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

摘要

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?Rddocumentclass[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>0documentclass[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).