Finite Time Extinction Research Articles

Positivity, decay, and extinction for a singular diffusion equation with gradient absorption

We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption∂tu−Δpu+|∇u|q=0in (0,∞)×RN, where N⩾1, p∈(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→∞ for q>p−N/(N+1), optimal decay estimates as t→∞ for p/2⩽q⩽p−N/(N+1), or extinction in finite time for 0<q<p/2. In addition, we show how the diffusion prevents extinction in finite time in some ranges of exponents where extinction occurs for the non-diffusive Hamilton–Jacobi equation.

Revisiting the Optimal Population Size Problem under Endogenous Growth: Minimal Utility Level and Finite Life

In this paper, we devise a social criterion in the spirit of the critical utility level of Blackorby-Donaldson (1984) to study an optimal population size problem in an endogenously growing economy populated by workers living a fixed amount of time and without capital accumulation. Population growth is endogenous. The problem is analytically solved, yielding closed-form solutions to optimal demographic and economic dynamics. It is shown that provided the economy is not driven to optimal finite time extinction, the optimal solution is egalitarian for appropriate choices of the critical utility levels: all individuals of any cohort are given the same consumption. The results obtained do not require any priori restriction of the values of the elasticity of intertemporal substitution unlike in several related papers.

Finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise

In this paper one surveys and sharpens some recent results obtained by the authors on the finite time extinction property of solutions to stochastic diffusion equations of the form d X − ρ Δ X m d t = X d W and d X − ρ Δ sgn X d t = X d W , where 0 < m < 1 , ρ > 0 . These equations arise as models for nonlinear diffusion processes in porous media, plasma and self-organized criticality under stochastic Gaussian perturbation. These equations can be also viewed as control systems governed by fast diffusion porous media equations with a stochastic feedback controller X d W .

Complete quenching for singular parabolic problems

We prove finite time extinction of the solution of the equation u t − Δ u + χ { u > 0 } ( u − β − λ f ( u ) ) = 0 in Ω × ( 0 , ∞ ) with boundary data u ( x , t ) = 0 on ∂ Ω × ( 0 , ∞ ) and initial condition u ( x , 0 ) = u 0 ( x ) in Ω, where Ω ⊂ R N is a bounded smooth domain, 0 < β < 1 and λ > 0 is a parameter. For every small enough λ > 0 there exists a time t 0 > 0 such that the solution is identically equal to zero.

Existence, uniqueness and behavior of solutions for a class of nonlinear parabolic problems

We prove existence, uniqueness, regularity results and estimates describing the behavior (both for large and small times) of a solution u of some nonlinear parabolic equations of Leray-Lions type including the p -Laplacian. In particular we show how the summability of the initial datum u 0 and the value of p influence the behavior of the solution u , producing ultracontractive or supercontractive estimates or extinction in finite time or different kinds of decay estimates.

Existence and localization of weak solutions of nonlinear parabolic equations with variable exponent of nonlinearity

The aim of this paper is to study the existence and uniqueness of weak solutions of the initial Neumann problem for \({u_{t}={\rm div}(|\nabla u|^{p(x,t)-2}\nabla u+\vec{F}(x,t))}\). First, the authors construct suitable function spaces to which the solution belongs and then applies Galerkin’s approximation technique to prove the existence of weak solutions with necessary uniform estimates and a compactness argument. Second, the authors obtain the properties of extinction in finite time of weak solutions under suitable conditions by proving some energy estimates and applying a comparison principle.

Variational approach to complicated similarity solutions of higher-order nonlinear PDEs. II

