-
Previous Article
Unique periodic orbits of a delay differential equation with piecewise linear feedback function
- DCDS Home
- This Issue
-
Next Article
Schauder estimates for a class of non-local elliptic equations
Thermal runaway for a nonlinear diffusion model in thermal electricity
1. | Department of Mathematics, Sichuan University, Chengdu 610064, China |
2. | Department of Mathematics, Jingcheng College of Sichuan University, Chengdu 611731, China |
References:
[1] |
D. G. Aronson, On the Green's function for second order parabolic differential equations with discontinuous coefficients, Bull. Amer. Math. Soc., 69 (1963), 841-847. |
[2] |
J. W. Bebernes and R. Ely, Existence and invariance for parabolic functional equations, Nonlinear Anal. TMA., 7 (1983), 1225-1235.
doi: 10.1016/0362-546X(83)90054-8. |
[3] |
J. W. Bebernes and D. Eberly, "Mathematical Problems From Combustion Theory," Applied Mathematical Sciences, 83, Springer-Verlag, New York, 1989. |
[4] |
J. W. Bebernes and P. Talaga, Nonlocal problems modelling shear banding, Comm. Appl. Nonlinear Anal., 3 (1996), 79-103. |
[5] |
E. Dibenedetto, "Degenerate Parabolic Equations," Springer, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[6] |
L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math., 202 (2007), 237-247.
doi: 10.1016/j.cam.2006.02.028. |
[7] |
L. Du, C. Mu and M. Fan, Global existence and non-existence for a quasilinear degenerate parabolic system with non-local source, Dyn. Syst., 20 (2005), 401-412.
doi: 10.1080/14689360500238818. |
[8] |
L. Du and Z. Yao, Localization of blow-up points for a nonlinear nonlocal porous medium equation, Commun. Pure Appl. Anal., 6 (2007), 183-190.
doi: 10.3934/cpaa.2007.6.183. |
[9] |
M. Fan and L. Du, Asymptotic behavior for an Ohmic heating model in thermal electricity, Appl. Math. Comput., 218 (2012), 10906-10913.
doi: 10.1016/j.amc.2012.04.053. |
[10] |
M. Fan, C. Mu, L. Du, Uniform blow-up profiles for a nonlocal degenerate parabolic system, Appl. Math. Sci., 1 (2007), 13-23. |
[11] |
N. I. Kavallaris, Asymptotic behaviour and blow-up for a nonlinear diffusion problem with a non-local source term, Proc. Edinb. Math. Soc., 47 (2004), 375-395.
doi: 10.1017/S0013091503000658. |
[12] |
A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating: Part I: Model derivation and some spacial cases, Eur. J. Appl. Math., 6 (1995), 127-144. |
[13] |
A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating: Part II: General proof of blow-up and asymptotics of runaway, Euro. J. Appl. Math., 6 (1995), 201-224.
doi: 10.1017/S0956792500001807. |
[14] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Trans. of Math. Monographs, 23, 1968. |
[15] |
P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems, Blow-Up, Global Existence and Steady States," Birkhäuser Verlag, Basel, 2007. |
[16] |
Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), 1301-1334.
doi: 10.1137/S0036141097318900. |
[17] |
Ph. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differ. Eqns, 153 (1999), 374-406.
doi: 10.1006/jdeq.1998.3535. |
show all references
References:
[1] |
D. G. Aronson, On the Green's function for second order parabolic differential equations with discontinuous coefficients, Bull. Amer. Math. Soc., 69 (1963), 841-847. |
[2] |
J. W. Bebernes and R. Ely, Existence and invariance for parabolic functional equations, Nonlinear Anal. TMA., 7 (1983), 1225-1235.
doi: 10.1016/0362-546X(83)90054-8. |
[3] |
J. W. Bebernes and D. Eberly, "Mathematical Problems From Combustion Theory," Applied Mathematical Sciences, 83, Springer-Verlag, New York, 1989. |
[4] |
J. W. Bebernes and P. Talaga, Nonlocal problems modelling shear banding, Comm. Appl. Nonlinear Anal., 3 (1996), 79-103. |
[5] |
E. Dibenedetto, "Degenerate Parabolic Equations," Springer, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[6] |
L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math., 202 (2007), 237-247.
doi: 10.1016/j.cam.2006.02.028. |
[7] |
L. Du, C. Mu and M. Fan, Global existence and non-existence for a quasilinear degenerate parabolic system with non-local source, Dyn. Syst., 20 (2005), 401-412.
doi: 10.1080/14689360500238818. |
[8] |
L. Du and Z. Yao, Localization of blow-up points for a nonlinear nonlocal porous medium equation, Commun. Pure Appl. Anal., 6 (2007), 183-190.
