We study the extinction behavior of solutions to the fast diffusion equation $u_t = Δu^m$ on $\R^N\times (0,T)$, in the range of exponents $m \in (0, \frac{N-2}{N})$, $N > 2$. We show that if the initial data $u_0$ is trapped in between two Barenblatt solutions vanishing at time $T$, then the vanishing behaviour of $u$ at $T$ is given by a Barenblatt solution. We also give an example showing that for such a behavior the bound from above by a Barenblatt solution $B$ (vanishing at $T$) is crucial: we construct a class of solutions $u$ with initial data $u_0 = B (1 + o(1))$, near $ |x| >> 1$, which live longer than $B$ and change behaviour at $T$. The behavior of such solutions is governed by $B(\cdot,t)$ up to $T$, while for $t >T$ the solutions become integrable and exhibit a different vanishing profile. For the Yamabe flow ($m = \frac{N-2}{N+2}$) the above means that these solutions $u$ develop a singularity at time $T$, when the Barenblatt solution disappears, and at $t >T$ they immediately smoothen up and exhibit the vanishing profile of a sphere.
In the appendix we show how to remove the assumption on the bound on $u_0$ from below by a Barenblatt.
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.
Optimal extinction rates near the extinction time are derived for non-negative solutions to a fast diffusion equation with strong absorption, the power of the absorption exceeding that of the diffusion.
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 find a continuum of extinction rates for solutions $u(y,τ)\ge 0$ of the fast diffusion equation $u_τ=Δu^m$ in a subrange of exponents $m\in (0,1)$. The equation is posed in $\ren$ for times up to the extinction time $T>0$. The rates take the form $\|u(\cdot,τ)\|_\infty\sim (T-τ)^θ$ \ for a whole interval of $θ>0$. These extinction rates depend explicitly on the spatial decay rates of initial data.
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.
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…
In this paper, the finite time extinction of solutions to the fast diffusion system $u_t=\mathrm{div}(|\nabla u|^{p-2}\nabla u)+v^m$, $v_t=\mathrm{div}(|\nabla v|^{q-2}\nabla v)+u^n$ is investigated, where $1<p,q<2$, $m,n>0$ and $Ω\subset \mathbb{R}^N\ (N\geq1)$ is a bounded smooth domain. After establishing the local existence of weak solutions, the authors show that if $mn>(p-1)(q-1)$, then any solution vanishes in finite time provided that the initial data are ``comparable"; if $mn=(p-1)(q-1)$ and $Ω$ is suitably small, then the existence of extinction solutions for small initial data is pr…
A classification of the behavior of the solutions $f(\cdot,a)$ to the ordinary differential equation $(|f'|^{p-2} f')' + f - |f'|^{p-1} = 0$ in $(0,\infty)$ with initial condition $f(0,a)=a$ and $f'(0,a)=0$ is provided, according to the value of the parameter $a>0$, the exponent $p$ ranging in $(1,2)$. There is a threshold value $a_*$ which separates different behaviors of $f(\cdot,a)$: if $a>a_*$ then $f(\cdot,a)$ vanishes at least once in $(0,\infty)$ and takes negative values while $f(\cdot,a)$ is positive in $(0,\infty)$ and decays algebraically to zero as $r\to\infty$ if $a\in (0,a_*)$. A…
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…
In a recent work, Duong, Ghoul and Zaag determined the gradient profile for blowup solutions of standard semilinear heat equation with power nonlinearities in the (supposed to be) generic case. Their method refines the constructive techniques introduced by Bricmont and Kupiainen and further developed by Merle and Zaag. In this paper, we extend their refinement to the problem about the reconnection of vortex lines with the boundary in a type-II superconductor under planar approximation, a physical model derived by Chapman, Hunton and Ockendon featuring the finite time quenching for the nonlinea…
Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $ρ_1>0$, $β\ge\frac{mρ_1}{n-2-nm}$ and $α=\frac{2β+ρ_1}{1-m}$. For any $λ>0$, we will prove the existence and uniqueness (for $β\ge\frac{ρ_1}{n-2-nm}$) of radially symmetric singular solution $g_λ\in C^{\infty}(R^n\setminus\{0\})$ of the elliptic equation $Δv^m+αv+βx\cdot\nabla v=0$, $v>0$, in $R^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{α/β}g_λ(x)=λ^{-\frac{ρ_1}{(1-m)β}}$. When $β$ is sufficiently large, we prove the higher order asymptotic behaviour of radially symmetric solutions of the above elliptic equation as $|x|\to\infty$. We als…
Let $u$ be the solution of $u_t=Δ\log u$ in $\R^N\times (0,T)$, N=3 or $N\ge 5$, with initial value $u_0$ satisfying $B_{k_1}(x,0)\le u_0\le B_{k_2}(x,0)$ for some constants $k_1>k_2>0$ where $B_k(x,t) =2(N-2)(T-t)_+^{N/(N-2)}/(k+(T-t)_+^{2/(N-2)}|x|^2)$ is the Barenblatt solution for the equation. We prove that the rescaled function $\4{u}(x,s)=(T-t)^{-N/(N-2)}u(x/(T-t)^{-1/(N-2)},t)$, $s=-\log (T-t)$, converges uniformly on $\R^N$ to the rescaled Barenblatt solution $\4{B}_{k_0}(x)=2(N-2)/(k_0+|x|^2)$ for some $k_0>0$ as $s\to\infty$. We also obtain convergence of the rescaled solution $\4{u…
We consider the Fast Diffusion Equation $u_t=Δu^m$ posed in a bounded smooth domain $Ω\subset \RR^d$ with homogeneous Dirichlet conditions; the exponent range is $m_s=(d-2)_+/(d+2)<m<1$. It is known that bounded positive solutions $u(t,x)$ of such problem extinguish in a finite time $T$, and also that such solutions approach a separate variable solution $u(t,x)\sim (T-t)^{1/(1-m)}S(x)$, as $t\to T^-$. Here we are interested in describing the behaviour of the solutions near the extinction time. We first show that the convergence $u(t,x)\,(T-t)^{-1/(1-m)}$ to $S(x)$ takes place uniformly in the …
For a large class of non-negative initial data, the solutions to the quasilinear viscous Hamilton-Jacobi equation $\partial\_t u-Δ\_p u+|\nabla u|^q=0$ in $(0,\infty)\times\real^N$ are known to vanish identically after a finite time when $2N/(N+1) \textless{} p \leq 2$ and $q\in(0,p-1)$. Further properties of this extinction phenomenon are established herein: \emph{instantaneous shrinking} of the support is shown to take place if the initial condition $u\_0$ decays sufficiently rapidly as $|x|\to\infty$, that is, for each $t \textgreater{} 0$, the positivity set of $u(t)$ is a bounded subset o…
We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation near the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on the spatial decay rates of initial data.
The behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption $\partial_t u - Δ_p u + |\nabla u|^{p-1} = 0$ in $(0, \infty) \times \mathbb{R}^N$, and fast diffusion $2N/(N + 1) < p < 2$. Given a non-negative and radially symmetric initial condition with a non-increasing profile which decays sufficiently fast as $|x| \to \infty$, it is shown that the corresponding solution $u$ to the above equation approaches a uniquely determined separate variable solution of the form $U(t, x) = (T_e - t)^{1/(2-p)} f_*…
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 …
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 finite time extinction for stochastic sign fast diffusion equations driven by linear multiplicative space-time noise, corresponding to the Bak-Tang-Wiesenfeld model for self-organized criticality. This solves a problem posed and left open in several works: [Barbu, Da Prato, Röckner, CMP, 2009], [Barbu, Da Prato, Röckner, CRMAS, 2009], [Barbu, Röckner, CMP, 2012], [Barbu, Da Prato, Röckner, JMAA, 2012], [Röckner, Wang, JMLS, 2013], [Barbu, MMAS, 2013]. The highly singular-degenerate nature of the drift in interplay with the stochastic perturbation causes the need for new methods in the…
We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres $(\Sn, [g_{\Sn}])$, it converges to the standard sphere up to a Möbius diffeomorphism. This result allows us to obtain extinction profiles of solutions of some fractional porous medium equations. In the end, we use this fractional fast diffusion equation, together with its extinction profile and some estimates of its extinction time, to improve a Sobolev inequality via a quantitative estimate of the rem…
We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption \partial_t u -Δ_{p}u+|\nabla u|^{q}=0\quad in\;\; (0,\infty)\times\RR^N, where $N\ge 1$, $p\in(1,2)$, and $q>0$. Based on gradient estimates for the solutions, we classify the behavior of the solutions for large times, obtaining either positivity as $t\to\infty$ for $q>p-N/(N+1)$, optimal decay estimates as $t\to\infty$ for $p/2\le q\le p-N/(N+1)$, or extinction in finite time for $0 < q < p/2$. In addition, we show how the diffusion prevents extinction in fi…
We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a critical exponent. After a suitable rescaling which yields a non--linear Fokker--Planck equation, we find a continuum of algebraic rates of convergence to a self--similar profile. These rates depend explicitly on the spatial decay rates of initial data. This improves a previous result on slow convergence for the critical fast diffusion equation ({\sc Bonforte et al}. in Arch Rat Mech Anal 196:631--680, 2010) and provides answers to some open problems.
We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a subcritical exponent. We show that separable solutions are stable in some suitable sense by finding a class of functions which belong to their domain of attraction. For solutions in this class we establish optimal rates of convergence to separable solutions.
We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $dX - ( νΔX + Δψ(X) ) dt = \sum_{i=1}^N \langle b_i, \nabla X \rangle \circ dβ_i$ in $]0,T[ \times \mathcal{O}$, with $X(0) = x(ξ)$ in $\mathcal{O}$ and $X = 0$ on $]0,T[ \times \partial \mathcal{O}$. Moreover, we prove extinction in finite time of the solutions in the special case of fast diffusion model and of self-organized criticality model.
We consider a class of nonlocal reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. By using explicit changes of unknown function, we show that they are equivalent to the heat equation and, therefore, compute their solution explicitly. Based on this, we then prove that, in the case of beneficial mutations in asexual populations, solutions dramatically depend on the tails of the initial data: they can be global, become extinct in finite time or, even, be defined for no positive time. In the former case, we prove that solutions are accelerating, and…
The stability of asymptotic profiles of solutions to the Cauchy-Dirichlet problem for Fast Diffusion Equation (FDE, for short) is discussed. The main result of the present paper is the stability of any asymptotic profiles of least energy. It is noteworthy that this result can cover non-isolated profiles, e.g., those for thin annular domain cases. The method of proof is based on the Lojasiewicz-Simon inequality, which is usually used to prove the convergence of solutions to prescribed limits, as well as a uniform extinction estimate for solutions to FDE. Besides, local minimizers of an energy f…
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 i…
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-…
This paper is concerned with a quantitative analysis of asymptotic behaviors of (possibly sign-changing) solutions to the Cauchy-Dirichlet problem for the fast diffusion equation posed on bounded domains with Sobolev subcritical exponents. More precisely, rates of convergence to non-degenerate asymptotic profiles will be revealed via an energy method. The sharp rate of convergence to \emph{positive} ones was recently discussed by Bonforte and Figalli (2021, CPAM) based on an entropy method. An alternative proof for their result will also be provided. Furthermore, dynamics of fast diffusion flo…
