A nonlinear diffusion problem with convection and anisotropic nonstandard growth conditions. (August 2019) - Liu, Bingchen; Dong, Mengzhen
Abstract: This paper deals with the homogeneous Dirichlet problem of a nonlinear diffusion equation involving anisotropic variable exponents and convection. Firstly, we prove the existence of weak solutions and obtain suitable energy estimate of solutions in anisotropic Orlicz–Sobolev spaces. Secondly, we apply the energy functional method and the convexity method to show blow-up criteria of solutions. Thirdly, the extinction or non-extinction of solutions is studied by using energy inequalities and comparison principle of ordinary differential equations. Fourthly, we show some result on global solutions without assumption on initial data. Moreover, some asymptotic estimate of blow-up and extinction solutions are determined.