In this paper, the finite time extinction of solutions to the fast diffusion system $u_t=mathrm{div}(| abla u|^{p-2} abla u)+v^m$, $v_t=mathrm{div}(| abla v|^{q-2} abla v)+u^n$ is investigated, where $1<p,q<2$, $m,n>0$ and $Omegasubset mathbb{R}^N (N
geq1)$ is a bounded smooth domain. After establishing the local existence of weak solutions, the authors show that if $mn>(p-1)(q-1)$, then any solution vanishes in finite time provided that the initial data are ``comparable; if $mn=(p-1)(q-1)$ and $Omega$ is suitably small, then the existence of extinction solutions for small initial data is proved by using the De Giorgi iteration process and comparison method. On the other hand, for $1<p=q<2$ and $mn<(p-1)^2$, the existence of at least one non-extinction solution for any positive smooth initial data is proved.
The authors of this paper deal with the existence and regularities of weak solutions to the homogenous $hbox{Dirichlet}$ boundary value problem for the equation $-hbox{div}(| abla u|^{p-2} abla u)+|u|^{p-2}u=frac{f(x)}{u^{alpha}}$. The authors apply
the method of regularization and $hbox{Leray-Schauder}$ fixed point theorem as well as a necessary compactness argument to prove the existence of solutions and then obtain some maximum norm estimates by constructing three suitable iterative sequences. Furthermore, we find that the critical exponent of $m$ in $|f|_{L^{m}(Omega)}$. That is, when $m$ lies in different intervals, the solutions of the problem mentioned belongs to different $hbox{Sobolev}$ spaces. Besides, we prove that the solution of this problem is not in $W^{1,p}_{0}(Omega)$ when $alpha>2$, while the solution of this problem is in $W^{1,p}_{0}(Omega)$ when $1<alpha<2$.
The authors of this paper study singular phenomena(vanishing and blowing-up in finite time) of solutions to the homogeneous $hbox{Dirichlet}$ boundary value problem of nonlinear diffusion equations involving $p(x)$-hbox{Laplacian} operator and a nonl
inear source. The authors discuss how the value of the variable exponent $p(x)$ and initial energy(data) affect the properties of solutions. At the same time, we obtain the critical extinction and blow-up exponents of solutions.
