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Sign-changing solutions of a quasilinear heat equation with a source term
Analysis of a scalar nonlocal peridynamic model with a sign changing kernel
1. | Department of Mathematics, Pennsylvania State University, University Park, PA 16802, United States |
References:
[1] |
B. Aksoylu and T. Mengesha, Results on nonlocal boundary value problems, Numerical Functional Analysis and Optimization, 31 (2010), 1301-1317.
doi: 10.1080/01630563.2010.519136. |
[2] |
B. Alali and R. Lipton, Multiscale analysis of heterogeneous media in the peridynamic formulation, Journal of Elasticity, 106 (2012), 71-103.
doi: 10.1007/s10659-010-9291-4. |
[3] |
F. Andreu-Vaillo, J. M. Mazn, Julio D. Rossi and J. J. Toledo-Melero, "Nonlocal Diffusion Problems," American Mathematical Society. Mathematical Surveys and Monographs, 2010. 165. |
[4] |
J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in "Optimal Control and Partial Differential Equations" (Editors, J. L. Menaldi, E. Rofman and A. Sulem), IOS Press (2001), 439-455. A volume in honour of A. Benssoussan's 60th birthday. |
[5] |
H. Brézis, "Analyse Fonctionnelle. Théorie et Applications," Masson, 1978. |
[6] |
H. Brézis, How to recognize constant functions. Connections with Sobolev spaces, Uspekhi Mat. Nauk 57, 59-74 (2002)(Russian). English version: Russian Math Surveys 57 (2002), 693-708, Volume in Honor of M. Vishik.
doi: 10.1070/RM2002v057n04ABEH000533. |
[7] |
R. Dautray and J-L. Lions, "Mathematical and Numerical Analysis for Science and Technology, Evolution Problems I," 5, Springer-Verlag, 1992.
doi: 10.1007/978-3-642-58090-1. |
[8] |
K. Dayal and K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics, J. Mech. Phys. Solids, 54 (2006), 1811-1842.
doi: 10.1016/j.jmps.2006.04.001. |
[9] |
Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, Analysis of the volume-constrained peridynamic Navier equation of linear elasticity, to appear in Journal of Elasticity, 2013.
doi: 10.1007/s10659-012-9418-x. |
[10] |
Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Mod. Meth. Appl. Sci., 23 (2013), 493-540.
doi: 10.1142/S0218202512500546. |
[11] |
Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54 (2012), 667-696.
doi: 10.1137/110833294. |
[12] |
Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory, ESIAM: Math. Mod. Numer. Anal., 45 (2011), 217-234.
doi: 10.1051/m2an/2010040. |
[13] |
E. Emmrich and O. Weckner, The peridynamic equation and its spatial discretization, Mathematical Modelling and Analysis, 12 (2007), 17-27.
doi: 10.3846/1392-6292.2007.12.17-27. |
[14] |
E. Emmrich and O. Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the {Navier equation of linear elasticity}, Commun. Math. Sci., 5 (2007), 851-864. |
[15] |
G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling and Simulation, 7 (2008), {1005-1028}.
doi: 10.1137/070698592. |
[16] |
M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598.
doi: 10.1137/090766607. |
[17] |
T. Mengesha and Q. Du, The bond-based peridynamic system with Dirichlet-type volume constraint, Proceeding of Royal Soc. Edinburgh A, to appear 2013. |
[18] |
T. Mengesha and Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation, preprint, 2013. |
[19] |
S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Physics, 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
[20] |
S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48 (2000), {175-209}.
doi: 10.1016/S0022-5096(99)00029-0. |
[21] |
E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[22] |
A. Ponce, An estimate in the spirit of Poincare's inequality, J. Eur. Math. Soc., 6 (2004), {1-15}. |
[23] |
S. Silling, O. Weckner, E. Askari and F. Bobaru, Crack nucleation in a peridynamic solid, Int. J. Fract., 162 (2010), 219-227. |
[24] |
K. Zhou and Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary, SIAM J. Numer. Anal., 48 (2010), 1759-1780.
doi: 10.1137/090781267. |
show all references
References:
[1] |
B. Aksoylu and T. Mengesha, Results on nonlocal boundary value problems, Numerical Functional Analysis and Optimization, 31 (2010), 1301-1317.
