Abstract
The type of problem under consideration is u t = ∇ ⋅ ( a ( u ) b ( x ) ∇ u ) + g ( x , t ) f ( u ) , i n D × ( 0 , T ) , ∂ u ∂ n + σ ( x , t ) u = 0 , o n ∂ D × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) > 0 , i n D _ where D is a smooth bounded domain of R N . By constructing an auxiliary function and using Hopf's maximum principles on it, existence theorems of blow-up solutions, upper bound of ‘blow-up time”, upper estimates of “blow-up rate”, existence theorems of global solutions and upper estimates of global solutions are given under suitable assumptions on a, b, f, g, σ, and initial data u 0( x). The obtained results are applied to some examples in which a, b, f, g, and σ are power functions or exponential functions.
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