Asymptotic behaviour for the porous medium equation posed in the whole space. (Q1565970)

The paper investigates the asymptotic behavior as \(t\to \infty\) of nonnegative solutions \(u(x,t)\) of \(n\)-dimensional Cauchy problems for the equation \(u_t= \Delta(u^m)\), \(m>1\), subject mostly to nonnegative integrable data. An existence and uniqueness theory exists so that for given initial data, there corresponds an orbit \(\{u(\cdot,t): t>0\}\) in \(L^1(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)\). The solution is not classical for \(m>1\), but it is proved that there exists a unique weak solution for all \(m>0\). It is also unique in the sense of a mild solution, obtained as the limit of the implicit time-discretization scheme. Properties of the solution are discussed. The paper also extends the asymptotic study to cover (i) signed solutions for \(u_t= \Delta (| u|^{m-1}u)\), \(m>1\), (ii) \(u_t= \Delta(| u|^{m-1}u)+ f\), \(m>1\) and \(f\in L^1 (\mathbb{R}^n\times \mathbb{R}^+)\), and (iii) \(m<1\).