The Cauchy problem for the nonlinear filtration equation in an inhomogeneous medium (Q919531)

The author considers the Cauchy problem \[ (1)\quad \rho (x)u_ t=\Delta G(u),\quad (x,t)\in {\mathbb{R}}^ N\times \{t>0\},\quad N\geq 3,\quad u|_{t=0}=\phi (x) \] with a density \(\rho\) (x) which is rapidly decreasing at infinity, i.e. \(0<\rho (x)\leq c_ 1(1+| x|)^{- \alpha}\), \(\alpha >N\), where \(0\leq \phi (x)\leq c_ 2\), \(x\in {\mathbb{R}}^ N\) and \(G(u)\in C'[0,\infty)\) satisfies \(G(0)=0\), \(G'(0)\geq 0\), \(G'(u)>0\) for all \(u>0\). He defines weak solutions of problem (1) in a function space where \[ 0\leq u(x,t)\leq c_ 3,\quad (x,t)\in {\mathbb{R}}^ N\times \{t>0\},\quad \int^{T}_{0}G(u)dt\leq c(T)(1+| x|)^{-(N- 2)},\quad x\in {\mathbb{R}}^ N,\quad T>0 \] and proves that there exists one and only one weak solution. Furthermore he studies the behaviours of solutions as \(t\to \infty\).