Blow-up estimates for a higher-order reaction–diffusion equation with a special diffusion process

Let d∈{1,2,3,…}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \in \{1,2,3,\ldots \}$$\end{document} and Ω⊂Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^d$$\end{document} be open bounded with Lipschitz boundary. Consider the reaction–diffusion parabolic problem (P)ut|x|4+Δ2u=k(t)|u|p-1uinΩ×(0,T),u(x,t)=∂u∂ν(x,t)=0if(x,t)∈∂Ω×(0,T),u(x,0)=u0(x)ifx∈Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (P) \quad \left\{ \begin{array}{ll} \displaystyle \frac{u_t}{|x|^4} + \Delta ^2 u = k(t) \, |u|^{p-1}u &{} \text{ in } \Omega \times (0,T), \\ u(x,t) = \displaystyle \frac{\partial u}{\partial \nu }(x,t) = 0 &{} \text{ if } (x,t) \in \partial \Omega \times (0,T), \\ u(x,0) = u_0(x) &{} \text{ if } x \in \Omega , \end{array}\right. \end{aligned}$$\end{document}where T>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T > 0$$\end{document}, p∈(1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in (1,\infty )$$\end{document} and 0≠u0∈H02(Ω)∩Lp+1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \ne u_0 \in H^2_0(\Omega ) \cap L^{p+1}(\Omega)$$\end{document}. We investigate the upper and lower bounds on the blow-up time of a weak solution to (P).