On the Existence of Extremal Periodic Solutions for Nonlinear Parabolic Problems with Discontinuities

✍ Scribed by Evgenios P. Avgerinos; Nikolas S. Papageorgiou

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 751 KB
• Volume: 132
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.0176

✦ Synopsis

We consider a very general second order nonlinear parabolic boundary value problem. Assuming the existence of an upper solution . and a lower solution satisfying

., we show that the problem has extremal periodic solutions in the order interval K=[ , .]. Our proof is based on a general surjectivity result for the sum of two operators of monotone type and on truncation and penalization techniques. In addition we use a result of independent interest which we prove here and which says that the pseudomonotonicity property of A(t, } ) can be lifted to its Nemitsky operator. Finally when we impose stronger conditions on the data, we show that the extremal solutions can be obtained with a monotone iterative process.