On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow (which is only possible in the presence of non-integrable zero modes). In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes up to at least twice the gap; this refines and confirms a 1980 conjecture of Berryman and Holland. We also improve on a result of Bonforte and Figalli, by providing a new and simpler approach which is able to accommodate the presence of zero modes, such as those that occur when the vanishing profile fails to be isolated (and possibly belongs to a continuum of such profiles).
We study a Sobolev critical fast diffusion equation in bounded domains with the Brezis-Nirenberg effect. We obtain extinction profiles of its positive solutions, and show that the convergence rates of the relative error in regular norms are at least polynomial. Exponential decay rates are proved for generic domains. Our proof makes use of its regularity estimates, a curvature type evolution equation, as well as blow up analysis. Results for Sobolev subcritical fast diffusion equations are also obtained.
We study extinction profiles of solutions to fast diffusion equations with some initial data in the Marcinkiewicz space. The extinction profiles will be the singular solutions of their stationary equations.
We prove the \textit{finite time extinction property} $(u(t)\equiv 0$ on $Ω$ for any $t\ge T_\star,$ for some $T_\star>0)$ for solutions of the nonlinear Schrödinger problem ${\rm i} u_t+Δu+a|u|^{-(1-m)}u=f(t,x),$ on a bounded domain $Ω$ of $\mathbb{R}^N,$ $N\le 3,$ $a\in\mathbb{C}$ with $\Im(a)>0$ (the damping case) and under the crucial assumptions $0<m<1$ and the dominating condition $2\sqrt m\,\Im(a)\ge(1-m)|\Re(a)|.$ We use an energy method as well as several a priori estimates to prove the main conclusion. The presence of the non-Lipschitz nonlinear term in the equation introduces a lack…
We establish both extinction and non-extinction self-similar profiles for the following fast diffusion equation with a weighted source term $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$, $N\geq3$, in the sub-critical range of the fast diffusion equation $0<m<m_c=(N-2)/N$. We consider $σ>0$ and $\max\{p_c(σ),1\}<p<p_L(σ)$, where $$ p_c(σ)=\frac{m(N+σ)}{N-2}, \qquad p_L(σ)=1+\frac{σ(1-m)}{2}. $$ We show that, on the one hand, positive self-similar solutions at any time $t>0$, in the form $$ u(x,t)=t^αf(|x|t^β), \qquad f(ξ)\sim Cξ^{-(N-2)/m}, \qquad α>0, \ β>0 $$ ex…
We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=Δu^m$, posed in a smooth bounded domain $Ω\subset \mathbb{R}^N$, in the exponent range $m_s=(N-2)_+/(N+2)<m<1$. It is known that bounded positive solutions extinguish in a finite time $T>0$, and also that they approach a separate variable solution $u(t,x)\sim (T-t)^{1/(1-m)}S(x)$, as $t\to T^-$. It has been shown recently that $v(x,t)=u(t,x)\,(T-t)^{-1/(1-m)}$ tends to $S(x)$ as $t\to T^-$, uniformly in the relative error norm. Starting from this result, we investigate the fine asymptotic behaviour and prove …
For a smooth bounded domain $Ω\subseteq\mathbb{R}^n$, $n\geq 3$, we consider the fast diffusion equation with critical sobolev exponent $$\frac{\partial w}{\partialτ} =Δw^{\frac{n-2}{n+2}}$$ under Dirichlet boundary condition $w(\cdot, τ) = 0$ on $\partialΩ$. Using the parabolic gluing method, we prove existence of an initial data $w_0$ such that the corresponding solution has extinction rate of the form $$\|w(\cdot, τ)\|_{L^\infty(Ω)} = γ_0(T-τ)^{\frac{n+2}{4}}\left|\ln(T-τ)\right|^{\frac{n+2}{2(n-2)}}(1+o(1))$$ as $t\to T^-$, here $T > 0$ is the finite extinction time of $w(x, τ)$. This gene…
The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption $$\partial_t u-Δu^m+|x|^σu^q=0, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ with $m\geq1$, $q\in(0,1)$ and $σ>0$, is addressed. Introducing the critical exponent $σ^* := 2(1-q)/(m-1)$ for $m>1$ and $σ_*=\infty$ for $m=1$, extinction in finite time is known to take place for $σ\in [0,σ^*)$ and an alternative proof is provided therein. When $m>1$ and $σ\ge σ^*$, the occurrence of finite time extinction is proved for a specific class of initial conditions, thereby sup…
This paper discusses finite time extinction for a perturbed fast diffusion equation with dynamic boundary conditions. The fast diffusion equation has the characteristic property of decay, such as the solution decays to zero in a finite amount of time depending upon the initial data. In the target problem, some $p$-th or $q$-th order perturbation term may work to blow up within this period. The problem arises from the conflict between the diffusion and the blow up, in the bulk and on the boundary. Firstly, the local existence and uniqueness of the solution are obtained. Finally, a result of fin…
We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity.
We study the homogeneous Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $\partial_t u =-\mathcal{L} u^m$ posed on a bounded Euclidean domain $Ω\subset\mathbb{R}^N$ with smooth boundary and $N\ge 1$. The linear diffusion operator $\mathcal{L}$ is a sub-Markovian operator, allowed to be of nonlocal type, while the nonlinearity is of singular type, namely $u^m=|u|^{m-1}u$ with $0<m<1$. The prototype equation is the Fractional Fast Diffusion Equation (FFDE), when $\mathcal{L}$ is one of the three possible Dirichlet Fractional Laplacians …
Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation with a Critical Exponent
研究了具有临界指数的快速扩散方程的柯西问题正解的渐近性态。经过适当的重新标度，产生一个非线性福克-普朗克方程，我们发现一个连续的代数收敛率的自相似的配置文件。这些速率明确地依赖于初始数据的空间衰减速率。这改进了以前关于临界快扩散方程慢收敛的结果（{ \sc Bonforte et al}. in Arch Rat Mech Anal 196：631- 680，2010），并提供了一些开放问题的答案。
We prove global Hölder gradient estimates for bounded positive weak solutions of fast diffusion equations in smooth bounded domains with the homogeneous Dirichlet boundary condition, which then lead us to establish their optimal global regularity. This solves a problem raised by Berryman and Holland in 1980.
We establish existence, uniqueness as well as quantitative estimates for solutions to the fractional nonlinear diffusion equation, $\partial_t u +{\mathcal L}_{s,p} (u)=0$, where ${\mathcal L}_{s,p}=(-Δ)_p^s$ is the standard fractional $p$-Laplacian operator. We work in the range of exponents $0<s<1$ and $1<p<2$, and in some sections $sp<1$. The equation is posed in the whole space $x\in {\mathbb R}^N$. We first obtain weighted global integral estimates that allow establishing the existence of solutions for a class of large data that is proved to be roughly optimal. We study the class of self-…
We prove existence and uniqueness of the branch of the so-called \emph{anomalous eternal solutions} in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed in $\real^N$ with $N\geq3$, where $$ 0<m<m_c=\frac{N-2}{N}, \qquad p>1, $$ and the critical value for the weight $$ σ=\frac{2(p-1)}{1-m}. $$ The branch of exponential self-similar solutions behaves similarly as the well-established anomalous solutions to the pure fast diffusion equation, but without a finite time extinction or a finite time blow-up, an…
Branching explosion of Brownian Fleming-Viot systems
School of Mathematical Sciences, South China Normal University（华南师范大学数学科学学院）；Information and Artificial Intelligence, Anhui Agricultural University（安徽农业大学信息与人工智能学院）
