Abstract and Applied Analysis

Global and Blow-Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition

Lingling Zhang and Hui Wang

Full-text: Open access

Abstract

We discuss the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions: ( b ( u ) ) t = · ( h ( t ) k ( x ) a ( u ) u ) + f ( x , u , | u | 2 , t ) , in D × ( 0 , T ) , ( u / n ) + γ u = 0 , on D × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) > 0 , in D ¯ , where D R N   ( N 2 ) is a bounded domain with smooth boundary D . Under some appropriate assumption on the functions f , h , k , b , and a and initial value u 0 , we obtain the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for “blow-up time,” and an upper estimate of “blow-up rate.” Our approach depends heavily on the maximum principles.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 241650, 9 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425048222

Digital Object Identifier
doi:10.1155/2014/241650

Mathematical Reviews number (MathSciNet)
MR3266300

Zentralblatt MATH identifier
07021982

Citation

Zhang, Lingling; Wang, Hui. Global and Blow-Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 241650, 9 pages. doi:10.1155/2014/241650. https://projecteuclid.org/euclid.aaa/1425048222


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