Skip to main content
SpringerLink
Log in

Analysis of a Free Boundary Problem Modeling Tumor Growth

  • ORIGINAL ARTICLES
  • Published:
Acta Mathematica Sinica Aims and scope Submit manuscript

Abstract

In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions: R = R(t), the radius of the tumor, and u = u(r, t), the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 < r < R(t), t > 0, and the function R satisfies a nonlinear integrodi fferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t → ∞.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adam, J., Bellomo, N.: A Survey of Models for Tumor–Immune System Dynamics, Birkhäuser, 1997

  2. Byrne, H. M., Chaplain, M. A.: Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math. Biosci., 130, 151–181 (1995)

    Article  Google Scholar 

  3. Byrne, H. M., Chaplain, M. A.: Free boundary value problems associated with the growth and development of multicellular spheroids. Euro. J. Appl. Math., 8, 639–658 (1997)

    Article  MathSciNet  Google Scholar 

  4. Cui, S.: Analysis of a mathematical model for the growth of tumors under the action of external inhibitors. J. Math. Biol. , 44, 395–426 (2002)

    Article  MathSciNet  Google Scholar 

  5. Cui, S., Friedman, A.: Analysis of a mathematical model of the effect of inbibitors on the growth of tumors. Math. Biosci., 164, 103–137 (2000)

    Article  MathSciNet  Google Scholar 

  6. Cui, S., Friedman, A.: Analysis of a mathematical model of the growth of necrotic tumors. J. Math. Anal. Appl., 255, 636–677 (2001)

    Article  MathSciNet  Google Scholar 

  7. Friedman, A., Reitich, F.: Analysis of a mathematical model for the growth of tumors. J. Math. Biol., 38, 262–284 (1999)

    Article  MathSciNet  Google Scholar 

  8. Greenspan, H. P.: Models for the growth of a solid tumor by diffusion. Stud. Appl. Math., 52, 317–340 (1972)

    Google Scholar 

  9. Ward, J. P., King, J. R.: Mathematical modelling of avascular–tumor growth. IMA J. Math. Appl. Medicine & Biol., 14, 39–70 (1997)

    Article  Google Scholar 

  10. Friedman, A.: Partial Differential Equations of Parabolic Type, Englewood Cliffs, 1964

  11. Ladyzenskaya, O. A., Solonnikov, V. A., Ural’tzva, N. N.: Linear and Quasi–Linear Equations of Parabolic Type, Transl. Math. Mono., 23, AMS, Providence, 1968

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China

    Shang Bin Cui

Authors
  1. Shang Bin Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shang Bin Cui.

Additional information

Project supported by the China National Natural Science Foundation, Grant number: 10171112

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cui, S.B. Analysis of a Free Boundary Problem Modeling Tumor Growth. Acta Math Sinica 21, 1071–1082 (2005). https://doi.org/10.1007/s10114-004-0483-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-004-0483-3

Keywords

MR (2000) Subject Classification

Search

Navigation