Dead Core Research Articles

Viscosity solutions to the inhomogeneous reaction–diffusion equation involving the infinity Laplacian

In this paper, we study the inhomogeneous reaction–diffusion equation involving the infinity Laplacian: [Formula: see text] where the continuous function [Formula: see text] satisfies [Formula: see text] a positive function [Formula: see text] [Formula: see text] [Formula: see text] and [Formula: see text]. Such a model permits existence of solutions with dead core zones, i.e. a priori unknown regions where non-negative solutions vanish identically. For [Formula: see text] and the non-positive inhomogeneous term [Formula: see text] we establish the existence, uniqueness and stability of the viscosity solution of the corresponding continuous Dirichlet problem. Under additional structure conditions on [Formula: see text] and [Formula: see text] we obtain the optimal [Formula: see text] regularity across the free boundary [Formula: see text] Moreover, we establish the porosity of the free boundary and Liouville type theorem for entire solutions. Finally, we prove that the dead core vanishes in the limit case [Formula: see text]

3D Biomimetic in Vitro Model of Self-induced Cardiac Ischemia Based on the Co-culture of Cardiomyocytes Derived from hiPSC and Primary Human Cardiac Fibroblast

Large cardiac spheroids (i.e., > 500 µm diameter) that self-induce an ischemic core were generated by co-culture of cardiomyocytes derived from induced pluripotent cells (hiPSC-CMs) and primary cardiac fibroblast (hCF). Without modification of culture conditions, spheroids’ large size allowed the recreation of a dead core, fibrotic remodelling and tissue stiffening.

Entire Minimizers of Allen–Cahn Systems with Sub-Quadratic Potentials

We study entire minimizers of the Allen–Cahn systems. The specific feature of our systems are potentials having a finite number of global minima, with sub-quadratic behaviour locally near their minima. The corresponding formal Euler–Lagrange equations are supplemented with free boundaries. We do not study regularity issues but focus on qualitative aspects. We show the existence of entire solutions in an equivariant setting connecting the minima of W at infinity, thus modeling many coexisting phases, possessing free boundaries and minimizing energy in the symmetry class. We also present a very modest result of existence of free boundaries under no symmetry hypotheses. The existence of a free boundary can be related to the existence of a specific sub-quadratic feature, a dead core, whose size is also quantified.

Continuation and Bifurcation in Nonlinear PDEs \u2013 Algorithms, Applications, and Experiments

Numerical continuation and bifurcation methods can be used to explore the set of steady and time–periodic solutions of parameter dependent nonlinear ODEs or PDEs. For PDEs, a basic idea is to first convert the PDE into a system of algebraic equations or ODEs via a spatial discretization. However, the large class of possible PDE bifurcation problems makes developing a general and user–friendly software a challenge, and the often needed large number of degrees of freedom, and the typically large set of solutions, often require adapted methods. Here we review some of these methods, and illustrate the approach by application of the package pde2path to some advanced pattern formation problems, including the interaction of Hopf and Turing modes, patterns on disks, and an experimental setting of dead core pattern formation.

Uniqueness and positivity issues in a quasilinear indefinite problem

We consider the problem $$\begin{aligned} (P_\lambda )\quad -\Delta _{p}u=\lambda u^{p-1}+a(x)u^{q-1},\quad u\ge 0\quad \text{ in } \Omega \end{aligned}$$under Dirichlet or Neumann boundary conditions. Here \(\Omega \) is a smooth bounded domain of \({\mathbb {R}}^{N}\) (\(N\ge 1\)), \(\lambda \in {\mathbb {R}}\), \(1<q<p\), and \(a\in C({\overline{\Omega }})\) changes sign. These conditions enable the existence of dead core solutions for this problem, which may admit multiple nontrivial solutions. We show that for \(\lambda <0\) the functional $$\begin{aligned} I_{\lambda }(u):=\int _{\Omega }\left( \frac{1}{p}|\nabla u|^{p}-\frac{\lambda }{p}|u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \end{aligned}$$defined in \(X=W_{0}^{1,p}(\Omega )\) or \(X=W^{1,p}(\Omega )\), has exactly one nonnegative global minimizer, and this one is the only solution of \((P_{\lambda })\) being positive in \(\Omega _{a}^{+}\) (the set where \(a>0\)). In particular, this problem has at most one positive solution for \(\lambda <0\). Under some condition on a, the above uniqueness result fails for some values of \(\lambda >0\) as we obtain, besides the ground state solution, a second solution positive in \(\Omega _{a}^{+}\). We also provide conditions on \(\lambda \), a and q such that these solutions become positive in \(\Omega \), and analyze the formation of dead cores for a generic solution.

