Blow up for Porous Medium Equations
Authors : Burhan SELÇUK
Pages : 22-27
Doi:10.36753/mathenot.686065
View : 38 | Download : 8
Publication Date : 2021-03-01
Article Type : Research Paper
Abstract :In various branches of applied sciences, porous medium equations exist where this basic model occurs in a natural fashion. It has been used to model fluid flow, chemical reactions, diffusion or heat transfer, population dynamics, etc.. Nonlinear diffusion equations involving the porous medium equations have also been extensively studied. However, there has not been much research effort in the parabolic problem for porous medium equations with two nonlinear boundary sources in the literature. This paper adresses the following porous medium equations with nonlinear boundary conditions. Firstly, we obtain finite time blow up on the boundary by using the maximum principle and blow up criteria and existence criteria by using steady state of the equation $k_{t}=k_{xx}^{n}insert ignore into journalissuearticles values(x,t);\in insert ignore into journalissuearticles values(0,L);\times insert ignore into journalissuearticles values(0,T);\ $with $ k_{x}^{n}insert ignore into journalissuearticles values(0,t);=k^{\alpha }insert ignore into journalissuearticles values(0,t);$, $k_{x}^{n}insert ignore into journalissuearticles values(L,t);=k^{\beta }insert ignore into journalissuearticles values(L,t);$,$\ t\in insert ignore into journalissuearticles values(0,T);\ $and initial function $k\leftinsert ignore into journalissuearticles values( x,0\right); =k_{0}\leftinsert ignore into journalissuearticles values( x\right); $,$\ x\in \lbrack 0,L]\ $where $n>1$, $\alpha \ $and $\beta \ $and positive constants.Keywords : Heat equation, Nonlinear parabolic equation, nonlinear boundary condition, blow up, maximum principles
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