doi: 10.3934/cpaa.2007.6.183. |
[9] |
M. Fan and L. Du, Asymptotic behavior for an Ohmic heating model in thermal electricity, Appl. Math. Comput., 218 (2012), 10906-10913.
doi: 10.1016/j.amc.2012.04.053. |
[10] |
M. Fan, C. Mu, L. Du, Uniform blow-up profiles for a nonlocal degenerate parabolic system, Appl. Math. Sci., 1 (2007), 13-23. |
[11] |
N. I. Kavallaris, Asymptotic behaviour and blow-up for a nonlinear diffusion problem with a non-local source term, Proc. Edinb. Math. Soc., 47 (2004), 375-395.
doi: 10.1017/S0013091503000658. |
[12] |
A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating: Part I: Model derivation and some spacial cases, Eur. J. Appl. Math., 6 (1995), 127-144. |
[13] |
A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating: Part II: General proof of blow-up and asymptotics of runaway, Euro. J. Appl. Math., 6 (1995), 201-224.
doi: 10.1017/S0956792500001807. |
[14] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Trans. of Math. Monographs, 23, 1968. |
[15] |
P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems, Blow-Up, Global Existence and Steady States," Birkhäuser Verlag, Basel, 2007. |
[16] |
Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), 1301-1334.
doi: 10.1137/S0036141097318900. |
[17] |
Ph. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differ. Eqns, 153 (1999), 374-406.
doi: 10.1006/jdeq.1998.3535. |
[1] |
Aníbal Rodríguez-Bernal, Silvia Sastre-Gómez. Nonlinear nonlocal reaction-diffusion problem with local reaction. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1731-1765. doi: 10.3934/dcds.2021170 |
[2] |
Yuan Lou, Wei-Ming Ni, Linlin Su. An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 643-655. doi: 10.3934/dcds.2010.27.643 |
[3] |
Dong Li, Xiaoyi Zhang. On a nonlocal aggregation model with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 301-323. doi: 10.3934/dcds.2010.27.301 |
[4] |
Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174 |
[5] |
Costică Moroşanu, Bianca Satco. Qualitative and quantitative analysis for a nonlocal and nonlinear reaction-diffusion problem with in-homogeneous Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022042 |
[6] |
Asim Aziz, Wasim Jamshed, Yasir Ali, Moniba Shams. Heat transfer and entropy analysis of Maxwell hybrid nanofluid including effects of inclined magnetic field, Joule heating and thermal radiation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2667-2690. doi: 10.3934/dcdss.2020142 |
[7] |
Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022039 |
[8] |
V. S. Manoranjan, Hong-Ming Yin, R. Showalter. On two-phase Stefan problem arising from a microwave heating process. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1155-1168. doi: 10.3934/dcds.2006.15.1155 |
[9] |
Renata Bunoiu, Claudia Timofte. Homogenization of a thermal problem with flux jump. Networks and Heterogeneous Media, 2016, 11 (4) : 545-562. doi: 10.3934/nhm.2016009 |
[10] |
Qingguang Guan, Max Gunzburger. Stability and convergence of time-stepping methods for a nonlocal model for diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1315-1335. doi: 10.3934/dcdsb.2015.20.1315 |
[11] |
Martin Burger, Marco Di Francesco. Large time behavior of nonlocal aggregation models with nonlinear diffusion. Networks and Heterogeneous Media, 2008, 3 (4) : 749-785. doi: 10.3934/nhm.2008.3.749 |
[12] |
Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171 |
[13] |
Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128 |
[14] |
José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure and Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85 |
[15] |
Dinh-Ke Tran, Tran-Phuong-Thuy Lam. Nonlocal final value problem governed by semilinear anomalous diffusion equations. Evolution Equations and Control Theory, 2020, 9 (3) : 891-914. doi: 10.3934/eect.2020038 |
[16] |
Dinh-Ke Tran, Nhu-Thang Nguyen. On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations. Communications on Pure and Applied Analysis, 2022, 21 (3) : 817-835. doi: 10.3934/cpaa.2021200 |
[17] |
Jonathan E. Rubin. A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 925-940. doi: 10.3934/dcds.2004.10.925 |
[18] |
Peng Zhou, Jiang Yu, Dongmei Xiao. A nonlinear diffusion problem arising in population genetics. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 821-841. doi: 10.3934/dcds.2014.34.821 |
[19] |
Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169 |
[20] |
Dieter Schmidt, Lucas Valeriano. Nonlinear stability of stationary points in the problem of Robe. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1917-1936. doi: 10.3934/dcdsb.2016029 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]