Finite Time Extinction Research Articles

Existence and extinction in finite time for Stratonovich gradient noise porous media equations

We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $dX - ( \nu \Delta X + \Delta \psi (X) ) dt = \sum_{i=1}^N \langle b_i, \nabla X \rangle \circ d\beta_i$ in $]0,T[ \times \mathcal{O}$, with $X(0) = x(\xi)$ 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.

Finite time collapsing of the Kähler-Ricci flow on threefolds

We show that if on a compact Kahler threefold there is a solution of the Kahler-Ricci flow which encounters a finite time collapsing singularity, then the manifold admits a Fano fibration. Furthermore, if there is finite time extinction then the manifold is Fano and the initial class is a positive multiple of the first Chern class.

Total variation flow perturbed by gradient linear multiplicative noise

We consider stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative gradient-type noise. We study existence and uniqueness of non-negative variational solutions in terms of stochastic variational inequalities. We also show the positivity preserving property of the solutions and extinction in finite time with probability one. These kinds of equations arise, e.g., in the use for simulation of image restoring techniques or for modeling turbulence.

Extinction for a Singular Diffusion Equation with Strong Gradient Absorption Revisited

Abstract When 2 ⁢ N / ( N + 1 ) < p < 2 {2N/(N+1)<p<2} and 0 < q < p / 2 {0<q<p/2} , non-negative solutions to the singular diffusion equation with gradient absorption ∂ t ⁡ u - Δ p ⁢ u + | ∇ ⁡ u | q = 0 in ⁢ ( 0 , ∞ ) × ℝ N \partial_{t}u-\Delta_{p}u+|\nabla u|^{q}=0\quad\text{in }(0,\infty)\times% \mathbb{R}^{N} vanish after a finite time. This phenomenon is usually referred to as finite-time extinction and takes place provided the initial condition u 0 {u_{0}} decays sufficiently rapidly as | x | → ∞ {|x|\to\infty} . On the one hand, the optimal decay of u 0 {u_{0}} at infinity guaranteeing the occurrence of finite-time extinction is identified. On the other hand, assuming further that p - 1 < q < p / 2 {p-1<q<p/2} , optimal extinction rates near the extinction time are derived.

Optimal Curvature Estimates for Homogeneous Ricci Flows

AbstractWe prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n)({\mathrm{scal}}(g(t)) - {\mathrm{scal}}(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, with constants depending only on the dimension $n$. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on that space. The above curvature estimates follow from a gap theorem for Ricci-flatness on homogeneous spaces. This theorem is proved by contradiction, using a local $W^{2,p}$ convergence result which holds without symmetry assumptions.

Self-similar extinction for a diffusive Hamilton–Jacobi equation with critical absorption

The behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption ∂_t u − ∆_p u + |∇u|^{p−1} = 0 in (0, ∞) × 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| → ∞, 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_* (|x|), (t, x) ∈ (0, T_e) × R^N , as t → T_e , where T_e denotes the finite extinction time of u. A cornerstone of the convergence proof is an underlying variational structure of the equation. Also, the selected profile f_* is the unique non-negative solution to a second order ordinary differential equation which decays exponentially at infinity. A complete classification of solutions to this equation is provided, thereby describing all separate variable solutions of the original equation. One important difficulty in the uniqueness proof is that no monotonicity argument seems to be available and it is overcome by the construction of an appropriate Pohozaev functional.

One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity

We consider positive classical solutions of $$\begin{aligned} v_t=(v^{m-1}v_x)_x, \qquad x\in {\mathbb {R}}, \ t>0, \qquad (\star ) \end{aligned}$$ in the super-fast diffusion range \(m<-1\). Our main interest is in smooth positive initial data \(v_0=v(\cdot ,0)\) which decay as \(x\rightarrow +\infty \), but which are possibly unbounded as \(x\rightarrow -\infty \), having in mind monotonically decreasing data as prototypes. It is firstly proved that if \(v_0\) decays sufficiently fast only in one direction by satisfying $$\begin{aligned} v_0(x) \le cx^{-\beta } \qquad \text{ for } \text{ all } ~x>0 \quad \hbox { with some }\quad \beta >\frac{2}{1-m} \end{aligned}$$ and some \(c>0\), then the so-called proper solution of (\(\star \)) vanishes identically in \({\mathbb {R}}\times (0,\infty )\), and accordingly no positive classical solution exists in any time interval in this case. Complemented by some sufficient criteria for solutions to remain positive either locally or globally in time, this condition for instantaneous extinction is shown to be optimal at least with respect to algebraic decay of the initial data. This partially extends some known nonexistence results for (\(\star \)) (Daskalopoulos and Del Pino in Arch Rat Mech Anal 137(4):363–380, 1997) in that it does not require any knowledge on the behavior of \(v_0(x)\) for \(x<0\). Next focusing on the phenomenon of extinction in finite time, we show that in this respect a mass influx from \(x=-\infty \) can interact with mass loss at \(x=+\infty \) in a nontrivial manner. Namely, we shall detect examples of monotone initial data, with critical decay as \(x\rightarrow +\infty \) and exponential growth as \(x\rightarrow -\infty \), that lead to solutions of (\(\star \)) which become extinct at a finite positive time, but which have empty extinction sets. This is in sharp contrast to known extinction mechanisms which are such that the corresponding extinction sets coincide with all of \({\mathbb {R}}\).

