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May  2020, 25(5): 1671-1698. doi: 10.3934/dcdsb.2019246

A nonlinear Stefan problem with variable exponent and different moving parameters

School of Mathematics, Southeast University, Nanjing 210096, China

* Corresponding author: Huiling Li

Received  April 2019 Revised  June 2019 Published  May 2020 Early access  November 2019

Fund Project: The work is supported by NSFC Grants 11171064 and 11871148, and by the Natural Science Foundation of Jiangsu Province BK20161412.

In this paper, we consider a nonlinear diffusion problem with variable exponent, accompanied by double free boundaries possessing different moving parameters, where the variable exponent function $ m(x) $ satisfies that $ m(x)-1 $ can change its sign. Local existence and uniqueness of solution are established firstly, and then, some sufficient conditions are achieved for finite time blowup, and as well for global existence. Asymptotic behavior is further investigated for global solution, and existences of fast solution and slow solution are presented by making use of upper-sub solutions, energy and scaling arguments.

Citation: Huiling Li, Xiaoliu Wang, Xueyan Lu. A nonlinear Stefan problem with variable exponent and different moving parameters. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1671-1698. doi: 10.3934/dcdsb.2019246
References:
[1]

T. Aiki, Behavior of free boundaries of blow up solutions to one-phase Stefan problems, Nonlinear Anal., 26 (1996), 707-723.  doi: 10.1016/0362-546X(94)00311-5.

[2]

T. Aiki and H. Imai, Blow-up points to one phase Stefan problems with Dirichlet boundary conditions, Modelling and Optimization of Distributed Parameter Systems, IFIP - The International Federation for Information Processing, Chapman & Hall, New York, 1996, 83–89. doi: 10.1007/978-0-387-34922-0_6.

[3]

T. Aiki and H. Imai, Global existence of solutions to one-phase Stefan problems for semilinear parabolic equations(*), Ann. Mat. Pura Appl., 175 (1998), 327-337.  doi: 10.1007/BF01783691.

[4]

T. AikiH. ImaiN. Ishimura and Y. Yamada, Well-posedness of one-phase Stefan problems for sublinear heat equations, Nonlinear Anal., 51 (2002), 587-606.  doi: 10.1016/S0362-546X(01)00845-8.

[5]

X. L. Bai and S. N. Zheng, A semilinear parabolic system with coupling variable exponents, Ann. Mat. Pura Appl., 190 (2011), 525-537.  doi: 10.1007/s10231-010-0161-2.

[6]

A. Bensoussan and J. L. Lions, Problémes de temps d'arr${\hat e}$t optimal et inéquations variationelles paraboliques, Applicable Anal., 3 (1973), 267-294.  doi: 10.1080/00036817308839070.

[7]

G. BuntingY. H. Du and K. Krakowski, Spreading speed revisted: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.

[8]

J. F. CaoW. T. Li and M. Zhao, A nonlocal diffusion model with free boundaries in spatial heterogeneous environment, J. Math. Anal. Appl., 449 (2017), 1015-1035.  doi: 10.1016/j.jmaa.2016.12.044.

[9]

Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.

[10]

Y. H. DuZ. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.

[11]

Y. H. Du and Z. G. 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.

[12]

Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.

[13]

Y. H. DuH. Matsuzawa and M. L. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.  doi: 10.1137/130908063.

[14]

X. L. Fan and D. Zhao, On the spaces $L^p(x)(\Omega)$ and $W^{k, \, p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.

[15]

A. Fasano and M. Primicerio, General free-boundary problems for the heat equation. I., J. Math. Anal. Appl., 57 (1977), 694–723; II. 58 (1977), 202–231; III. 59 (1977), l–14. doi: 10.1016/0022-247X(77)90256-6.

[16]

R. FerreiraA. de PabloM. Pérez-LLanos and J. D. Rossi, Critical exponents for a semilinear parabolic equation with variable reaction, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1027-1042.  doi: 10.1017/S0308210510000399.

[17]

M. Fila and P. Souplet, Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem, Interfaces Free Bound., 3 (2001), 337-344.  doi: 10.4171/IFB/43.

[18]

A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[19]

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

[20]

H. GuB. D. Lou and M. L. Zhou, Long time behavior for solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002.

[21]

H. M. Huang and M. X. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050.  doi: 10.3934/dcdsb.2015.20.2039.

[22]

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330.  doi: 10.1002/cpa.3160160307.

[23]

Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differential Equations, 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017.

[24]

H. A. Levine, Some noexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au+F(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.

[25]

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

[26]

B. C. Liu and F. J. Li, Non-simultaneous blowup in heat equations with nonstandard growth conditions, J. Differential Equations, 252 (2012), 4481-4502.  doi: 10.1016/j.jde.2012.01.001.

