January  2012, 17(1): 429-444. doi: 10.3934/dcdsb.2012.17.429

Blowup, global fast and slow solutions to a parabolic system with double fronts free boundary

1. 

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China, China

Received  October 2010 Revised  January 2011 Published  October 2011

The purpose of this paper is to investigate a reaction-diffusion model with double fronts free boundary. Our approach to the local existence and uniqueness of the solution is based on the contraction mapping theorem. Also we study the blowup property of the solution. The result shows that blowup occurs if the initial datum is large enough. Finally the long-time behavior of global solutions is presented. It is proved that the solution is global and fast, which decays uniformly at an exponential rate if the initial datum is small, while there is a global and slow solution provided that the initial value is suitably large.
Citation: Qunying Zhang, Zhigui Lin. Blowup, global fast and slow solutions to a parabolic system with double fronts free boundary. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 429-444. doi: 10.3934/dcdsb.2012.17.429
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A. Amadori and J. Vazquez, Singular free boundary problem from image processing, Math. Mod. Meth. Appl. Sci., 15 (2005), 689-715. doi: 10.1142/S0218202505000509.

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X. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693.

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X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388.

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J. Crank, "Free and Moving Boundary Problems," Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1984.

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Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089.

[6]

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

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A. Friedman and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194. doi: 10.1137/060656292.

[8]

A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds, SIAM J. Math. Anal., 42 (2010), 2013-2040. doi: 10.1137/090772630.

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H. Fujita, On the blowing-up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo, Sect. I, 13 (1966), 109-124.

[10]

H. Ghidouche, P. Souplet and D. Tarzia, Decay of global solutions, stability and blow-up for a reaction-diffusion problem with free boundary, Proc. Am. Math. Soc., 129 (2001), 781-792. doi: 10.1090/S0002-9939-00-05705-1.

[11]

J. Goodman and D. N. Ostrov, On the early exercise boundary of the American put option, SIAM J. Appl. Math., 62 (2002), 1823-1835. doi: 10.1137/S0036139900378293.

[12]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-505. doi: 10.3792/pja/1195519254.

[13]

D. Hilhorst, M. Mimura and R. Schätzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285. doi: 10.1016/S1468-1218(02)00009-3.

[14]

H. Imai and H. Kawarada, Numerical analysis of a free boundary problem involving blow-up phenomena, Proc. Joint Symp. Appl. Math., 2 (1987), 277-280.

[15]

L. Jiang and M. Dai, Convergence of binomial tree methods for European/American path-dependent options, SIAM J. Numer.Anal., 42 (2004), 1094-1109. doi: 10.1137/S0036142902414220.

[16]

O. Ladyženskaja, V. Solonnikov and N. Ural'ceva, "Lineĭnye i Kvazilineĭnye Uravneniya Parabolicheskogo Tipa," (Russian) [Linear and Quasi-Linear Equations of Parabolic Type], Izdat. "Nauka," Moscow, 1968.

[17]

J. Liang, B. Hu and L. Jiang, Optimal convergence rate of the binomial tree scheme for American options with jump diffusion and their free boundaries, SIAM J. Financial Math., 1 (2010), 30-65. doi: 10.1137/090746239.

[18]

Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004.

[19]

G. Marinoschi, Well posedness of a time-difference scheme for a degenerate fast diffusion problem, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 435-454. doi: 10.3934/dcdsb.2010.13.435.

[20]

W. Merz and P. Rybka, A free boundary problem describing reaction-diffusion problems in chemical vapor infiltration of pyrolytic carbon, J. Math. Anal. Appl., 292 (2004), 571-588. doi: 10.1016/j.jmaa.2003.12.025.

[21]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280.

[22]

R. Ricci and D. Tarzia, Asymptotic behavior of the solutions of the dead-core problem, Nonlinear Anal., 13 (1989), 405-411. doi: 10.1016/0362-546X(89)90047-3.

[23]

P. Souplet, Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations, Commun. Part. Diff. Equations, 24 (1999), 951-973.