doi: 10.1080/01630563.2010.519136. |
[2] |
B. Alali and R. Lipton, Multiscale analysis of heterogeneous media in the peridynamic formulation, Journal of Elasticity, 106 (2012), 71-103.
doi: 10.1007/s10659-010-9291-4. |
[3] |
F. Andreu-Vaillo, J. M. Mazn, Julio D. Rossi and J. J. Toledo-Melero, "Nonlocal Diffusion Problems," American Mathematical Society. Mathematical Surveys and Monographs, 2010. 165. |
[4] |
J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in "Optimal Control and Partial Differential Equations" (Editors, J. L. Menaldi, E. Rofman and A. Sulem), IOS Press (2001), 439-455. A volume in honour of A. Benssoussan's 60th birthday. |
[5] |
H. Brézis, "Analyse Fonctionnelle. Théorie et Applications," Masson, 1978. |
[6] |
H. Brézis, How to recognize constant functions. Connections with Sobolev spaces, Uspekhi Mat. Nauk 57, 59-74 (2002)(Russian). English version: Russian Math Surveys 57 (2002), 693-708, Volume in Honor of M. Vishik.
doi: 10.1070/RM2002v057n04ABEH000533. |
[7] |
R. Dautray and J-L. Lions, "Mathematical and Numerical Analysis for Science and Technology, Evolution Problems I," 5, Springer-Verlag, 1992.
doi: 10.1007/978-3-642-58090-1. |
[8] |
K. Dayal and K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics, J. Mech. Phys. Solids, 54 (2006), 1811-1842.
doi: 10.1016/j.jmps.2006.04.001. |
[9] |
Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, Analysis of the volume-constrained peridynamic Navier equation of linear elasticity, to appear in Journal of Elasticity, 2013.
doi: 10.1007/s10659-012-9418-x. |
[10] |
Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Mod. Meth. Appl. Sci., 23 (2013), 493-540.
doi: 10.1142/S0218202512500546. |
[11] |
Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54 (2012), 667-696.
doi: 10.1137/110833294. |
[12] |
Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory, ESIAM: Math. Mod. Numer. Anal., 45 (2011), 217-234.
doi: 10.1051/m2an/2010040. |
[13] |
E. Emmrich and O. Weckner, The peridynamic equation and its spatial discretization, Mathematical Modelling and Analysis, 12 (2007), 17-27.
doi: 10.3846/1392-6292.2007.12.17-27. |
[14] |
E. Emmrich and O. Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the {Navier equation of linear elasticity}, Commun. Math. Sci., 5 (2007), 851-864. |
[15] |
G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling and Simulation, 7 (2008), {1005-1028}.
doi: 10.1137/070698592. |
[16] |
M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598.
doi: 10.1137/090766607. |
[17] |
T. Mengesha and Q. Du, The bond-based peridynamic system with Dirichlet-type volume constraint, Proceeding of Royal Soc. Edinburgh A, to appear 2013. |
[18] |
T. Mengesha and Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation, preprint, 2013. |
[19] |
S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Physics, 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
[20] |
S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48 (2000), {175-209}.
doi: 10.1016/S0022-5096(99)00029-0. |
[21] |
E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[22] |
A. Ponce, An estimate in the spirit of Poincare's inequality, J. Eur. Math. Soc., 6 (2004), {1-15}. |
[23] |
S. Silling, O. Weckner, E. Askari and F. Bobaru, Crack nucleation in a peridynamic solid, Int. J. Fract., 162 (2010), 219-227. |
[24] |
K. Zhou and Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary, SIAM J. Numer. Anal., 48 (2010), 1759-1780.
doi: 10.1137/090781267. |
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