Lifespan Estimates Research Articles

The lifespan of solutions of semilinear wave equations with the scale-invariant damping in one space dimension

The critical constant $\mu$ (see (1.1)) of time-decaying damping in the scale-invariant case is recently conjectured. It also has been expected that the lifespan estimate is the same as for the associated semilinear heat equations if the constant is in the “heat-like” domain. In this paper, we point out that this is not true if the total integral of the sum of initial position and speed vanishes. In such a case, we have a new type of the lifespan estimates which is closely related to the non-damped case in shifted space dimensions.

Controls on the lifespans of Icelandic ice caps

The increasingly common disappearance of glaciers is striking, and it is often used to highlight the effects of anthropogenic climate change to the public. This approach is more effective when modern glacier and climate change is placed within a frame of reference. Here we present a new, efficient method to assess the timing of glacier birth and death. We model equilibrium line altitudes (ELAs) of 22 Icelandic glaciers through time, and in doing so produce the first regional estimates of glacier lifespans. We force the model with three Holocene temperature reconstructions and five future-climate projections, while also exploring the effects of glacial-isostatic adjustment and variable precipitation. Mass balance parameters are tuned for each ice cap using modern data. Geologically-constrained inception ages are used to validate individual simulations. The timing of Icelandic glacier inception likely ranged from before the Holocene to as late as the Little Ice Age. Anthropogenic warming is expected to cut short the lifespans of glaciers that have existed for thousands of years. By quantifying the controlling factors we show that topography and the rate of summer temperature change are the primary controls of glacier lifespans in Iceland. Topography, represented by the difference between the modern ELA and the highest local topography, is ten-times as important as ELA sensitivity to temperature. This easy-to-measure topographic metric exerts a first-order control on the timing of inception and the future loss of accumulation area and it can be used to contextualize ongoing glacier disappearance.

Model parameter estimation and residual life span prediction of pneumatic diaphragm pump based on hidden Markov model in intelligent spraying

Pneumatic diaphragm pump is an important part in intelligent spraying. When pneumatic diaphragm pump does not work normally, the entire intelligent spraying product line will be malfunctioned. To maintain and manage pneumatic diaphragm pump effectively, the grade analysis of the health status of pneumatic diaphragm pump is generally used according to its working state. Due to the effects of condition monitoring and random faults, some observable health predictions are often inaccurate. There are very few papers dealing with the health monitoring of pneumatic diaphragm pump and their estimation of residual life span. In this article, a method with vector autoregressive model and continuous-time hidden Markov model was proposed to analyze and evaluate the life span of pneumatic diaphragm pump based on the estimation error of the health condition and the cumulative deterioration of pneumatic diaphragm pump. It is modeled through a continuous-time Markov chain with three states, which includes unobservable healthy state 0, unobservable warning state 1, and observable fault state 2. The expectation–maximization algorithm is used to estimate the model parameters of the fitted hidden Markov. Through the posterior probability of pneumatic diaphragm pump in warning state 1, the derived conditional reliability function and mean residual life span formula can be calculated to evaluate the residual life span of pneumatic diaphragm pump. The results showed that the method can effectively predict the residual life span of pneumatic diaphragm pump, illustrate the effectiveness of the model, and improve the accuracy of the health status rating.

Sharp lifespan estimates of blowup solutions to semi-linear wave equations with time-dependent effective damping

We consider the initial value problem for a semi-linear wave equation with a time-dependent effective damping term. The interest is the behavior of lifespan of solutions in view of the asymptotic profile of the coefficient of damping as time tends to infinity. A simple case of effective damping and a threshold case in the sense of overdamping were discussed in Ikeda and Wakasugi (2015) and Ikeda and Inui (2019), respectively. We discuss here general damping terms under a certain assumption that is discussed. The result of this paper is the sharp lifespan estimates of blowup solutions including the typical cases treated in previous studies. The proof of the upper bound of lifespan is a modification of the test-function method by Ikeda–Sobajima (2019) and the one of lower bound is based on the technique of scaling variables introduced by Gallay and Raugel (1998) for the one-dimensional case and Wakasugi (2017) for the higher-dimensional case.

Lifespan estimate for semilinear wave equation in Schwarzschild spacetime

In this paper we prove the blow-up result for small data Cauchy problem of semilinear wave equation in Schwarzschild spacetime with p∈[32,2], without the assumption that the supports of the initial data should be far away from the black hole. What is more, we establish the upper bound of the lifespan estimate.

