We analyze long-time behavior of solutions to a class of problems related to very fast and singular diffusion porous medium equations having non-homogeneous in space and time source terms with zero mean. In dimensions two and three, we determine critical values of porous medium exponent for the asymptotic H 1 -convergence of the solutions to a unique non-homogeneous positive steady state generally to hold.

Abstract The group of combinatorial self-similarities of a pseudometric space ( X , d ) \left(X,d) is the maximal subgroup of the symmetric group Sym ( X ) {\rm{Sym}}\left(X) whose elements preserve the four-point equality d ( x , y ) = d ( u , v ) d\left(x,y)=d\left(u,v) . Let us denote by ℐP {\mathcal{ {\mathcal I} P}} the class of all pseudometric spaces ( X , d ) \left(X,d) for which every combinatorial self-similarity Φ : X → X \Phi :X\to X satisfies the equality d ( x , Φ ( x ) ) = 0 , d\left(x,\Phi \left(x))=0, but all permutations of metric reflection of ( X , d ) \left(X,d) are combinatorial self-similarities of this reflection. The structure of ℐP {\mathcal{ {\mathcal I} P}} -spaces is fully described.

Titanium drill pipe have promising application prospects in ultra-deep drilling, but the nonlinear dynamic characteristics of composite drill strings are very complex. It is very important to use titanium drill pipe safely, economically, and effectively in the drilling process. In this paper, different schemes of the titanium/steel drill pipe composite drill string was designed, and the statics and dynamics characteristics of these composite drill strings were analyzed. The design scheme of titanium/steel composite drill strings were optimized by using whirling characteristics, dynamic stress and vibration acceleration. The results show that the titanium drill pipe can effectively improve the tensile allowance of the wellhead and reduce the whirling velocity, dynamic stress and vibration acceleration of the drill string. By comparing and analyzing the dynamic characteristics of 3 kinds of composite drill strings, the use of titanium drill pipe for the lower 2000 m of the 4′′ drill pipe have the smaller dynamic stress and lateral vibration, and less collision with the wellbore. Based on the analysis of vibration and dynamic characteristics, an optimal scheme of drilling operation parameters for the titanium/steel composite drill string was formed finally, which can provide guidance for the use of the drill string composed of titanium/steel drill pipe in natural gas drilling of ultra-deep well.

We prove continuity and Harnack’s inequality for bounded solutions to the equation div(∣∇u∣p(x)-2∇u)=0,p(x)=p¯+Lloglog1∣x-x0∣log1∣x-x0∣,p¯>1,L>0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{div}\\big (\\mid \ abla u\\mid ^{p(x)-2}\\,\ abla u \\big )&=0, \\quad p(x)= {\\bar{p}} + L\\frac{\\log \\log \\frac{1}{\\mid x-x_{0}\\mid }}{\\log \\frac{1}{\\mid x-x_{0}\\mid }}, \\\\ {\\bar{p}}&>1, \\quad L>0, \\end{aligned}$$\\end{document}under the precise non-logarithmic condition on the function p(x).