American Institute of Mathematical Sciences

$u_t = \Delta \Phi(u)$ in R$^n \times [0,T]$

$u(x)=u_0(x)$ on R$^n$

with the nonlinearity $\Phi(u)$. For the case of $\Phi(u)=\sum_{i=1}^m c_i u^{\alpha_i}$, we show the existence of a solution which smoothness depends on the exponents $\alpha_i$. Regardless of the regularity of the solution, we show the free-boundary is smooth. We also extend similar results for $\Phi(u)$ as an infinite sum.

$u_{t t}-\Delta u+\int^t_0g(t-s)\Delta u(s)ds + \alpha u_{t}=0$ in $\hat Q$

where $\hat Q$ is a non cylindrical domains of $\mathbb R^{n+1}$ $(n\ge1)$ with the lateral boundary $\hat{\sum}$ and $\alpha$ is a positive constant.

-div $( |x|^{-p\gamma}|\nabla u|^{p-2}\nabla u)=f(x, u)\in L^1(\Omega),\quad x\in \Omega$

$u(x)=0$ on $\partial \Omega,$

where $-\infty<\gamma<\frac{N-p}{p}$, $\Omega$ is a bounded domain in $\mathbb R^N$ such that $0\in\Omega$ and $f(x,u)$ is a Caratheodory function under suitable conditions that will be stated in each section.