Estimation of lifespan of diesel locomotive engine

The operating time of a diesel locomotive indicates its mechanical states to a certain scale. Though, even the same variant of diesel locomotive with the same types of operating hours shows different technical states in different types of operative environments. Therefore, it is quite intricate to obtain the data for entire life or general life prediction methods are required to study the physical failure mechanism. In pursuant of the discussed cases, mathematical model of diesel locomotive’s life estimation with respect to downward trend and neural networks is evolved in the study. Primarily, the parameters corresponds to downward trends are selected and the sample data is to be standardized & then the principal component analysis method is being used to simplify different parameters to a pervasive parameter. Method of interpolation is being used for the parameter’s time series data as the train data of neural network. Finally, the life span estimation model of the locomotive engine based on neural network is evolved.

The end of life on Earth is not the end of the world: converging to an estimate of life span of the biosphere?

AbstractEnvironmental conditions have changed in the past of our planet but were not hostile enough to extinguish life. In the future, an aged Earth and a more luminous Sun may lead to harsh or even uninhabitable conditions for life. In order to estimate the life span of the biosphere we built a minimal model of the co-evolution of the geosphere, atmosphere and biosphere of our planet, taking into account temperature boundaries, CO2partial pressure lower limits for C3 and C4 plants, and the presence of enough surface water. Our results indicate that the end of the biosphere will happen long before the Sun becomes a red giant, as the biosphere faces increasingly more difficult conditions in the future until its collapse due to high temperatures. The lower limit for CO2partial pressure for C3 plants will be reached in 170(+ 320, − 110) Myr, followed by the C4 plants limit in 840(+ 270, − 100) Myr. The mean surface temperature will reach 373 K in 1.63(+ 0.14, − 0.05) Gyr, a point that would mark the extinction of the biosphere. Water loss due to internal geophysical processes will not be dramatic, implying almost no variation in the surface ocean mass and ocean depth for the next 1.5 billion years. Our predictions show qualitative convergence and some quantitative agreement with results found in the literature, but there is considerable scattering in the scale of hundreds of millions of years for all the criteria devised. Even considering these uncertainties, the end of the biosphere will hardly happen sooner than 1.5 Gyr.

Blow-up phenomena of semilinear wave equations and their weakly coupled systems

In this paper we consider the wave equations with power type nonlinearities including time-derivatives of unknown functions and their weakly coupled systems. We propose a framework of test function methods and give a simple proof of the derivation of sharp upper bounds for lifespan of solutions to nonlinear wave equations and their systems. We point out that for respective critical cases, we use a family of self-similar solutions to the linear wave equation including Gauss's hypergeometric functions, which are originally introduced by Zhou [59]. We emphasize that our framework does not require the pointwise positivity of the initial data even in the high dimensional case N≥4. Moreover, we find a new (p,q)-curve for the system ∂t2u−Δu=|v|q, ∂t2v−Δv=|∂tu|p with lifespan estimates for small solutions in a new region.

Global existence of the self-interacting scalar field in the de Sitter universe

We present some sufficient conditions for the global in time existence of solutions of the semilinear Klein-Gordon equation of the self-interacting scalar field with complex mass. The coefficients of the equation depend on spatial variables as well, which makes results applicable, in particular, to the spacetime with the time slices being Riemannian manifolds. The least lifespan estimate is given for the class of equations including the Higgs boson equation, which according to physics has a finite lifetime.

Estimates of Lifespan and Blow-up Rates for the Wave Equation with a Time-dependent Damping and a Power-type Nonlinearity

We study blow-up behavior of solutions for the Cauchy problem of the semilinear wave equation with time-dependent damping. When the damping is effective, and the nonlinearity is subcritical, we show the blow-up rates and the sharp lifespan estimates of solutions. Upper estimates are proved by an ODE argument, and lower estimates are given by a method of scaling variables.

Lifespan of Solutions to MHD Boundary Layer Equations with Analytic Perturbation of General Shear Flow

In this paper, we consider the lifespan of solution to the MHD boundary layer system as an analytic perturbation of general shear flow. By using the cancellation mechanism in the system observed in [12], the lifespan of solution is shown to have a lower bound in the order of e−2− if the strength of the perturbation is of the order of e. Since there is no restriction on the strength of the shear flow and the lifespan estimate is larger than the one obtained for the classical Prandtl system in this setting, it reveals the stabilizing effect of the magnetic field on the electrically conducting fluid near the boundary.

