Skip to main content
SpringerLink
Log in

A semilinear parabolic system with a free boundary

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This paper deals with a semilinear parabolic system with reaction terms \({v^p, u^q}\) and a free boundary \({x = s(t)}\) in one space dimension, where \({s(t)}\) evolves according to the free boundary condition \({s'(t) = -\mu(u_x + \rho v_x)}\). The main aim of this paper was to study the existence, uniqueness, regularity and long-time behavior of positive solution (maximal positive solution). Firstly, we prove that this problem has a unique positive solution when \({p, q \geq 1}\), and a (unique) maximal positive solution when \({p < 1}\) or \({q < 1}\). Then, we study the regularity of \({(u,v)}\) and \({s}\). At last, we discuss the global existence, finite-time blowup of the unique positive solution (maximal positive solution) and long-time behavior of bounded global solution.

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. Bunting G., Du Y.H., Krakowski K.: Spreading speed revisited: analysis of a free boundary model. Netw. Heterog. Media 7, 583–603 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Caffarelli, L., Salsa, S.: A Geometric Approach to Free Boundary Problems. Graduate Studies in Mathematics, vol. 68. American Mathematical Society, Providence (2005)

  3. Caristi G., Mitidieri E.: Blow-up estimates of positive solutions of a parabolic system. J. Differ. Equ. 113, 265–271 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen X.F., Friedman A.: A free boundary problem arising in a model of wound healing. SIAM J. Math. Anal. 32(4), 778–800 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Crank J.: Free and Moving Boundary Problem. Clarendon Press, Oxford (1984)

    Google Scholar 

  6. Du Y.H., Guo Z.M.: Spreading–vanishing dichotomy in a diffusive logistic model with a free boundary, II. J. Differ. Equa. 250, 4336–4366 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Du Y.H., Guo Z.M., Peng R.: A diffusive logistic model with a free boundary in time-periodic environment. J. Funct. Anal. 265, 2089–2142 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Du Y.H., Lin Z.G.: Spreading–vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42, 377–405 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Du Y.H., Lin Z.G.: A diffusive competition model with a free boundary: invasion of a superior or inferior competition. Discrete Cont. Dyn. Syst. Ser. B 19(10), 3105–3132 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  10. Escobedo M., Herrero M.A.: A uniqueness result for a semilinear reaction–diffusion system. Proc. Am. Math. Soc. 112(1), 175–185 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  11. Escobedo M., Herrero M.A.: Boundedness and blow up for a semilinear reaction–diffusion system. J. Differ. Equ. 89, 176–202 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Escobedo M., Herrero M.A.: A semilinear parabolic system in a bounded domain. Annali di Matematica pura ed applicata (IV) CLXV, 315–336 (1993)

    Article  MathSciNet  Google Scholar 

  13. Fila M., Souplet P.: Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem. Interfaces Free Bound. 3, 337–344 (2001)

    MATH  MathSciNet  Google Scholar 

  14. Friedman A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  15. Friedman A., Giga Y.: A single point blow-up for solutions of semilinear parabolic systems. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34, 65–79 (1987)

    MATH  MathSciNet  Google Scholar 

  16. Friedman A., Hu B., Xue C.: Analysis of a mathematical model of ischemic cutaneous wounds. SIAM J. Math. Anal. 42, 2013–2040 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ghidouche H., Souplet P., Tarzia D.: Decay of global solutions, stability and blow-up for a reaction–diffusion problem with free boundary. Proc. Am. Math. Soc. 129(3), 781–792 (2000)

    Article  MathSciNet  Google Scholar 

  18. Guo J.S., Wu C.H.: On a free boundary problem for a two-species weak competition system. J. Dyn. Differ. Equ. 24, 873–895 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Guyonne V., Lorenzi L.: Instability in a flame ball problem. Discrete Contin. Dyn. Syst. B 7, 315–355 (2007)

    MATH  MathSciNet  Google Scholar 

  20. Kaneko Y.: Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction–diffusion equations. Nonlinear Anal.: Real World Appl. 18, 121–140 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kaneko Y., Yamada Y.: A free boundary problem for a reaction diffusion equation appearing in ecology. Adv. Math. Sci. Appl. 21(2), 467–492 (2011)

