On a Higher-order Reaction-diffusion Equation with a Special Medium Void via Potential Well Method

Let d ∈ {1, 2, 3, . . .} and Ω ⊂ ℝ d be open bounded with Lipschitz boundary. Consider the reaction-diffusion parabolic problem u t | x | 4 + Δ 2 u = k ( t ) | u | p - 1 u , ( x , t ) ∈ Ω × ( 0 , T ) , u ( x , t ) = ∂ u ∂ ν ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω , where T > 0, p ∈ (1,∞), 0 ≠ u 0 ∈ H 0 2 (Ω) and ν is the outward normal vector to ∂Ω. We investigate the existence of a global weak solution to the problem together with the decaying and blow-up properties using the potential well method.