Existence and non-existence of a fast diffusion equation in \(\mathbb{R}^ n\)
For the Cauchy problem to a fast diffusion equation of the form \(u_t = \Delta \log u, x\in \mathbb{R}^n, t>0\) the author studies the global existence, finite time extinction, and asymptotic behaviour of solutions. The main results are summarized as follows: Let \(n\geq 3\) then (a) there exists a non-trivial global solution \(u\) of the Cauchy problem which has the property that \(u(\cdot,t) \in L^p(\mathbb{R}^n),\) for any \(p>n/2\) and for all \(t>0\) but it does not belong to any \(L^p(\mathbb{R}^n)\) if \(p\leq n/2;\) (b) for any \(T>0,\) there exists a solution \(u(\cdot,t)\) which has a finite maximal existence time \(T\) and such that for any \(p>n/2, u(\cdot,t)\in L^p(\mathbb{R}^n), 0<t<T\); moreover, \(u\) becomes extinct at \(t=T,\) i.e., \(u(\cdot,T)\equiv 0;\) (c) any non-trivial solution \(u\) satisfies \(\int_{R^n}u^p(x,t)dx = \infty, 0<t<t_{\max},\) where \(1<p\leq n/2,\) and \(t_{\max}>0\) is the maximal existence time.
scientific article; zbMATH DE number 1025925
scientific article