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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.
|
[2] |
J. W. Bebernes and R. Ely, Existence and invariance for parabolic functional equations,, Nonlinear Anal. TMA., 7 (1983), 1225.
doi: 10.1016/0362-546X(83)90054-8. |
[3] |
J. W. Bebernes and D. Eberly, "Mathematical Problems From Combustion Theory,", Applied Mathematical Sciences, 83 (1989).
|
[4] |
J. W. Bebernes and P. Talaga, Nonlocal problems modelling shear banding,, Comm. Appl. Nonlinear Anal., 3 (1996), 79.
|
[5] |
E. Dibenedetto, "Degenerate Parabolic Equations,", Springer, (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.
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.
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.
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.
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.
|
[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.
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. Google Scholar |
[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.
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, (2007).
|
[16] |
Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations,, SIAM J. Math. Anal., 29 (1998), 1301.
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.
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.
|
[2] |
J. W. Bebernes and R. Ely, Existence and invariance for parabolic functional equations,, Nonlinear Anal. TMA., 7 (1983), 1225.
doi: 10.1016/0362-546X(83)90054-8. |
[3] |
J. W. Bebernes and D. Eberly, "Mathematical Problems From Combustion Theory,", Applied Mathematical Sciences, 83 (1989).
|
[4] |
J. W. Bebernes and P. Talaga, Nonlocal problems modelling shear banding,, Comm. Appl. Nonlinear Anal., 3 (1996), 79.
|
[5] |
E. Dibenedetto, "Degenerate Parabolic Equations,", Springer, (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.
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.
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.
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.
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.
|
[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.
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. Google Scholar |
[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.
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, (2007).
|
[16] |
Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations,, SIAM J. Math. Anal., 29 (1998), 1301.
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.
doi: 10.1006/jdeq.1998.3535. |
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