This paper continues the study that began in [1,2] of the Cauchy problem for ( x , t ) ∈ R N × R + for three higher-order degenerate quasilinear partial differential equations (PDEs), as basic models, u t = ( − 1 ) m + 1 Δ m ( | u | n u ) + | u | n u , u t t = ( − 1 ) m + 1 Δ m ( | u | n u ) + | u | n u , u t = ( − 1 ) m + 1 [ Δ m ( | u | n u ) ] x 1 + ( | u | n u ) x 1 , where n > 0 is a fixed exponent and Δ m is the ( m ≥ 2 ) th iteration of the Laplacian. A diverse class of degenerate PDEs from various areas of applications of three types: parabolic, hyperbolic, and nonlinear dispersion, is dealt with. General local, global, and blow-up features of such PDEs are studied on the basis of their blow-up similarity or traveling wave (for the last one) solutions. In [1,2], the Lusternik–Schnirel’man category theory of variational calculus and fibering methods were applied. The case m = 2 and n > 0 was studied in greater detail analytically and numerically. Here, more attention is paid to a combination of a Cartesian approximation and fibering to get new compactly supported similarity patterns. Using numerics, such compactly supported solutions are constructed for m = 3 and for higher orders. The “smother” case of negative n < 0 is included, with a typical “fast diffusion–absorption” parabolic PDE: u t = ( − 1 ) m + 1 Δ m ( | u | n u ) − | u | n u , where n ∈ ( − 1 , 0 ) , which admits finite-time extinction rather than blow-up. Finally, a homotopy approach is developed for some kind of classification of various patterns obtained by variational and other methods. Using a variety of analytic, variational, qualitative, and numerical methods allows us to justify that the above PDEs admit an infinite countable set of countable families of compactly supported blow-up (extinction) or traveling wave solutions.

Finite Time Extinction by Nonlinear Damping for the Schrödinger Equation

We consider the Schrödinger equation on a compact manifold, in the presence of a nonlinear damping term, which is homogeneous and sublinear. For initial data in the energy space, we construct a weak solution, defined for all positive time, which is shown to be unique. In the one-dimensional case, we show that it becomes zero in finite time. In the two and three-dimensional cases, we prove the same result under the assumption of extra regularity on the initial datum.

Extinction and Positivity of the Solutions for a p‐Laplacian Equation with Absorption on Graphs

We deal with the extinction of the solutions of the initial‐boundary value problem of the discrete p‐Laplacian equation with absorption ut = Δp,ωu − uq with p &gt; 1, q &gt; 0, which is said to be the discrete p‐Laplacian equation on weighted graphs. For 0 &lt; q &lt; 1, we show that the nontrivial solution becomes extinction in finite time while it remains strictly positive for p ≥ 2, q ≥ 1 and q ≥ p − 1. Finally, a numerical experiment on a simple graph with standard weight is given.

On the Existence of Min-Max Minimal Torus

In this paper, we will study the existence problem of min-max minimal torus. We use classical conformal invariant geometric variational methods. We prove a theorem about the existence of min-max minimal torus in Theorem 5.1. First we prove a strong uniformization result (Proposition 3.1) using the method of Ahlfors and Bers (Ann. Math. 72(2):385–404, 1960). Then we use this proposition to choose good parameterization for our min-max sequences. We prove a compactification result (Lemma 4.1) similar to that of Colding and Minicozzi (Width and finite extinction time of Ricci flow, 0707.0108 [math.DG], 2007), and then give bubbling convergence results similar to that of Ding et al. (Invent. math. 165:225–242, 2006). In fact, we get an approximating result similar to the classical deformation lemma (Theorem 1.1).

Self-organized criticality and convergence to equilibrium of solutions to nonlinear diffusion equations

This article is a survey on some recent results on longtime behaviour of solutions to nonlinear singular diffusion equations with main emphasis on fast diffusion porous media equations and the sand-pile model of self-organized criticality. Deterministic as well stochastic equations are presented. One of the key problem considered here is the extinction in finite time of solutions.

Pointwise decay for the solutions of degenerate and singular parabolic equations

We study the asymptotic behavior, as $t\to\infty$, of the solutions to the evolutionary $p$-Laplace equation \[ v_t={\operatorname{div}}( | {\nabla v} | ^{p-2}\nabla v), \] with time-independent lateral boundary values. We obtain the sharp decay rate of $\max_{x\in{\Omega}} | {v(x,t)-u(x)}| $, where $u$ is the stationary solution, both in the degenerate case $p > 2$ and in the singular case $1 < p > 2$. A key tool in the proofs is the Moser iteration, which is applied to the difference $v(x,t)-u(x)$. In the singular case, we construct an example proving that the celebrated phenomenon of finite extinction time, valid for $v(x,t)$ when $u\equiv 0$, does not have a counterpart for $v(x,t)-u(x)$.

Finite time extinction for solutions to fast diffusion stochastic porous media equations

We prove that the solutions to fast diffusion stochastic porous media equations have finite time extinction with strictly positive probability. To cite this article: V. Barbu et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Extinction Versus Persistence in Strong Oscillating Flows

In this paper, we give some conditions for finite-time extinction or persistence of the solutions of diffusion–advection equations in strong and oscillating flows under Dirichlet boundary conditions. The enhancement of the diffusion rate depends on the interplay between strong advection and time-homogenization, and in particular on the ratio between the strength of the flow and its frequency parameter. Quantitative estimates of this ratio, which depend on the geometry of the domain, are provided in the case of a uniform flow. In the general time–space dependent case, the finite-time behavior of the solutions is related to the existence of first integrals of the flow.

