On the existence of extremal weak solutions for a class of quasilinear parabolic problems
Siegfried Carl · Differential and Integral Equations · 1993
The existence of the greatest and least solution (in the sense of an underlying partial ordering) enclosed by assumed upper and lower solutions is proved by showing that the solution set between the given upper and lower solution is directed and by applying Zorn's lemma.Our approach allows treatment of rather general quasilinear parabolic initial boundary value problems under low regularity assumptions on the data.In this way our result extends numerous earlier results on extremal solutions obtained by using monotone iteration technique on the one hand and Ak6's method on the other hand.