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September 2018, 17(5): 1945-1956. doi: 10.3934/cpaa.2018092

A quasilinear parabolic problem with a source term and a nonlocal absorption

School of Mathematics, Southeast University, Nanjing 210096, China

* Corresponding author

Received  August 2017 Revised  December 2017 Published  April 2018

Fund Project: The first author is supported by the NSF of China 11171064. The third author is supported by the Postdoctoral Science Foundation of Jiangsu Province 1402026C, etc.

We investigate a quasi-linear parabolicproblem with nonlocal absorption, for which the comparison principle is not always available. Thesufficient conditions are established via energy method to guaranteesolution to blow up or not, and the long time behavior is alsocharacterized for global solutions.

Citation: Hui-Ling Li, Heng-Ling Wang, Xiao-Liu Wang. A quasilinear parabolic problem with a source term and a nonlocal absorption. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1945-1956. doi: 10.3934/cpaa.2018092
References:
[1]

J. Bebernes and A. Bressan, Thermal behavior for a confined reactive gas, J. Differential Equations, 44 (1982), 118-133.

[2]

S. Boussa${\rm{\ddot i}}$dD. Hilhorst and T. N. Nguyen, Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation, Evol. Equ. Control Theory, 4 (2015), 39-59.

[3]

C. BuddB. Dold and A. Stuart, Blowup in a partial differential equation with conserved first integral, SIAM J. Appl. Math., 53 (1993), 718-742.

[4]

K. L. Cheung and Z. Y. Zhang, Nonexistence of global solutions for a family of nonlocal or higher-order parabolic problems, Differential Integral Equations, 25 (2012), 787-800.

[5]

W. B. DengY. X. Li and C. H. Xie, Existence and nonexistence of global solutions of some nonlocal degenerate parabolic equations, App. Math. Lett., 16 (2003), 803-808.

[6]

A. El SoufiM. Jazar and R. Monneau, A gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 17-39.

[7]

P. Freitas, Stability of stationary solutions for a scalar non-local reaction-diffusion equation, Quart. J. Mech. Appl. Math., 48 (1995), 557-582.

[8]

P. Freitas and M. P. Vishnevskii, Stability of stationary solutions of nonlocal reaction-diffusion equations in $m$-dimensional space, Differential Integral Equations, 13 (2000), 265-288.

[9]

W. J. Gao and Y. Z. Han, A degenerate parabolic equation with a nonlocal source and an absorption term, Appl. Anal., 89 (2010), 1917-1930.

[10]

W. J. Gao and Y. Z. Han, Blow-up of a nonlocal semilinear parabolic equation with positive initial energy, Appl. Math. Lett., 24 (2011), 784-788.

[11]

B. Hu and H. M. Yin, Semilinear parabolic equations with prescribed energy, Rend. Circ. Mat. Palermo (2), 44 (1995), 479-505.

[12]

M. Jazar and R. Kiwan, Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 215-218.

[13]

A. Khelghati and K. Baghaei, Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy, Comput. Math. Appl., 70 (2015), 896-902.

[14]

O. A. Lady$\check{\rm{z}}$enskaja, V. A Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R. I. 1967.

[15]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.

[16]

Y. C. Lin and D. H. Tsai, On a simple maximum principle technique applied to equations on the circle, J. Differential Equations, 245 (2008), 377-391.

[17]

Y. Y. Mao, S. L. Pan and Y. L. Wang, An area-preserving flow for convex closed plane curves, Int. J. Math., 24 (2013), 1350029 (31 pages).

[18]

P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkh$\ddot{\mathit{\rm{a}}}$user Advanced Texts: Basler Lehrb$\ddot{\mathit{\rm{u}}}$cher. Birkh$\ddot{\mathit{\rm{a}}}$user Verlag, Basel, 2007.

[19]

J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math., 48 (1992), 249-264.

[20]

P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), 1301-1334.

[21]

D. H. Tsai and X. L. Wang, On length-preserving and area-preserving nonlocal flow of convex closed plane curves, Calc. Var. Partial Differential Equations, 54 (2015), 3603-3622.

[22]

M. X. Wang and Y. M. Wang, Properties of positive solutions for non-local reaction-diffusion problems, Math. Methods Appl. Sci., 19 (1996), 1141-1156.

[23]

X. L. WangF. Z. Tian and G. Li, Nonlocal parabolic equation with conserved spatial integral, Archiv der Mathmatik, 105 (2015), 93-100.

[24]

S. WangM. X. Wang and C. H. Xie, A nonlinear degenerate diffusion equation not in divergence form, Z. Angew. Math. Phys., 51 (2000), 149-159.

