Blow-up time estimate for porous-medium problems with gradient terms under Robin boundary conditions

Abstract

AbstractThis paper deals with the blow-up phenomena connected to the following porous-medium problem with gradient terms under Robin boundary conditions: $$ \textstyle\begin{cases} u_{t}=\Delta u^{m}+k_{1}u^{p}-k_{2} \vert \nabla u \vert ^{q} & \text{in } \Omega\times(0,t^{*}), \\ \frac{\partial u}{\partial\nu}+\gamma u=0 &\text{on } \partial\Omega\times(0,t^{*}), \\ u(x,0)=u_{0}(x)\geq0 &\text{in } \overline{\Omega}, \end{cases} $$ { u t = Δ u m + k 1 u p − k 2 | ∇ u | q in  Ω × ( 0 , t ∗ ) , ∂ u ∂ ν + γ u = 0 on  ∂ Ω × ( 0 , t ∗ ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 in  Ω ‾ , where $\Omega\subset\mathbb{R}^{n}$ Ω ⊂ R n ($n\geq3$ n ≥ 3 ) is a bounded and convex domain with smooth boundary ∂Ω. The constants p, q, m are positive, and $p>q>m>1$ p > q > m > 1 , $q>2$ q > 2 . By making use of the Sobolev inequality and the differential inequality technique, we obtain a lower bound for the blow-up time of the solution. In addition, an example is given as an application of the abstract results obtained in this paper. Our results can be regarded as an answer to the open question raised by Li et al. in (Z. Angew. Math. Phys. 70:1–18, 2019).