Explicit exponential decay bounds in quasilinear parabolic problems

• Main Authors: Piro S Vernier, Philippin GA
• Format: Article
• Language: English
• Published: SpringerOpen 1999-01-01
• Series: Journal of Inequalities and Applications
• Subjects: Maximum principles for quasilinear parabolic equations
• Online Access: http://www.journalofinequalitiesandapplications.com/content/3/152404

<p/> <p>This paper deals with classical solutions <inline-formula><graphic file="1029-242X-1999-152404-i1.gif"/></inline-formula> of some initial boundary value problems involving the quasilinear parabolic equation <inline-formula><graphic file="1029-242X-1999-152404-i2.gif"/></inline-formula> where <inline-formula><graphic file="1029-242X-1999-152404-i3.gif"/></inline-formula> are given functions. In the case of one space variable, i.e. when <inline-formula><graphic file="1029-242X-1999-152404-i4.gif"/></inline-formula>, we establish a maximum principle for the auxiliary function <inline-formula><graphic file="1029-242X-1999-152404-i5.gif"/></inline-formula> where a is an arbitrary nonnegative parameter. In some cases this maximum principle may be used to derive explicit exponential decay bounds for <inline-formula><graphic file="1029-242X-1999-152404-i6.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-1999-152404-i7.gif"/></inline-formula>. Some extensions in <inline-formula><graphic file="1029-242X-1999-152404-i8.gif"/></inline-formula> space dimensions are indicated. This work may be considered as a continuation of previous works by Payne and Philippin (<it>Mathematical Models and Methods in Applied Sciences</it>, 5 (1995), 95&#8211;110; Decay bounds in quasilinear parabolic problems, In: <it>Nonlinear Problems in Applied Mathematics</it>, Ed. by T.S. Angell, L. Pamela, Cook, R.E., SIAM, 1997).</p>