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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[15]

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

[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. Google Scholar

[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).Google Scholar

[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.Google Scholar

[19]

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

[20]

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

[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. Google Scholar

[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. Google Scholar

[23]

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

[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. Google Scholar

[25]

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

[26]

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

[27]

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

show all references

References:
[1]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[15]

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

[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. Google Scholar

[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).Google Scholar

[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.Google Scholar

[19]

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

[20]

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

[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. Google Scholar

[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. Google Scholar

[23]

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

[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. Google Scholar

[25]

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

[26]

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

[27]

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

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