[27]

S. Y. LiuH. M. Huang and M. X. Wang, Asymptotic spreading of a diffusive competition model with different free boundaries, J. Differential Equations, 266 (2019), 4769-4799.  doi: 10.1016/j.jde.2018.10.009.

[28]

E. Magenes, Topics in parabolic equations: Some typical free boundary problems, in Boundary value problems for linear evolution partial differential equations, NATO Advanced Study Institutes Series, 29 (1977), 239–312. doi: 10.1007/978-94-010-1205-8_5.

[29]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031.  doi: 10.3934/dcds.2013.33.2007.

[30]

J. P. Pinasco, Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear Anal., 71 (2009), 1094-1099.  doi: 10.1016/j.na.2008.11.030.

[31]

R. Ricci and D. A. 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.

[32]

L. I. Rubinstein, The Stefan problem, translated from the Russian by A. D. Solomon, in Translations of Mathematical Monographs, 27, American Mathematical Society, Providence, R.I., 1971.

[33]

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

[34]

J. P. Wang and M. X. Wang, The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary, Math. Methods Appl. Sci., 41 (2018), 6741-6762.  doi: 10.1002/mma.5189.

[35]

M. X. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508.  doi: 10.1016/j.jfa.2015.10.014.

[36]

M. X. Wang, Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 415-421.  doi: 10.3934/dcdsb.2018179.

[37]

M. X. Wang and Q. Y. Zhang, Dynamics for the diffusive Leslie-Gower model with double free boundaries, Discrete Contin. Dyn. Syst., 38 (2018), 2591-2607.  doi: 10.3934/dcds.2018109.

[38]

M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with two free boundaries, J. Differential Equations, 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.

[39]

M. X. Wang and Y. Zhang, Note on a two-species competition-diffusion model with two free boundaries, Nonlinear Anal., 159 (2017), 458-467.  doi: 10.1016/j.na.2017.01.005.

[40]

M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal. Real World Appl., 24 (2015), 73-82.  doi: 10.1016/j.nonrwa.2015.01.004.

[41]

M. X. Wang and J. F. Zhao, A free boundary problem for a prey-predator model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.

[42]

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

[43]

Y. Zhang and M. X. Wang, A free boundary problem of the ratio-dependent prey-predator model, Appl. Anal., 94 (2015), 2147-2167.  doi: 10.1080/00036811.2014.979806.

[44]

Y. G. Zhao and M. X. Wang, Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients, IMA J. Appl. Math., 81 (2016), 255-280.  doi: 10.1093/imamat/hxv035.

[45]

P. ZhouJ. Bao and Z. G. Lin, Global existence and blowup of a localized problem with free boundary, Nonlinear Anal., 74 (2011), 2523-2533.  doi: 10.1016/j.na.2010.11.047.

[46]

P. Zhou and D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954.  doi: 10.1016/j.jde.2013.12.008.

show all references

References:
[1]

T. Aiki, Behavior of free boundaries of blow up solutions to one-phase Stefan problems, Nonlinear Anal., 26 (1996), 707-723.  doi: 10.1016/0362-546X(94)00311-5.

[2]

T. Aiki and H. Imai, Blow-up points to one phase Stefan problems with Dirichlet boundary conditions, Modelling and Optimization of Distributed Parameter Systems, IFIP - The International Federation for Information Processing, Chapman & Hall, New York, 1996, 83–89. doi: 10.1007/978-0-387-34922-0_6.

[3]

T. Aiki and H. Imai, Global existence of solutions to one-phase Stefan problems for semilinear parabolic equations(*), Ann. Mat. Pura Appl., 175 (1998), 327-337.  doi: 10.1007/BF01783691.

[4]

T. AikiH. ImaiN. Ishimura and Y. Yamada, Well-posedness of one-phase Stefan problems for sublinear heat equations, Nonlinear Anal., 51 (2002), 587-606.  doi: 10.1016/S0362-546X(01)00845-8.

[5]

X. L. Bai and S. N. Zheng, A semilinear parabolic system with coupling variable exponents, Ann. Mat. Pura Appl., 190 (2011), 525-537.  doi: 10.1007/s10231-010-0161-2.

[6]

A. Bensoussan and J. L. Lions, Problémes de temps d'arr${\hat e}$t optimal et inéquations variationelles paraboliques, Applicable Anal., 3 (1973), 267-294.  doi: 10.1080/00036817308839070.

[7]

G. BuntingY. H. Du and K. Krakowski, Spreading speed revisted: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.

[8]

J. F. CaoW. T. Li and M. Zhao, A nonlocal diffusion model with free boundaries in spatial heterogeneous environment, J. Math. Anal. Appl., 449 (2017), 1015-1035.  doi: 10.1016/j.jmaa.2016.12.044.

