Blow-up phenomena for a class of nonlinear reaction-diffusion equations under nonlinear boundary conditions. Issue 16 (10th December 2019) - Zhang, Ling-Ling; Tian, Hui-Min
ABSTRACT: The paper considers the blow-up solution for the following nonlinear reaction-diffusion equations under nonlinear boundary conditions: ( h ( u ) ) t = ∇ ⋅ ( ρ ( | ∇ u | p ) | ∇ u | p − 2 ∇ u ) + b ( x ) k ( t ) f ( u ), ( x, t ) ∈ Ω × ( 0, T ), ρ ( | ∇ u | p ) | ∇ u | p − 2 ∂ u ∂ n = g ( u ), ( x, t ) ∈ ∂Ω × ( 0, T ), u ( x, 0 ) = u 0 ( x ) ≥ 0, x ∈ Ω ¯, where p >2 and Ω ⊂ R N ( N > 2 ) is a bounded convex domain with smooth boundary ∂Ω . By means of constructing auxiliary functions and a first-order differential inequality technique, we obtain the upper and lower bounds on blow-up time under some appropriate assumptions. Moreover, some examples are also presented as application of our main results.