ABSTRACT
Chapter 3
Uniqueness, comparison, and
well-posedness results for
quasilinear dierential equations
The purpose of this chapter is to present comparison, uniqueness, and well-
posedness results for initial and boundary value problems of rst and second
order quasilinear dierential equations. We begin section 3.1 with a com-
parison principle, which is then applied to prove comparison and uniqueness
results for a boundary value problem of the dierential equation
d
dt
'(t; u(t)) = g(t; u(t)):
In section 3.2 we apply an analogous procedure to provide comparison and
uniqueness results for initial value problems of the above-mentioned dier-
ential equation and the implicit dierential equation
d
dt
'(t; u(t)) = g(t; u(t)) + f(t; u(t);
d
dt
'(t; u(t)) g(t; u(t))):
The assumptions imposed on the functions ', g, and f allow them to be
discontinuous in all their variables. The so-obtained results, combined with
existence results derived in chapter 2, yield also existence and uniqueness
results for the above problems with '(t; u(t)) replaced by '(u(t)).