Estimating the lifespan of the expansion of a landfill in Brazil by means of a model equation based on socio-economic indicators

This study aimed to develop an equation with the aid of socio-economic indicators to estimate the amount of MSW to be disposed of in ASMOC Landfill and its lifespan. Thereby enabling a comparison with the estimate of MSW and lifespan calculated in the environmental impact report (EIR) of the ASMOC, which did not take economic indicators into account. The data GDP, HDI, population, water consumption, electricity consumption and amount of MSW were used in the research. Pearson correlation analysis and multiple regressions were performed with the SPSS program to generate the equation. Therefore, predicted values were obtained for the amounts of MSW and their respective volumes, making it possible to estimate the lifespan of the expansion of the ASMOC to 12 years and two months, a difference of four years and six months from the projection stipulated by the EIR of the ASMOC (16 years and eight months).

$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

We study the Cauchy problem of the damped wave equation \begin{align*} \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1\le q \le p < \infty\ (p\neq 1)$ with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in $(H^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r)$ with $r \in (1,2]$, $s\ge 0$, and $\beta = (n-1)|\frac{1}{2}-\frac{1}{r}|$, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power $1+\frac{2r}{n}$, while it is known that the critical power $1+\frac{2}{n}$ belongs to the blow-up region when $r=1$. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan and blow-up rates by an ODE argument.

Sharp upper bound for lifespan of solutions to some critical semilinear parabolic, dispersive and hyperbolic equations via a test function method

This paper is concerned with the blowup phenomena for initial–boundary value problem (0.1)τ∂t2u(x,t)−Δu(x,t)+a(x)∂tu(x,t)=λ|u(x,t)|p,(x,t)∈CΣ×(0,T),u(x,t)=0,(x,t)∈∂CΣ×(0,T),u(x,0)=εf(x),x∈CΣ,τ∂tu(x,0)=τεg(x),x∈CΣ,where CΣ is a cone-like domain in RN (N≥2) defined as CΣ=intrω∈RN;r≥0,ω∈Σ with a connected open set Σ in SN−1 with smooth boundary ∂Σ. If N=1, then we only consider two cases CΣ=(0,∞) and CΣ=R. Here a(x) is a non-zero coefficient of ∂tu which could be complex-valued and space-dependent, λ∈ℂ is a fixed constant, and ε>0 is a small parameter. The constants τ=0,1 switch the parabolicity and hyperbolicity of the problem (0.1). The result proposes an argument for sharp upper lifespan estimates of solutions to (0.1) by a test function method based on Mitidieri and Pokhozhaev (2001). The crucial idea is to introduce an ordinary differential inequality with a parameter as a variable.

Life span bias explains live–dead discordance in abundance of two common bivalves

AbstractLife span bias potentially alters species abundance in death assemblages through the overrepresentation of short-lived organisms compared with their long-lived counterparts. Although previous work found that life span bias did not contribute significantly to live–dead discordance in bivalve assemblages, life span bias better explained discordance in two groups: longer-lived bivalve species and species with known life spans. More studies using local, rather than global, species-wide life spans and mortality rates would help to determine the prevalence of life span bias, especially for long-lived species with known life spans. Here, we conducted a field study at two sites in North Carolina to assess potential life span bias between Mercenaria mercenaria and Chione elevata, two long-lived bivalve species that can be aged directly. We compared the ability of directly measured local life spans with that of regional and global life spans to predict live–dead discordance between these two species. The shorter-lived species (C. elevata) was overrepresented in the death assemblage compared with its live abundance, and local life span data largely predicted the amount of live–dead discordance; local life spans predicted 43% to 88% of discordance. Furthermore, the global maximum life span for M. mercenaria resulted in substantial overpredictions of discordance (1.4 to 1.6 times the observed live–dead discordance). The results of this study suggest that life span bias should be considered as a factor affecting proportional abundances of species in death assemblages and that using life span estimates appropriate to the study locality improves predictions of discordance based on life span compared with using global life span estimates.

