TY - JOUR
T1 - Unidirectional evolution equations of diffusion type
AU - Akagi, Goro
AU - Kimura, Masato
N1 - Funding Information:
The authors are grateful to the anonymous referees for careful reading and useful comments to improve the presentation of the paper. G.A. is supported by JSPS KAKENHI Grant Number JP16H03946 , JP16K05199 , JP17H01095 , by the Alexander von Humboldt Foundation and by the Carl Friedrich von Siemens Foundation . M.K. is supported by JSPS KAKENHI Grant Number JP17H02857 . Both authors are also supported by the JSPS–CNR bilateral joint research project: Innovative Variational Methods for Evolution Equations.
Publisher Copyright:
© 2018 Elsevier Inc.
PY - 2019/1/5
Y1 - 2019/1/5
N2 - This paper is concerned with the uniqueness, existence, partial smoothing effect, comparison principle and long-time behavior of solutions to the initial-boundary value problem for a unidirectional diffusion equation. The unidirectional evolution often appears in Damage Mechanics due to the strong irreversibility of crack propagation or damage evolution. The existence of solutions is proved in an L2-framework by employing a backward Euler scheme and by introducing a new method of a priori estimates based on a reduction of discretized equations to variational inequalities of obstacle type and by developing a regularity theory for such obstacle problems. The novel discretization argument will be also applied to prove the comparison principle as well as to investigate the long-time behavior of solutions.
AB - This paper is concerned with the uniqueness, existence, partial smoothing effect, comparison principle and long-time behavior of solutions to the initial-boundary value problem for a unidirectional diffusion equation. The unidirectional evolution often appears in Damage Mechanics due to the strong irreversibility of crack propagation or damage evolution. The existence of solutions is proved in an L2-framework by employing a backward Euler scheme and by introducing a new method of a priori estimates based on a reduction of discretized equations to variational inequalities of obstacle type and by developing a regularity theory for such obstacle problems. The novel discretization argument will be also applied to prove the comparison principle as well as to investigate the long-time behavior of solutions.
KW - Damage mechanics
KW - Discretization
KW - Regularity
KW - Subdifferential calculus
KW - Unidirectional diffusion equation
KW - Variational inequality of obstacle type
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U2 - 10.1016/j.jde.2018.05.022
DO - 10.1016/j.jde.2018.05.022
M3 - Article
AN - SCOPUS:85055669410
SN - 0022-0396
VL - 266
SP - 1
EP - 43
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 1
ER -