Numerical simulation of universal finite time behavior for parabolic IVP via geometric renormalization group

A numerical procedure based on the renormalization group (RG) is presented. This procedure will compute the spatial profile and blow up time for self-similar behavior. This will be generated by a family of parabolic IVPs, which includes the semilinear heat equation. This procedure also handles different diffusion structures, finite time extinction problems and exponential absorption with trivial modifications to the power law version. Convergence of the procedure is proved for the semilinear heat equation, which is a marginal perturbation to the heat equation, with a typical class of initial data. Numerical experiments show the accuracy of the method for various related problems. The main feature is simplicity: it will be shown that an explicit numerical method with a fixed mesh size is capable of computing very sharp approximations to these behaviors. This will include a priori detection of the logarithmic correction in the case of the semilinear heat equation.

Blow-up and finite time extinction for p(x,t)-curl systems arising in electromagnetism

We study a class of p(x,t)-curl systems arising in electromagnetism, with a nonlinear source term. Denoting by h the magnetic field, the source term considered is of the form λh(∫Ω|h|2)σ−22 where λ∈{−1,0,1}: when λ∈{−1,0} we consider 0<σ≤2 and for λ=1 we have σ≥1. We introduce a suitable functional framework and a convenient basis that allow us to apply the Galerkin's method and prove existence of local or global solutions, depending on the values of λ and σ. We study the finite time extinction or the stabilization towards zero of the solutions when λ∈{−1,0} and the blow-up of local solutions when λ=1.

Notes on a PDE system for biological network formation

We present new analytical and numerical results for the elliptic–parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transport networks. The model describes the pressure field using a Darcy’s type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. The analytical part extends the results of Haskovec et al. (2015) regarding the existence of weak and mild solutions to the whole range of meaningful relaxation exponents. Moreover, we prove finite time extinction or break-down of solutions in the spatially one-dimensional setting for certain ranges of the relaxation exponent. We also construct stationary solutions for the case of vanishing diffusion and critical value of the relaxation exponent, using a variational formulation and a penalty method.The analytical part is complemented by extensive numerical simulations. We propose a discretization based on mixed finite elements and study the qualitative properties of network structures for various parameter values. Furthermore, we indicate numerically that some analytical results proved for the spatially one-dimensional setting are likely to be valid also in several space dimensions.

Semi-classical analysis for proof extinction-property in finite time of solutions for parabolic equations with homogeneous main part and degenerate absorption potential

We study the behavior of solutions of parabolic equation with double nonlinearity and a degenerate absorption term. The main topic of interest is the property of finite time extinction, i.e., the solution vanish after finite time.

On the long time behavior of homogeneous Ricci flows

In this paper we prove the following structure results for homogeneous Ricci flow solutions: Any homogeneous Ricci flow solution with finite extinction time develops a Type I singularity. Any homogeneous Ricci flow solution on a compact homogeneous space, not diffeomorphic to a torus, has finite extinction time. Any immortal homogeneous Ricci flow solution develops a Type III singularity and the natural blow downs subconverge to an immortal locally homogeneous Ricci flow solution.

On uniform and decay estimates for unbounded solutions of partial differential equations

We prove that if a function u satisfies certain integral estimates (of energy type) then it verifies estimates of the type‖u(t)‖Lr(Ω)≤c‖u(0)‖Lr0(Ω)γ0tγ1,t>0, where r and r0 are exponents that appear in these integral estimates and γ1 and γ0 are positive constants that can be expressed in terms of these constants. We will see that in some cases r>r0 (supercontractive estimates) and hence a regularizing effect on u appears while in other cases this improvement of regularity does not appear since r<r0; in any case, we prove that the Lr(Ω)-norm of u decays in time (for t large) since it verifies the previous estimate.We show how to apply this result to obtain new estimates of this kind for the solutions of many nonlinear parabolic equations. This new method allows also to explain the reason of the similar behavior of the solutions of very different parabolic problems.Finally, we study sufficient conditions for extinction in finite time.

