 455469. doi: 10.3934/dcds.2020380 
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]