Evaluating the residual life of aged railway bridges

The UK is home to a very expansive railway network. The network includes a significant number of bridges that were constructed in the Victorian era. The aim of this study is to estimate the remaining lifespan of a unique aged railway bridge, the Windsor Railway Bridge in the UK. This research encompassed several steps: analysis of past and current traffic, prediction of future traffic trends, fatigue life analysis, estimation of lifespan consumption and estimation of remaining fatigue life. The finite-element analysis results showed that the most highly stressed members in the structure were the arch stringer and arch vertical bracing. By using the finite-element method together with the cumulative fatigue theory, these members are predicted to have failed in 5–7 years’ time, depending on the future traffic trends. Under a less conservative design class, some members are shown to have already failed sometime in the 1920s. It is found that a number of major conservative design assumptions were made. The failure mode and mechanism of the aged railway bridge are highlighted in this paper.

Estimates of the Heritability of Human Longevity Are Substantially Inflated due to Assortative Mating.

Human life span is a phenotype that integrates many aspects of health and environment into a single ultimate quantity: the elapsed time between birth and death. Though it is widely believed that long life runs in families for genetic reasons, estimates of life span "heritability" are consistently low (∼15-30%). Here, we used pedigree data from Ancestry public trees, including hundreds of millions of historical persons, to estimate the heritability of human longevity. Although "nominal heritability" estimates based on correlations among genetic relatives agreed with prior literature, the majority of that correlation was also captured by correlations among nongenetic (in-law) relatives, suggestive of highly assortative mating around life span-influencing factors (genetic and/or environmental). We used structural equation modeling to account for assortative mating, and concluded that the true heritability of human longevity for birth cohorts across the 1800s and early 1900s was well below 10%, and that it has been generally overestimated due to the effect of assortative mating.

Lifespan of semilinear wave equation with scale invariant dissipation and mass and sub-Strauss power nonlinearity

In this paper, we study the blow-up of solutions for semilinear wave equations with scale-invariant dissipation and mass in the case in which the model is somehow “wave-like”. A Strauss type critical exponent is determined as the upper bound for the exponent in the nonlinearity in the main theorems. Two blow-up results are obtained for the subcritical case and for the critical case, respectively. In both cases, an upper bound lifespan estimate is given.

Existence, lifespan, and transfer rate of Ricci flows on manifolds with small Ricci curvature

We show that in dimension 4 and above, the lifespan of Ricci flows depends on the relative smallness of the Ricci curvature compared to the Riemann curvature on the initial manifold. We can generalize this lifespan estimate to the local Ricci flow, using which we prove the short-time existence of Ricci flow solutions on noncompact Riemannian manifolds with at most quadratic curvature growth, whose Ricci curvature and its first two derivatives are sufficiently small in regions where the Riemann curvature is large. Those Ricci flow solutions may have unbounded curvature. Moreover, our method implies that, under some appropriate assumptions, the spatial transfer rate (the rate at which high curvature regions affect low curvature regions) of the Ricci flow resembles that of the heat equation.

An experimental loading study of a pitching wind turbine airfoil in near- and post-stall regions

A series of experimental tests was conducted to present an insight into lift characteristics of an oscillating wind turbine airfoil at nearand post-stall regions. Due to the unsteady nature of the flow around a wind turbine, the blades are subjected to oscillating motions and consequently, unsteady phenomena occur, dynamic stall in particular. Therefore, the unsteady lift of wind turbine blades is of considerable importance in performance and estimation of a wind turbine lifespan. In the current study, at Re = 420000, a detailed survey of parameters affecting lift characteristics in the hysteresis loops are carried out for the critical section of a 660 kW wind turbine blade. To investigate the effects of reduced frequency, mean angle of attack (AOA) and amplitude of oscillation, the characteristics of lift hysteresis loops including maximum lift, width of the loop, crossover point (if exists) and the normal force defect (NFD) are compared qualitatively and quantitatively. The results indicate that increase of reduced frequency leads to decrease in lift curve slope and delay in maximum lift occurrence. Furthermore, the lift curves are the evidence of strong dependence of lift characteristics on mean AOA and amplitude of pitching motion. Increase of amplitude makes the airfoil enter the post-stall region and hence, it delays the maximum lift occurrence. Entering the post-stall region, the airfoil encounters deeper dynamic stall. Also, approaching to post-stall region makes the clockwise part of the loops more dominant.