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

Journal of Inequalities and Applications

Xuhui Shen¹ 

[1] School of Applied Mathematics, Shanxi University of Finance and Economics;

关键词: Porous-medium problem;    Blow-up;    Gradient terms;    Lower bound;   

DOI  :  10.1186/s13660-022-02779-z

来源: DOAJ

【 摘 要 】

Abstract This paper deals with the blow-up phenomena connected to the following porous-medium problem with gradient terms under Robin boundary conditions: { 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 ‾ , $$ \textstyle\begin{cases} u_{t}=\Delta u^{m}+k_{1}u^{p}-k_{2} \vert \nabla u \vert ^{q} & \text{in } \Omega\times(0,t^{*}), \\ \frac{\partial u}{\partial\nu}+\gamma u=0 &\text{on } \partial\Omega\times(0,t^{*}), \\ u(x,0)=u_{0}(x)\geq0 &\text{in } \overline{\Omega}, \end{cases} $$ where Ω ⊂ R n $\Omega\subset\mathbb{R}^{n}$ ( n ≥ 3 $n\geq3$ ) 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).

【 授权许可】

Unknown