Extended solutions for general fast diffusion equations with optimal measure data

Abstract

We study the theory of existence and uniqueness of nonlinear diffusion equations of the form \[ u_{t}=\Delta \varphi(u)\quad\text{in}\quad\mathbb{R}^{N} \times(0,\infty), \] where $\varphi:{{\mathbb{R}^N}}_+\to{{\mathbb{R}^N}}_+$ is a continuous and increasing function. We focus on the fast diffusion type by imposing the growth condition : $$ \qquad {\displaystyle} m_1\le \dfrac{s\varphi'(s)}{\varphi(s)} \le m_2, $$ for some constants $0 < m_1\le m_2 < 1$ and all $s>0$. Moreover, $m_1>(N-2)/N$. Existence is obtained for an optimal class of initial data, namely, for any nonnegative Borel measure (not necessarily a locally finite measure). Uniqueness is proven for concave $\varphi$'s. The results extend to the general equation the optimal theory with measure-valued data now available for the equation with power functions $\varphi(s)=s^{m}$. The asymptotic behaviour is also studied.