[9]

Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.

[10]

Y. H. DuZ. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.

[11]

Y. H. Du and Z. G. 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.

[12]

Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.

[13]

Y. H. DuH. Matsuzawa and M. L. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.  doi: 10.1137/130908063.

[14]

X. L. Fan and D. Zhao, On the spaces $L^p(x)(\Omega)$ and $W^{k, \, p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.

[15]

A. Fasano and M. Primicerio, General free-boundary problems for the heat equation. I., J. Math. Anal. Appl., 57 (1977), 694–723; II. 58 (1977), 202–231; III. 59 (1977), l–14. doi: 10.1016/0022-247X(77)90256-6.

[16]

R. FerreiraA. de PabloM. Pérez-LLanos and J. D. Rossi, Critical exponents for a semilinear parabolic equation with variable reaction, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1027-1042.  doi: 10.1017/S0308210510000399.

[17]

M. Fila and P. Souplet, Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem, Interfaces Free Bound., 3 (2001), 337-344.  doi: 10.4171/IFB/43.

[18]

A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[19]

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

[20]

H. GuB. D. Lou and M. L. Zhou, Long time behavior for solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002.

[21]

H. M. Huang and M. X. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050.  doi: 10.3934/dcdsb.2015.20.2039.

[22]

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330.  doi: 10.1002/cpa.3160160307.

[23]

Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differential Equations, 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017.

[24]

H. A. Levine, Some noexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au+F(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.

[25]

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

[26]

B. C. Liu and F. J. Li, Non-simultaneous blowup in heat equations with nonstandard growth conditions, J. Differential Equations, 252 (2012), 4481-4502.  doi: 10.1016/j.jde.2012.01.001.

[27]

S. Y. LiuH. M. Huang and M. X. Wang, Asymptotic spreading of a diffusive competition model with different free boundaries, J. Differential Equations, 266 (2019), 4769-4799.  doi: 10.1016/j.jde.2018.10.009.

[28]

E. Magenes, Topics in parabolic equations: Some typical free boundary problems, in Boundary value problems for linear evolution partial differential equations, NATO Advanced Study Institutes Series, 29 (1977), 239–312. doi: 10.1007/978-94-010-1205-8_5.

[29]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031.  doi: 10.3934/dcds.2013.33.2007.

[30]

J. P. Pinasco, Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear Anal., 71 (2009), 1094-1099.  doi: 10.1016/j.na.2008.11.030.

[31]

R. Ricci and D. A. 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.

[32]

L. I. Rubinstein, The Stefan problem, translated from the Russian by A. D. Solomon, in Translations of Mathematical Monographs, 27, American Mathematical Society, Providence, R.I., 1971.

[33]

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

[34]

J. P. Wang and M. X. Wang, The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary, Math. Methods Appl. Sci., 41 (2018), 6741-6762.  doi: 10.1002/mma.5189.

[35]

M. X. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508.  doi: 10.1016/j.jfa.2015.10.014.

[36]

M. X. Wang, Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 415-421.  doi: 10.3934/dcdsb.2018179.

[37]

M. X. Wang and Q. Y. Zhang, Dynamics for the diffusive Leslie-Gower model with double free boundaries, Discrete Contin. Dyn. Syst., 38 (2018), 2591-2607.  doi: 10.3934/dcds.2018109.

[38]

M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with two free boundaries, J. Differential Equations, 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.

[39]

M. X. Wang and Y. Zhang, Note on a two-species competition-diffusion model with two free boundaries, Nonlinear Anal., 159 (2017), 458-467.  doi: 10.1016/j.na.2017.01.005.

[40]

M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal. Real World Appl., 24 (2015), 73-82.  doi: 10.1016/j.nonrwa.2015.01.004.

[41]

M. X. Wang and J. F. Zhao, A free boundary problem for a prey-predator model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.

[42]

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

[43]

Y. Zhang and M. X. Wang, A free boundary problem of the ratio-dependent prey-predator model, Appl. Anal., 94 (2015), 2147-2167.  doi: 10.1080/00036811.2014.979806.

[44]

Y. G. Zhao and M. X. Wang, Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients, IMA J. Appl. Math., 81 (2016), 255-280.  doi: 10.1093/imamat/hxv035.

[45]

P. ZhouJ. Bao and Z. G. Lin, Global existence and blowup of a localized problem with free boundary, Nonlinear Anal., 74 (2011), 2523-2533.  doi: 10.1016/j.na.2010.11.047.

[46]

P. Zhou and D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954.  doi: 10.1016/j.jde.2013.12.008.

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Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017

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