[24]

Y. Tao, A free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids, J. Differential Equations, 247 (2009), 49-68.

[25]

F. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.

[26]

S. Xu, Analysis of a delayed free boundary problem for tumor growth, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 293-308. doi: 10.3934/dcdsb.2011.15.293.

[27]

F. Yi, One dimensional combustion free boundary problem, Glasgow Mathematical Journal, 46 (2004), 63-75. doi: 10.1017/S0017089503001538.

show all references

References:
[1]

A. Amadori and J. Vazquez, Singular free boundary problem from image processing, Math. Mod. Meth. Appl. Sci., 15 (2005), 689-715. doi: 10.1142/S0218202505000509.

[2]

X. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693.

[3]

X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388.

[4]

J. Crank, "Free and Moving Boundary Problems," Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1984.

[5]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089.

[6]

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

[7]

A. Friedman and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194. doi: 10.1137/060656292.

[8]

A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds, SIAM J. Math. Anal., 42 (2010), 2013-2040. doi: 10.1137/090772630.

[9]

H. Fujita, On the blowing-up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo, Sect. I, 13 (1966), 109-124.

[10]

H. Ghidouche, P. Souplet and D. Tarzia, Decay of global solutions, stability and blow-up for a reaction-diffusion problem with free boundary, Proc. Am. Math. Soc., 129 (2001), 781-792. doi: 10.1090/S0002-9939-00-05705-1.

[11]

J. Goodman and D. N. Ostrov, On the early exercise boundary of the American put option, SIAM J. Appl. Math., 62 (2002), 1823-1835. doi: 10.1137/S0036139900378293.

[12]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-505. doi: 10.3792/pja/1195519254.

[13]

D. Hilhorst, M. Mimura and R. Schätzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285. doi: 10.1016/S1468-1218(02)00009-3.

[14]

H. Imai and H. Kawarada, Numerical analysis of a free boundary problem involving blow-up phenomena, Proc. Joint Symp. Appl. Math., 2 (1987), 277-280.

[15]

L. Jiang and M. Dai, Convergence of binomial tree methods for European/American path-dependent options, SIAM J. Numer.Anal., 42 (2004), 1094-1109. doi: 10.1137/S0036142902414220.

[16]

O. Ladyženskaja, V. Solonnikov and N. Ural'ceva, "Lineĭnye i Kvazilineĭnye Uravneniya Parabolicheskogo Tipa," (Russian) [Linear and Quasi-Linear Equations of Parabolic Type], Izdat. "Nauka," Moscow, 1968.

[17]

J. Liang, B. Hu and L. Jiang, Optimal convergence rate of the binomial tree scheme for American options with jump diffusion and their free boundaries, SIAM J. Financial Math., 1 (2010), 30-65. doi: 10.1137/090746239.

[18]

Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004.

[19]

G. Marinoschi, Well posedness of a time-difference scheme for a degenerate fast diffusion problem, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 435-454. doi: 10.3934/dcdsb.2010.13.435.

[20]

W. Merz and P. Rybka, A free boundary problem describing reaction-diffusion problems in chemical vapor infiltration of pyrolytic carbon, J. Math. Anal. Appl., 292 (2004), 571-588. doi: 10.1016/j.jmaa.2003.12.025.

[21]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280.

[22]

R. Ricci and D. Tarzia, Asymptotic behavior of the solutions of the dead-core problem, Nonlinear Anal., 13 (1989), 405-411. doi: 10.1016/0362-546X(89)90047-3.

[23]

P. Souplet, Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations, Commun. Part. Diff. Equations, 24 (1999), 951-973.

[24]

Y. Tao, A free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids, J. Differential Equations, 247 (2009), 49-68.

[25]

F. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.

[26]

S. Xu, Analysis of a delayed free boundary problem for tumor growth, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 293-308. doi: 10.3934/dcdsb.2011.15.293.

[27]

F. Yi, One dimensional combustion free boundary problem, Glasgow Mathematical Journal, 46 (2004), 63-75. doi: 10.1017/S0017089503001538.

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