Non-existence of dead cores in fully nonlinear elliptic models

We investigate non-existence of non-negative dead core solutions for the problem [Formula: see text] Here, [Formula: see text] is a bounded smooth domain, [Formula: see text] is a fully nonlinear elliptic operator, [Formula: see text] is a sign-changing weight, [Formula: see text], and [Formula: see text]. We show that this problem has no nontrivial dead core solutions if either [Formula: see text] is close enough to [Formula: see text] or the negative part of [Formula: see text] is sufficiently small. In addition, we obtain the existence and uniqueness of a positive solution under these conditions on [Formula: see text] and [Formula: see text]. Our results extend previous ones established in the semilinear case, and are new even for the simple model [Formula: see text], where [Formula: see text] is a uniformly elliptic and non-negative matrix.

Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a subcontinuum

Let $$\Omega \subset {\mathbb {R}}^{N}$$ ( $$N\ge 1$$ ) be a smooth bounded domain, $$a\in C({\overline{\Omega }})$$ a sign-changing function, and $$0\le q<1$$ . We investigate the Robin problem where $$\alpha \in [-\infty ,\infty )$$ and $$\nu $$ is the unit outward normal to $$\partial \Omega $$ . Due to the lack of strong maximum principle structure, this problem may have dead core solutions. However, for a large class of weights a we recover a positivity property when q is close to 1, which considerably simplifies the structure of the solution set. Such property, combined with a bifurcation analysis and a suitable change of variables, enables us to show the following exactness result for these values of q: $$(P_{\alpha })$$ has exactly one nontrivial solution for $$\alpha \le 0$$ , exactly two nontrivial solutions for $$\alpha >0$$ small, and no such solution for $$\alpha >0$$ large. Assuming some further conditions on a, we show that these solutions lie in a subcontinuum. These results partially rely on (and extend) our previous work (Kaufmann et al. in J Differ Equ 263:4481–4502, 2017), where the cases $$\alpha =-\infty $$ (Dirichlet) and $$\alpha =0$$ (Neumann) have been considered. We also obtain some results for arbitrary $$q\in \left[ 0,1\right) $$ . Our approach combines mainly bifurcation techniques, the sub-supersolutions method, and a priori lower and upper bounds.

Fully nonlinear parabolic dead core problems

Fully nonlinear parabolic dead core problems

Dead-Core Solutions to Simple Catalytic Reaction Problems in Chemical Engineering

The catalytic chemical reaction is usually carried out in a pellet where the catalyst is distributed throughout its porous structure. The selectivity, yield and productivity of the catalytic reactor often depend on the rates of chemical reactions and the rates of diffusion of species involved in the reactions in the pellet porous space. In such systems, the fast reaction can lead to the consumption of reactants close to the external pellet surface and creation of the dead core where no reaction occurs. This will result in an inefficient use of expensive catalyst. In the discussed simplified diffusion-reaction problems a nonlinear reaction term is of power-law type with a small positive reaction exponent. Such reaction term represents the kinetics of catalytic reaction accompanied by a strong adsorption of the reactant. The ways to calculate the exact solutions possessing dead cores are presented. It was also proved analytically that the exact solution of the nonlinear two-point boundary value problem satisfies physical a-priori bounds. Furthermore, the approximate solutions were obtained using the orthogonal collocation method for pellets of planar, spherical and cylindrical geometries. Numerical results confirmed that the length of the dead core increases for the more active catalysts due to the larger values of the reaction rate constant. The dead core length also depends on the pellet geometry.

Regularity properties for p−dead core problems and their asymptotic limit as p→∞

Fil: Da Silva, Joao Vitor. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matematica; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

Sharp regularity estimates for quasi-linear elliptic dead core problems and applications

In this manuscript we study geometric regularity estimates for quasi-linear elliptic equations of p-Laplace type ( $$1< p< \infty $$ ) with strong absorption condition: $$\begin{aligned} -\mathrm {div}(\Phi (x, u, \nabla u)) + \lambda _0(x) u_{+}^q(x) = 0 \quad \hbox {in} \quad \Omega \subset \mathbb {R}^N, \end{aligned}$$ where $$\Phi : \Omega \times \mathbb {R}_{+} \times \mathbb {R}^N \rightarrow \mathbb {R}^N$$ is a vector field with an appropriate p-structure, $$\lambda _0$$ is a non-negative and bounded function and $$0\le q<p-1$$ . Such a model permits existence of solutions with dead core zones, i.e, a priori unknown regions where non-negative solutions vanish identically. We establish sharp and improved $$C^{\gamma }$$ regularity estimates along free boundary points, namely $$\mathfrak {F}_0(u, \Omega ) = \partial \{u>0\} \cap \Omega $$ , where the regularity exponent is given explicitly by $$\gamma = \frac{p}{p-1-q} \gg 1$$ . Some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density and porosity of free boundary are proved. As an application, a Liouville-type result for entire solutions is established provided that their growth at infinity can be controlled in an appropriate manner. Finally, we obtain finiteness of $$(N-1)$$ -Hausdorff measure of free boundary for a particular class of dead core problems. The approach employed in this article is novel even to dead core problems governed by the p-Laplace operator $$-\Delta _p u + \lambda _0 u^q\chi _{\{u>0\}} = 0$$ for any $$\lambda _0>0$$ .