Lyapunov pairs for continuous perturbations of nonlinear evolutions

This paper deals with the Lyapunov pairs for a nonlinear evolution system in Banach spaces. Applications on a priori estimates, finite extinction time, null controllability and regularity of the minimal time function are given.

Width and finite extinction time of Ricci flow

This is an expository article with complete proofs intended for a general nonspecialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2–spheres. For instance, when M is a homotopy 3–sphere, the width is loosely speaking the area of the smallest 2–sphere needed to ‘pull over’ M. Second, we use this to conclude that Hamilton’s Ricci flow becomes extinct in finite time on any homotopy 3–sphere.

Stochastic Porous Media Equations and Self-Organized Criticality

The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time extinction of solutions with high probability is also proven in 1-D. The results are relevant for self-organized criticality behavior of stochastic nonlinear diffusion equations with critical states.

Scaling Laws for the Decay of Multiqubit Entanglement

We investigate the decay of entanglement of generalized N-particle Greenberger-Horne-Zeilinger (GHZ) states interacting with independent reservoirs. Scaling laws for the decay of entanglement and for its finite-time extinction (sudden death) are derived for different types of reservoirs. The latter is found to increase with N. However, entanglement becomes arbitrarily small, and therefore useless as a resource, much before it completely disappears, around a time which is inversely proportional to the number of particles. We also show that the decay of multiparticle GHZ states can generate bound entangled states.

Large time behavior of nonlocal aggregation models with nonlinear diffusion

The aim of this paper is to establish rigorous results on thelarge time behavior of nonlocal models for aggregation, includingthe possible presence of nonlinear diffusion terms modeling localrepulsions. We show that, as expected from the practicalmotivation as well as from numerical simulations, one obtainsconcentrated densities (Dirac $\delta$ distributions) asstationary solutions and large time limits in the absence ofdiffusion. In addition, we provide a comparison for aggregationkernels with infinite respectively finite support. In the firstcase, there is a unique stationary solution corresponding toconcentration at the center of mass, and all solutions of theevolution problem converge to the stationary solution for largetime. The speed of convergence in this case is just determined bythe behavior of the aggregation kernels at zero, yielding eitheralgebraic or exponential decay or even finite time extinction. Forkernels with finite support, we show that an infinite number ofstationary solutions exist, and solutions of the evolution problemconverge only in a measure-valued sense to the set of stationarysolutions, which we characterize in detail.Moreover, we also consider the behavior in the presence of nonlinear diffusion terms,the most interesting case being the one of small diffusion coefficients. Via the implicit function theorem we give aquite general proof of a rather natural assertion for such models, namely that there exist stationary solutionsthat have the form of a local peak around the center of mass. Our approach even yields the order of the size ofthe support in terms of the diffusion coefficients.All these results are obtained via a reformulation of theequations considered using the Wasserstein metric for probabilitymeasures, and are carried out in the case of a single spatialdimension.

The evolution of dead cores in strongly degenerate diffusion problems

This paper studies the large time behavior of nonnegative solutions of $$ u_t = u^p \Delta u - u^{-q} \chi_{\{u>0\}} \qquad \mbox{in } \Omega \times (0,\infty), \qquad p \ge 1, q>-1, $$ with prescribed boundary values $u|_{{\partial\Omega}}=1$ in smoothly bounded domains $\Omega \subset \mathbb R^n$. The particular interest here is in the question of how the interplay of strongly degenerate diffusion and strong absorption influences the evolution of the dead core set $\{u(t)=0\}$ (which is enforced to be nontrivial for all time by the assumption that the initial data $u_0$ vanish in some ball). The main results state that there exist four parameter regimes in which, widely independent of $u_0$, all solutions undergo (i)a complete extinction at time $t=0$ ($q>p-1$) (ii) a total extinction in finite but positive time ($0 <q\le p-1$), (iii) an extinction in infinite time ($1-p <q <0$), or (iv) no extinction ($q <1-p$), respectively. Specifically, the first two types of behavior are in sharp contrast to the non-degenerate and weakly degenerate cases $p=0$ and $p \in (0,1)$, respectively, where only (iv) occurs.