Quasi-variational structure approach to systems with degenerate diffusions
Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 169-235.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

We consider the following system consisting of one strongly nonlinear partial differential inclusion, one linear PDE, and one ODE, which describes a tumor invasion phenomenon with a haptotaxis effect and was originally proposed by Chaplain and Anderson in 2003:

u t - · ( d u ( v ) β ( v ; u ) - u λ ( v ) ) + β ( v ; u ) 0 , v t = - a v w , w t = d w Δ w - b w + c u .

This system has two interesting characteristics. One is that the diffusion coefficient d u for the unknown function u in the partial differential inclusion depends on the function v, which is also unknown in this system. The other one is that the diffusion flux β(v;u) of u also depends on v. Moreover, we are especially interested in the case that β(v;u) is nonsmooth and degenerate under suitable assumptions. These facts make it difficult for us to treat this system mathematically. In order to overcome these mathematical difficulties, we apply the theory of evolution inclusions on the real Hilbert space V * , the dual space of V, with a quasi-variational structure for the inner products, which was established by the author in 2019, and show the existence of time global solutions to the initial-boundary value problem for this system.

DOI : 10.4171/rsmup/96
Classification : 34, 47, 49, 58
@article{RSMUP_2022__147__169_0,
     author = {Akio Ito},
     title = {Quasi-variational structure approach to systems with degenerate diffusions},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {169--235},
     volume = {147},
     year = {2022},
     doi = {10.4171/rsmup/96},
     mrnumber = {4450789},
     zbl = {1495.35222},
     language = {en},
     url = {http://www.numdam.org/articles/10.4171/rsmup/96/}
}
TY  - JOUR
AU  - Akio Ito
TI  - Quasi-variational structure approach to systems with degenerate diffusions
JO  - Rendiconti del Seminario Matematico della Università di Padova
PY  - 2022
SP  - 169
EP  - 235
VL  - 147
UR  - http://www.numdam.org/articles/10.4171/rsmup/96/
DO  - 10.4171/rsmup/96
LA  - en
ID  - RSMUP_2022__147__169_0
ER  - 
%0 Journal Article
%A Akio Ito
%T Quasi-variational structure approach to systems with degenerate diffusions
%J Rendiconti del Seminario Matematico della Università di Padova
%D 2022
%P 169-235
%V 147
%U http://www.numdam.org/articles/10.4171/rsmup/96/
%R 10.4171/rsmup/96
%G en
%F RSMUP_2022__147__169_0
Akio Ito. Quasi-variational structure approach to systems with degenerate diffusions. Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 169-235. doi : 10.4171/rsmup/96. http://www.numdam.org/articles/10.4171/rsmup/96/

Cité par Sources :