Existence and regularity of nonlinear advection problems

This paper examines the existence and regularity of classical positive solutions of −Δu=|∇u|p on a bounded domain in RN with 0<p<1. This appears to be the first paper to discuss dead core solutions for a PDE with a nonlinear advection term. We also give a Liouville-type result for supersolutions on an exterior domain, asymptotics of solutions as p↗1 and p↘0, as well as various extensions of the PDE.

Modeling de novo granulation of anaerobic sludge

BackgroundA unique combination of mechanical, physiochemical and biological forces influences granulation during processes of anaerobic digestion. Understanding this process requires a systems biology approach due to the need to consider not just single-cell metabolic processes, but also the multicellular organization and development of the granule.ResultsIn this computational experiment, we address the role that physiochemical and biological processes play in granulation and provide a literature-validated working model of anaerobic granule de novo formation. The agent-based model developed in a cDynoMiCs simulation environment successfully demonstrated a de novo granulation in a glucose fed system, with the average specific methanogenic activity of 1.11 ml CH4/g biomass and formation of a 0.5 mm mature granule in 33 days. The simulated granules exhibit experimental observations of radial stratification: a central dead core surrounded by methanogens then encased in acidogens. Practical application of the granulation model was assessed on the anaerobic digestion of low-strength wastewater by measuring the changes in methane yield as experimental configuration parameters were systematically searched.ConclusionsIn the model, the emergence of multicellular organization of anaerobic granules from randomly mixed population of methanogens and acidogens was observed and validated. The model of anaerobic de novo granulation can be used to predict the morphology of the anaerobic granules in a alternative substrates of interest and to estimate methane potential of the resulting microbial consortia. The study demonstrates a successful integration of a systems biology approach to model multicellular systems with the engineering of an efficient anaerobic digestion system.

Occurrence of dead core in catalytic particles containing immobilized enzymes: analysis for the Michaelis-Menten kinetics and assessment of numerical methods.

In this article, the occurrence of dead core in catalytic particles containing immobilized enzymes is analyzed for the Michaelis-Menten kinetics. An assessment of numerical methods is performed to solve the boundary value problem generated by the mathematical modeling of diffusion and reaction processes under steady state and isothermal conditions. Two classes of numerical methods were employed: shooting and collocation. The shooting method used the ode function from Scilab software. The collocation methods included: that implemented by the bvode function of Scilab, the orthogonal collocation, and the orthogonal collocation on finite elements. The methods were validated for simplified forms of the Michaelis-Menten equation (zero-order and first-order kinetics), for which analytical solutions are available. Among the methods covered in this article, the orthogonal collocation on finite elements proved to be the most robust and efficient method to solve the boundary value problem concerning Michaelis-Menten kinetics. For this enzyme kinetics, it was found that the dead core can occur when verified certain conditions of diffusion-reaction within the catalytic particle. The application of the concepts and methods presented in this study will allow for a more generalized analysis and more accurate designs of heterogeneous enzymatic reactors.

Parabolic Minkowski convolutions of solutions to parabolic boundary value problems

We introduce a new kind of convolution, which is a sort of parabolic version of the classical supremal convolution of convex analysis. This operation allows us to compare solutions of different parabolic problems in different domains. As examples of applications of our main result, we study the parabolic concavity of solutions to parabolic boundary value problems, analyzing in particular the case of heat equation with an inhomogeneous term and with a nonlinear reaction term. We also apply our technique to the study of the dead core problem obtaining new results about necessary conditions for the existence of a dead core and estimates of the dead core time, proving some optimality of the ball.

Physiological mechanism of the overproduction of ε-poly-L-lysine by acidic pH shock in fed-batch fermentation.

