TY - JOUR
T1 - Porous medium equation with a blow-up nonlinearity and a non-decreasing constraint
AU - Akagi, Goro
AU - Melchionna, Stefano
N1 - Funding Information:
GA is supported by JSPS KAKENHI Grant Numbers JP16H03946, JP16K05199, JP17H01095, JP18K18715 and by the Alexander von Humboldt Foundation and by the Carl Friedrich von Siemens Foundation. SM acknowledges the support of the Austrian Science Fund (FWF) Project P27052-N25.
Publisher Copyright:
© 2019, Springer Nature Switzerland AG.
PY - 2019/4/1
Y1 - 2019/4/1
N2 - The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage Mechanics, is reformulated as a mixed form of two different types of doubly nonlinear evolution equations. Global (in time) solutions to some approximate problems are constructed by performing a time discretization argument and by taking advantage of energy techniques based on specific structures of the equation. Moreover, a variational comparison principle for (possibly non-unique) approximate solutions is established and it also enables us to obtain a local solution as a limit of approximate ones.
AB - The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage Mechanics, is reformulated as a mixed form of two different types of doubly nonlinear evolution equations. Global (in time) solutions to some approximate problems are constructed by performing a time discretization argument and by taking advantage of energy techniques based on specific structures of the equation. Moreover, a variational comparison principle for (possibly non-unique) approximate solutions is established and it also enables us to obtain a local solution as a limit of approximate ones.
KW - Blow-up in finite time
KW - Mixed doubly nonlinear equations
KW - Porous medium equation
KW - Unidirectional evolution
KW - Variational comparison principle
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U2 - 10.1007/s00030-019-0551-0
DO - 10.1007/s00030-019-0551-0
M3 - Article
AN - SCOPUS:85062558112
SN - 1021-9722
VL - 26
JO - Nonlinear Differential Equations and Applications
JF - Nonlinear Differential Equations and Applications
IS - 2
M1 - 10
ER -