Blow-up analysis for parabolic p-Laplacian equations with a gradient source term

Abstract

Abstract In this work, we deal with the blow-up solutions of the following parabolic p-Laplacian equations with a gradient source term: { ( b ( u ) ) t = ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) + f ( x , u , | ∇ u | 2 , t ) in Ω × ( 0 , t ∗ ) , ∂ u ∂ n = 0 on ∂ Ω × ( 0 , t ∗ ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 in Ω ‾ , $$ \textstyle\begin{cases} (b(u) )_{t} =\nabla \cdot ( \vert \nabla u \vert ^{p-2}\nabla u )+f(x,u, \vert \nabla u \vert ^{2},t) &\text{in } \varOmega \times (0,t^{*}), \\ \frac{\partial u}{\partial n}=0 &\text{on } \partial \varOmega \times (0,t^{*}),\\ u(x,0)=u_{0}(x)\geq 0 & \text{in } \overline{\varOmega }, \end{cases} $$ where p > 2 $p>2$ , the spatial domain Ω ⊂ R N $\varOmega \subset \mathbb{R}^{N}$ ( N ≥ 2 $N\geq 2$ ) is bounded, and the boundary ∂Ω is smooth. Our research relies on the creation of some suitable auxiliary functions and the use of the differential inequality techniques and parabolic maximum principles. We give sufficient conditions to ensure that the solution blows up at a finite time t ∗ $t^{*}$ . The upper bounds of the blow-up time t ∗ $t^{*}$ and the upper estimates of the blow-up rate are also obtained.

Keywords

• Blow-up solution
• Parabolic p-Laplacian equation
• Gradient source term