The introduction of an environmental stress of acidic pH shock had successfully solved the common deficiency existed in ε-PL production, viz. the distinct decline of ε-PL productivity in the feeding phase of the fed-batch fermentation. To unravel the underlying mechanism, we comparatively studied the physiological changes of Streptomyces sp. M-Z18 during fed-batch fermentations with the pH shock strategy (PS) and pH non-shock strategy (PNS). Morphology investigation showed that pellet-shape change was negligible throughout both fermentations. In addition, the distribution of pellet size rarely changed in the PS, whereas pellet size and number decreased substantially with time in the PNS. This was consistent with the performances of ε-PL productivity in both strategies, demonstrating that morphology could be used as a predictor of ε-PL productivity during fed-batch fermentation. Furthermore, a second growth phase happened in the PS after pH shock, followed by the re-appearance of live mycelia in the dead core of the pellets. Meanwhile, mycelia respiration and key enzymes in the central metabolic and ε-PL biosynthetic pathways were overall strengthened until the end of the fed-batch fermentation. As a result, the physiological changes induced by the acidic pH shock have synergistically and permanently contributed to the stimulation of ε-PL productivity. However, this second growth phase and re-appearance of live mycelia were absent in the PNS. These results indicated that the introduction of a short-term suppression on mycelia physiological metabolism would guarantee the long-term high ε-PL productivity.

Distribution of live and dead cells in pellets of an actinomycete Amycolatopsis balhimycina and its correlation with balhimycin productivity

Secondary metabolites such as antibiotics are typically produced by actinomycetes as a response to growth limiting stress conditions. Several studies have shown that secondary metabolite production is correlated with changes observed in actinomycete pellet morphology. Therefore, we investigated the correlation between the production of balhimycin and the spatio-temporal distribution of live and dead cells in pellets of Amycolatopsis balhimycina in submerged cultures. To this end, we used laser scanning confocal microscopy to analyze pellets from balhimycin producing and nonproducing media containing 0.2 and 1.0gl(-1) of potassium di-hydrogen phosphate, respectively. We observed a substantially higher fraction of live cells in pellets from cultures yielding larger amounts of balhimycin. Moreover, in media that resulted in no balhimycin production, the pellets exhibit an initial death phase which commences from the centre of the pellet and extends in the radial direction. A second growth phase was observed in these pellets, where live mycelia are seen to appear in the dead core of the pellets. This secondary growth was absent in pellets from media producing higher amounts of balhimycin. These results suggest that distribution of live and dead cells and its correlation with antibiotic production in the non-sporulating A. balhimycina differs markedly than that observed in Streptomycetes.

A new idea about reducing reperfusion injury in ischemic stroke: Gradual reperfusion

Around the world, stroke is the second most common cause of death and a major cause of disability. The main direct cause of stroke is the occlusion of intracranial artery, which leads to cell death in the core suffered region, or cell functional impairment surrounding the dead core (termed ischemic penumbra). Opening the occluded artery to save the ischemic penumbra is the aim of thrombolysis therapy. But the reperfusion induced injury counteracts the potential profit by thrombolysis. Herein, we assume that gradual reperfusion can reduce the reperfusion injury by reducing the production of free radicals during reperfusion. The reason is: free radicals are critical in the reperfusion injury; free radicals come from the penumbra during reperfusion; the respiratory chain is the main source of free radical; the enzyme activity of the respiratory chain is upgraded during ischemia; once reperfused, the activity upgraded enzymes in the respiratory chain meet normal amount of oxygen and glucose, which produces exceeding intermediates (free radicals); while gradual reperfusion reduces the production of free radicals, because it can confine the amount of oxygen and glucose.

Are Exhalation Valves on N95 Filtering Facepiece Respirators Beneficial at Low-Moderate Work Rates: An Overview

Exhalation valves (EVs) are touted as useful in dissipating humidity, heat, and carbon dioxide from the dead space of N95 filtering facepiece respirators and decreasing exhalation resistance, thereby making the respirator more comfortable and less physiologically demanding. Despite decades of use, there is limited research on the psychophysiological impact of EVs on the wearer at the current, low-moderate work rates of many workers. The available literature indicates that past and current EVs on the market decrease exhalation resistance to a variable degree and may improve dead space heat dissipation and, consequently, the temperature of the skin covered by the respirator. EVs have little substantial effect on dead space humidity, respiratory rate, heart rate, core temperature, speech intelligibility, or dead space oxygen and carbon dioxide levels at the aforementioned work rates. The studies also indicate that EVs may impact comfort and tolerance when N95 filtering facepiece respirators are worn for extended periods or at high work rates. Because comfort and tolerance impact respirator use compliance and, by extension, protection, more research into the psychophysiological impact of EVs on wearers and the development of new EVs tailored for low-moderate work rates are warranted.

Transformation thermodynamics: cloaking and concentrating heat flux

We adapt tools of transformation optics, governed by a (elliptic) wave equation, to thermodynamics, governed by the (parabolic) heat equation. We apply this new concept to an invibility cloak in order to thermally protect a region (a dead core) and to a concentrator to focus heat flux in a small region. We finally propose a multilayered cloak consisting of 20 homogeneous concentric layers with a piecewise constant isotropic diffusivity working over a finite time interval (homogenization approach).