On a nonlocal degenerate parabolic problem

Conditions for the existence and uniqueness of weak solutions for a class of nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour of the solutions as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved.

Time regularity and long-time behavior of parabolic p-Laplace equations on infinite graphs

We consider the so-called discrete p-Laplacian, a nonlinear difference operator that acts on functions defined on the nodes of a possibly infinite graph. We study the corresponding Cauchy problem and identify the generator of the associated nonlinear semigroups. We prove higher order time regularity of the solutions. We investigate the long-time behavior of the solutions and discuss in particular finite extinction time and conservation of mass. Namely, on one hand, for small p if an infinite graph satisfies some isoperimetric inequality, then the solution to the parabolic p-Laplace equation vanishes in finite time; on the other hand, for large p, these parabolic p-Laplace equations always enjoy conservation of mass.

Extinction of solutions to a class of fast diffusion systems with nonlinear sources

Y. Wang In this paper, the finite time extinction of solutions to the fast diffusion system ut=div(|∇u|p − 2∇u) + vm, vt=div(|∇v|q − 2∇v) + un is investigated, where 1 < p,q < 2, m,n > 0 and 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 proved by using the De Giorgi iteration process and comparison method. On the other hand, for 1 < p = q < 2 and mn < (p − 1)2, the existence of at least one non-extinction solution for any positive smooth initial data is proved. Copyright © 2015 John Wiley & Sons, Ltd.

Finite Time Extinction for Nonlinear Schrödinger Equation in 1D and 2D

We consider a nonlinear Schrödinger equation with power nonlinearity, either on a compact manifold without boundary, or on the whole space in the presence of harmonic confinement, in space dimension one and two. Up to introducing an extra superlinear damping to prevent finite time blow up, we show that the presence of a sublinear damping always leads to finite time extinction of the solution in 1D, and that the same phenomenon is present in the case of small mass initial data in 2D.

On a logistic growth model with predation and a power‐type diffusion coefficient: I. Existence of solutions and extinction criteria

We consider a logistic growth model with a predation term and a stochastic perturbation yielding constant elasticity of variance. The resulting stochastic differential equation does not satisfy the standard assumptions for existence and uniqueness of solutions, namely, linear growth and the Lipschitz condition. Nevertheless, for any positive initial condition, we prove that a solution exists and is unique up to the first time it hits zero. Additionally, we provide alternative criteria for population extinction depending on the choice of parameters. More precisely, we provide criteria that guarantee the following: (i) population extinction with positive probability for a set of initial conditions with positive Lebesgue measure; (ii) exponentially fast population extinction with full probability for any positive initial condition; and (iii) population extinction in finite time with full probability for any positive initial condition. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Finite Time Extinction for Stochastic Sign Fast Diffusion and Self-Organized Criticality

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, R\"ockner, CMP, 2009], [Barbu, Da Prato, R\"ockner, CRMAS, 2009], [Barbu, R\"ockner, CMP, 2012], [Barbu, Da Prato, R\"ockner, JMAA, 2012], [R\"ockner, 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 analysis of mass diffusion and several new estimates and techniques are introduced.

Egalitarianism under population change: Age structure does matter

We study the compatibility of the optimal population size concepts produced by different social welfare functions and egalitarianism meant as “equal consumption for all individuals of all generations”. Social welfare functions are parameterized by an altruism parameter generating the Benthamite and Millian criteria as polar cases. The economy considered is in continuous time and is populated by homogeneous cohorts with a given life span. Production functions are linear in labor, (costly) procreation is the unique way to transfer resources forward in time. First, we show that egalitarianism is optimal whatever the degree of altruism in “perpetual youth” model, that is when lifetime span is finite but age structure does not matter: in this case egalitarianism does not discriminate between the social welfare functions considered. Then we show that, when life span is finite but age structure matters, egalitarianism does not arise systematically as an optimal outcome. In particular, in a growing economy, that is when population growth is optimal in the long-run, this egalitarian rule can only hold when the welfare function is Benthamite. When altruism is impure, egalitarianism is impossible in the context of a growing economy. Either in the Benthamite or impure altruism cases, procreation is never optimal for small enough life spans, leading to finite time extinction and maximal consumption for all existing individuals.