• Previous Article
    Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth
  • DCDS Home
  • This Issue
  • Next Article
    Age-dependent equations with non-linear diffusion
April  2010, 26(2): 713-736. doi: 10.3934/dcds.2010.26.713

Boundary dynamics of a two-dimensional diffusive free boundary problem

1. 

Mathematics and Computer Science Department, Goucher College, 1021 Dulaney Valley Road, Baltimore, MD, 21204, United States

2. 

Department of Mathematics, University of California, Irvine, CA 92697-3875

Received  February 2009 Revised  July 2009 Published  October 2009

Numerous models of industrial processes such as diffusion in glassy polymers or solidification phenomena, lead to general one-phase free boundary value problems with phase onset. In this paper we develop a framework viable to prove global existence and stability of planar solutions to one such multi-dimensional model whose application is in controlled-release pharmaceuticals. We utilize a boundary integral reformulation to allow for the use of maximal regularity. To this effect, we view the operators as pseudo-differential and exploit knowledge of the relevant symbols. Within this framework, we give a local existence and continuous dependence result necessary to prove planar solutions are locally exponentially stable with respect to two-dimensional perturbations.
Citation: Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713
[1]

Lars Diening, Michael Růžička. An existence result for non-Newtonian fluids in non-regular domains. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 255-268. doi: 10.3934/dcdss.2010.3.255

[2]

Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036

[3]

Wenming Hu, Huicheng Yin. Well-posedness of low regularity solutions to the second order strictly hyperbolic equations with non-Lipschitzian coefficients. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1891-1919. doi: 10.3934/cpaa.2019088

[4]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635

[5]

Birgit Jacob, Hafida Laasri. Well-posedness of infinite-dimensional non-autonomous passive boundary control systems. Evolution Equations and Control Theory, 2021, 10 (2) : 385-409. doi: 10.3934/eect.2020072

[6]

Mircea Sofonea, Yi-bin Xiao. Tykhonov well-posedness of a viscoplastic contact problem. Evolution Equations and Control Theory, 2020, 9 (4) : 1167-1185. doi: 10.3934/eect.2020048

[7]

Yuri Trakhinin. On well-posedness of the plasma-vacuum interface problem: the case of non-elliptic interface symbol. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1371-1399. doi: 10.3934/cpaa.2016.15.1371

[8]

Fujun Zhou, Shangbin Cui. Well-posedness and stability of a multidimensional moving boundary problem modeling the growth of tumor cord. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 929-943. doi: 10.3934/dcds.2008.21.929

[9]

Joachim Escher, Anca-Voichita Matioc. Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 573-596. doi: 10.3934/dcdsb.2011.15.573

[10]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3749-3778. doi: 10.3934/dcdsb.2021205

[11]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems and Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307

[12]

Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321

[13]

Hyungjin Huh, Bora Moon. Low regularity well-posedness for Gross-Neveu equations. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1903-1913. doi: 10.3934/cpaa.2015.14.1903

[14]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[15]

Hong Chen, Xin Zhong. Local well-posedness to the 2D Cauchy problem of non-isothermal nonhomogeneous nematic liquid crystal flows with vacuum at infinity. Communications on Pure and Applied Analysis, 2022, 21 (9) : 3141-3169. doi: 10.3934/cpaa.2022093

[16]

Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763

[17]

Iñigo U. Erneta. Well-posedness for boundary value problems for coagulation-fragmentation equations. Kinetic and Related Models, 2020, 13 (4) : 815-835. doi: 10.3934/krm.2020028

[18]

George Avalos, Pelin G. Geredeli, Justin T. Webster. Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1267-1295. doi: 10.3934/dcdsb.2018151

[19]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control and Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61

[20]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]