arXiv subjects

Matteo Muratori

Publications and source records attributed to Matteo Muratori.

39 records · Page 3Linked to original sources

arXiv2013

Radial Fast Diffusion on the Hyperbolic Space

We consider radial solutions to the fast diffusion equation $u_t=Δu^m$ on the hyperbolic space $\mathbb{H}^{N}$ for $N \ge 2$, $m\in(m_s,1)$, $m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the geodesic distance $r$ from a given point $o \in \mathbb{H}^N$. We investigate their fine asymptotics near the extinction time $T$ in terms of a separable solution of the form ${\mathcal V}(r,t)=(1-t/T)^{1/(1-m)}V^{1/m}(r)$, where $V$ is the unique positive energy solution, radial w.r.t. $o$, to $-ΔV=c\,V^{1/m}$ for a suitable $c>0$, a semilinear elliptic problem thoroughly studied in \cite{MS08}, \cite{BGGV}. We show that $u$ converges to ${\mathcal V}$ in relative error, in the sense that $\|{u^m(\cdot,t)}/{{\mathcal V}^m(\cdot,t)}-1\|_\infty\to0$ as $t\to T^-$. In particular the solution is bounded above and below, near the extinction time $T$, by multiples of $(1-t/T)^{1/(1-m)}e^{-(N-1)r/m}$.

math.AP↗

arXiv2012

Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincaré inequalities

We study weighted porous media equations on domains $Ω\subseteq{\mathbb R}^N$, either with Dirichlet or with Neumann homogeneous boundary conditions when $Ω\not={\mathbb R}^N$. Existence of weak solutions and uniqueness in a suitable class is studied in detail. Moreover, $L^{q_0}$-$L^\varrho$ smoothing effects ($1\leq q_0<\varrho<\infty$) are discussed for short time, in connection with the validity of a Poincaré inequality in appropriate weighted Sobolev spaces, and the long-time asymptotic behaviour is also studied. Particular emphasis is given to the Neumann problem, which is much less studied in the literature, as well as to the case $Ω={\mathbb R}^N$ when the corresponding weight makes its measure finite, so that solutions converge to their weighted average instead than to zero. Examples are given in terms of wide classes of weights.

math.AP↗

arXiv2012

Sharp short and long time $\mathbf L^{\boldsymbol \infty}$ bounds for solutions to porous media equations with Neumann boundary conditions

We study a class of nonlinear diffusion equations whose model is the classical porous media equation on domains $Ω\subseteq{\mathbb R}^N$, $N\ge3$, with homogeneous Neumann boundary conditions. Firstly we improve some known results in such model case, both as concerns sharp $L^{q_0}$-$L^\infty$ regularizing properties of the evolution for short time and as concerns sharp long time asymptotics in the sense of $L^\infty$ convergence of solutions to their mean value. The generality of the discussion allows to consider, almost at the same time, also weighted versions of the above equation provided an appropriate weighted Sobolev inequality is required to hold. \normalcolor In fact, we show that the validity of a slightly weaker functional inequality is equivalent to the validity of a suitable $L^{q_0}$-$L^\infty$ bound for solutions to the associated weighted porous media equation. The long time asymptotic analysis relies as well on the assumed weighted Sobolev inequality only, and allows to prove uniform convergence to the mean value, with the rate predicted by linearization, in such generality. This fact was not known even for the explicit classes of weights previously considered in the literature.

math.AP↗