Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights

Abstract

We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the fast diffusion equation with weights (WFDE) $u_t=|x{|}^{\gamma }\mathrm{div}(|x{|}^{-\beta }\nabla u^m)$ posed on $(0,+\infty )\times {\mathbb{R}}^d$, with $d\ge 3$, in the so-called good fast diffusion range $m_c<m<1$, within the range of parameters $\gamma ,\beta$ which is optimal for the validity of the so-called Caffarelli–Kohn–Nirenberg inequalities.

It is natural to ask in which sense such solutions behave like the Barenblatt $\mathfrak{B}$ (fundamental solution): for instance, asymptotic convergence, i.e. $\Vert u(t)-\mathfrak{B}(t){\Vert }_{{\mathrm{L}}^p({\mathbb{R}}^d)}\stackrel{t\to \infty }{\to }0$, is well known for all $1\le p\le \infty$, while only a few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data $\mathcal{X}\subset {\mathrm{L}}_{+}^1({\mathbb{R}}^d)$ that produces solutions which are pointwise trapped between two Barenblatt (global Harnack principle), and uniformly converge in relative error (UREC), i.e. ${\mathrm{d}}_{\infty }(u(t))=\Vert u(t)/\mathcal{B}(t)-1{\Vert }_{{\mathrm{L}}^{\infty }({\mathbb{R}}^d)}\stackrel{t\to \infty }{\to }0$. Such a characterization is in terms of an integral condition on $u(t=0)$.

To the best of our knowledge, analogous issues for the linear heat equation, $m=1$, do not possess such clear answers, but only partial results. Our characterization is also new for the classical, nonweighted FDE. We are able to provide minimal rates of convergence to $\mathcal{B}$ in different norms. Such rates are almost optimal in the nonweighted case, and become optimal for radial solutions. To complete the panorama, we show that solutions with data in ${\mathrm{L}}_{+}^1({\mathbb{R}}^d)\setminus \mathcal{X}$, preserve the same “fat” spatial tail for all times, hence UREC fails and ${\mathrm{d}}_{\infty }(u(t))=\infty$, even if $\Vert u(t)-\mathcal{B}(t){\Vert }_{{\mathrm{L}}^1({\mathbb{R}}^d)}\stackrel{t\to \infty }{\to }0$.