arXiv subjects

Giulia Meglioli

Publications and source records attributed to Giulia Meglioli.

22 records · Page 2Linked to original sources

arXiv2020

Global existence of solutions and smoothing effects for classes of reaction-diffusion equations on manifolds

We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on $p$ and $m$ in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincaré inequality hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in ${\mathbb R}^n$.

math.AP↗

arXiv2020

Blow-up and global existence for solutions to the porous medium equation with reaction and fast decaying density

We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density $ρ(x)$ and a power-like reaction term $u^p$ with $p>1$. The density decays {\it fast} at infinity, in the sense that $ρ(x)\sim |x|^{-q}$ as $|x|\to +\infty$ with $q\ge 2.$ In the case when $q=2$, if $p$ is bigger than $m$, we show that, for large enough initial data, solutions blow-up in finite time and for small initial datum, solutions globally exist. On the other hand, in the case when $q>2$, we show that existence of global in time solutions always prevails. The case of {\it slowly} decaying density at infinity, i.e. $q\in [0,2)$, is examined in [41].

math.AP↗

arXiv2020

Blow-up and global existence for the inhomogeneous porous medium equation with reaction

We study finite time blow-up and global existence of solutions to the Cauchy problem for the porous medium equation with a variable density $ρ(x)$ and a power-like reaction term. We show that for small enough initial data, if $ρ(x)\sim \frac{1}{\left(\log|x|\right)^α|x|^{2}}$ as $|x|\to \infty$, then solutions globally exist for any $p>1$. On the other hand, when $ρ(x)\sim\frac{\left(\log|x|\right)^α}{|x|^{2}}$ as $|x|\to \infty$, if the initial datum is small enough then one has global existence of the solution for any $p>m$, while if the initial datum is large enough then the blow-up of the solutions occurs for any $p>m$. Such results generalize those established in [27] and [28], where it is supposed that $ρ(x)\sim |x|^{-q}$ for $q>0$ as $|x|\to \infty$.

math.AP↗

arXiv2020

Blow-up and global existence for solutions to the porous medium equation with reaction and slowly decaying density

We study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density $ρ(x)$ and a power-like reaction term $ρ(x) u^p$ with $p>1$; this is a mathematical model of a thermal evolution of a heated plasma (see [25]). The density decays slowly at infinity, in the sense that $ρ(x)\lesssim |x|^{-q}$ as $|x|\to +\infty$ with $q\in [0, 2).$ We show that for large enough initial data, solutions blow-up in finite time for any $p>1$. On the other hand, if the initial datum is small enough and $p>\bar p$, for a suitable $\bar p$ depending on $ρ, m, N$, then global solutions exist. In addition, if $p<\underline p$, for a suitable $\underline p\leq \bar p$ depending on $ρ, m, N$, then the solution blows-up in finite time for any nontrivial initial datum; we need the extra hypotehsis that $q\in [0, ε)$ for $ε>0$ small enough, when $m\leq p<\underline p$. Observe that $\underline p=\bar p$, if $ρ(x)$ is a multiple of $|x|^{-q}$ for $|x|$ large enough. Such results are in agreement with those established in [41], where $ρ(x)\equiv 1$. The case of fast decaying density at infinity, i.e. $q\geq 2$, is examined in [31].

math.AP↗