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Global and blow‐up analysis for a class of nonlinear reaction diffusion model with Dirichlet boundary conditions
Author(s) -
Zhang Lingling,
Wang Hui,
Wang Xiaoqiang
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5241
Subject(s) - mathematics , bounded function , dirichlet boundary condition , reaction–diffusion system , dirichlet distribution , domain (mathematical analysis) , nonlinear system , mathematical analysis , boundary value problem , upper and lower bounds , boundary (topology) , work (physics) , class (philosophy) , partial differential equation , diffusion , mechanical engineering , physics , quantum mechanics , artificial intelligence , computer science , engineering , thermodynamics
The work is concerned with the following nonlinear reaction diffusion model with Dirichlet boundary conditions:( g ( u ) ) t = ∇ · ( ρ ( | ∇ u | p ) | ∇ u | p − 2 ∇ u ) + h ( x ) k ( t ) f ( u ) , in D × ( 0 , t ∗ ) ,u ( x , t ) = 0 , on ∂ D × ( 0 , t ∗ ) ,u ( x , 0 ) = u 0 ( x ) ≥ 0 , inD ¯ ,where p ≥ 2 is a real number and D ⊂ R N ( N ≥ 2 ) is a bounded domain with smooth boundary ∂ D . Under some appropriate assumptions on the functions f , h , k , g , ρ , and initial value u 0 , by defining auxiliary functions and using a first‐order differential inequality technique, we not only present that the solution exists globally or blows up in a finite time but also compute the upper and lower bound for blow‐up time when blow‐up occurs. Additionally, two examples are given to illustrate the main results.