In this paper we present some results on the uniqueness and existence of a class of weak solutions (the so called BV solutions) of the Cauchy-Dirichlet problem associated to the doubly nonlinear diffusion equation
b(u)t - div (|∇u - k(b(u))e|p-2 (∇u - k(b(u))e)) + g(x,u) = f(t,x).
This problem arises in the study of some turbulent regimes: flows of incompressible turbulent fluids through porous media, gases flowing in pipelines, etc. The solvability of this problem is established in the BVt(Q) space. We prove some comparison properties (implying uniqueness) when the set of jumping points of the BV solution has N-dimensional null measure and suitable additional conditions as, for instance, b-1 locally Lipschitz. The existence of this type of weak solution is based on suitable uniform estimates of the BV norm of an approximated solution.
@article{urn:eudml:doc:41258,
title = {Uniqueness and existence of solution in the BVt(Q) space to a doubly nonlinear parabolic problem.},
journal = {Publicacions Matem\`atiques},
volume = {40},
year = {1996},
pages = {527-560},
mrnumber = {MR1425634},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41258}
}
Díaz, Jesús Ildefonso; Padial, Juan Francisco. Uniqueness and existence of solution in the BVt(Q) space to a doubly nonlinear parabolic problem.. Publicacions Matemàtiques, Tome 40 (1996) pp. 527-560. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41258/