[25]

M. Winkler, Blow-up of solutions to a degenerate parabolic equation not in divergence form, J. Differential Equations, 192 (2003), 445-474.

[26]

M. Winkler, Blow-up in a degenerate parabolic equation, Indiana Univ. Math. J., 53 (2004), 1415-1442.

[27]

M. Winkler, A doubly critical degenerate parabolic problem, Math. Methods Appl. Sci., 27 (2004), 1619-1627.

show all references

References:
[1]

J. Bebernes and A. Bressan, Thermal behavior for a confined reactive gas, J. Differential Equations, 44 (1982), 118-133.

[2]

S. Boussa${\rm{\ddot i}}$dD. Hilhorst and T. N. Nguyen, Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation, Evol. Equ. Control Theory, 4 (2015), 39-59.

[3]

C. BuddB. Dold and A. Stuart, Blowup in a partial differential equation with conserved first integral, SIAM J. Appl. Math., 53 (1993), 718-742.

[4]

K. L. Cheung and Z. Y. Zhang, Nonexistence of global solutions for a family of nonlocal or higher-order parabolic problems, Differential Integral Equations, 25 (2012), 787-800.

[5]

W. B. DengY. X. Li and C. H. Xie, Existence and nonexistence of global solutions of some nonlocal degenerate parabolic equations, App. Math. Lett., 16 (2003), 803-808.

[6]

A. El SoufiM. Jazar and R. Monneau, A gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 17-39.

[7]

P. Freitas, Stability of stationary solutions for a scalar non-local reaction-diffusion equation, Quart. J. Mech. Appl. Math., 48 (1995), 557-582.

[8]

P. Freitas and M. P. Vishnevskii, Stability of stationary solutions of nonlocal reaction-diffusion equations in $m$-dimensional space, Differential Integral Equations, 13 (2000), 265-288.

[9]

W. J. Gao and Y. Z. Han, A degenerate parabolic equation with a nonlocal source and an absorption term, Appl. Anal., 89 (2010), 1917-1930.

[10]

W. J. Gao and Y. Z. Han, Blow-up of a nonlocal semilinear parabolic equation with positive initial energy, Appl. Math. Lett., 24 (2011), 784-788.

[11]

B. Hu and H. M. Yin, Semilinear parabolic equations with prescribed energy, Rend. Circ. Mat. Palermo (2), 44 (1995), 479-505.

[12]

M. Jazar and R. Kiwan, Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 215-218.

[13]

A. Khelghati and K. Baghaei, Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy, Comput. Math. Appl., 70 (2015), 896-902.

[14]

O. A. Lady$\check{\rm{z}}$enskaja, V. A Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R. I. 1967.

[15]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.

[16]

Y. C. Lin and D. H. Tsai, On a simple maximum principle technique applied to equations on the circle, J. Differential Equations, 245 (2008), 377-391.

[17]

Y. Y. Mao, S. L. Pan and Y. L. Wang, An area-preserving flow for convex closed plane curves, Int. J. Math., 24 (2013), 1350029 (31 pages).

[18]

P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkh$\ddot{\mathit{\rm{a}}}$user Advanced Texts: Basler Lehrb$\ddot{\mathit{\rm{u}}}$cher. Birkh$\ddot{\mathit{\rm{a}}}$user Verlag, Basel, 2007.

[19]

J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math., 48 (1992), 249-264.

[20]

P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), 1301-1334.

[21]

D. H. Tsai and X. L. Wang, On length-preserving and area-preserving nonlocal flow of convex closed plane curves, Calc. Var. Partial Differential Equations, 54 (2015), 3603-3622.

[22]

M. X. Wang and Y. M. Wang, Properties of positive solutions for non-local reaction-diffusion problems, Math. Methods Appl. Sci., 19 (1996), 1141-1156.

[23]

X. L. WangF. Z. Tian and G. Li, Nonlocal parabolic equation with conserved spatial integral, Archiv der Mathmatik, 105 (2015), 93-100.

[24]

S. WangM. X. Wang and C. H. Xie, A nonlinear degenerate diffusion equation not in divergence form, Z. Angew. Math. Phys., 51 (2000), 149-159.

[25]

M. Winkler, Blow-up of solutions to a degenerate parabolic equation not in divergence form, J. Differential Equations, 192 (2003), 445-474.

[26]

M. Winkler, Blow-up in a degenerate parabolic equation, Indiana Univ. Math. J., 53 (2004), 1415-1442.

[27]

M. Winkler, A doubly critical degenerate parabolic problem, Math. Methods Appl. Sci., 27 (2004), 1619-1627.

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