    MATH  MathSciNet  Google Scholar 

  22. Ladyzenskaja O.A., Solonnikov V.A., Uralceva N.N.: Linear and Quasilinear Equations of Parabolic Type. Academic Press, New York (1968)

    Google Scholar 

  23. Marinoschi G.: Well posedness of a time-difference scheme for a degenerate fast diffusion problem. Discrete Contin. Dyn. Syst. Ser. B 13, 435–454 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. Merz W., Rybka P.: A free boundary problem describing reaction–diffusion problems in chemical vapor infiltration of pyrolytic carbon. J. Math. Anal. Appl. 292, 571–588 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Peng R., Zhao X.Q.: The diffusive logistic model with a free boundary and seasonal succession. Discrete Cont. Dyn. Syst. A 33(5), 2007–2031 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Ricci R., Tarzia D.A.: Asymptotic behavior of the solutions of the dead-core problem. Nonlinear Anal. 13, 439–456 (1983)

    MathSciNet  Google Scholar 

  27. Rubinstein L.I.: The Stefan Problem. American Mathematical Society, Providence (1971)

    Google Scholar 

  28. Schaeffer D.G.: A new proof of the infinite differentiability of the free boundary in the Stefan problem. J. Differ. Equ. 20, 266–269 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  29. Souplet P.: Stability and continuous dependence of solutions of one-phase Stefan problems for semilinear parabolic equations. Port. Math. 59, 315–323 (2002)

    MATH  MathSciNet  Google Scholar 

  30. Wang M.X.: On some free boundary problems of the prey–predator model. J. Differ. Equ. 256(10), 3365–3394 (2014)

    Article  MATH  Google Scholar 

  31. Wang M.X.: The diffusive logistic equation with a free boundary and sign-changing coefficient. J. Differ. Equ. 258(4), 1252–1266 (2015)

    Article  MATH  Google Scholar 

  32. Wang M.X.: Spreading and vanishing in the diffusive prey–predator model with a free boundary. Commun. Nonlinear Sci. Numer. Simul. 23, 311–327 (2015)

    Article  MathSciNet  Google Scholar 

  33. Wang M.X., Zhang Y.: Two kinds of free boundary problems for the diffusive prey–predator model. Nonlinear Anal.: Real World Appl. 24, 73–82 (2015)

    Article  MathSciNet  Google Scholar 

  34. Wang M.X., Zhao J.F.: Free boundary problems for a Lotka–Volterra competition system. J. Dyn. Differ. Equ. 26(3), 655–672 (2014)

    Article  MATH  Google Scholar 

  35. Wang, M.X., Zhao, J.F.: A Free Boundary Problem for a Predator–Prey Model with Double Free Boundaries. arXiv:1312.7751 [math.DS]

  36. Zhang Q.Y., Lin Z.G.: Blowup, global fast and slow solutions to a parabolic system with double fronts free boundary. Discrete Contin. Dyn. Syst. B 17(1), 429–444 (2012)

    MATH  Google Scholar 

  37. Zhao J.F., Wang M.X.: A free boundary problem of a predator–prey model with higher dimension and heterogeneous environment. Nonlinear Anal.: Real World Appl. 16, 250–263 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  38. Zhou P., Lin Z.G.: Global existence and blowup of a nonlocal problem in space with free boundary. J. Funct. Anal. 262, 3409–3429 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Natural Science Research Center, Harbin Institute of Technology, Harbin, 150080, People’s Republic of China

    Mingxin Wang & Yonggang Zhao

Authors
  1. Mingxin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yonggang Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mingxin Wang.

Additional information

This work was supported by NSFC Grant 11371113.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, M., Zhao, Y. A semilinear parabolic system with a free boundary. Z. Angew. Math. Phys. 66, 3309–3332 (2015). https://doi.org/10.1007/s00033-015-0582-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-015-0582-2

Mathematics Subject Classification

Keywords

Search

Navigation