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We study the following version of Hardy-type inequality on a domain Ω in a Riemannian manifold $ (M,g) $ (M , g) : $$\begin{align*} &\int_{\Omega}|\nabla u|_g^p\rho^\alpha dV_g \geq \left(\frac{|p-1+\beta|}{p}\right)^p\int_{\Omega}\frac{|u|^p|\nabla \rho|_g^p}{|\rho|^p}\rho^\alpha dV_g\\ &\quad+\int_{\Omega} V|u|^p\rho^\alpha dV_g, \quad \forall\ u\in C_c^\infty (\Omega). \end{align*} $$ ∫ Ω | ∇ u | g p ρ α d V g ≥ ( | p − 1 + β | p) p ∫ Ω | u | p | ∇ρ | g p | ρ | p ρ α d V g + ∫ Ω V | u | p ρ α d V g , ∀ u ∈ C c ∞ (Ω). We provide sufficient conditions on $ p, \alpha, \beta,\rho $ p , α , β , ρ and V for which the above inequality holds. This generalizes earlier well-known works on Hardy inequalities on Riemannian manifolds. The functional setup covers a wide variety of particular cases, which are discussed briefly: for example, $ \mathbb {R}^N $ R N with p

The present paper studies the homogenization of the parabolic equation stated in a domain perforated by "tiny" balls. On the boundary of these perforations, a unilateral dynamic boundary constraints are specified. We address the so-called "critical" case that is characterized by a relation between the coefficient in the boundary condition, the period of the structure and the size of the holes. In this case, the homogenized equation contains a nonlocal "strange" term. This term is obtained as a solution to the variational problem involving an ordinary differential operator. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a multitype branching process with random migration following the research initiated with the Galton–Watson process with migration introduced in [N. M. Yanev and K. V. Mitov, Controlled branching processes: The case of random migration, C. R. Acad. Bulgare Sci. 33 1980, 4, 473–475]. We focus our attention in what we call the critical case. Sufficient conditions are provided for the process to have unlimited growth or not. Furthermore, using suitable normalizing sequences, we study the asymptotic distribution of the process. Finally, we obtain a Feller-type diffusion approximation. [ABSTRACT FROM AUTHOR]

Constructive solvability conditions and a scheme for constructing solutions of an autonomous nonlinear boundary value problem unsolved with respect to the derivative have been found. A convergent iterative scheme for Finding approximations to the solutions of a nonlinear autonomous nonlinear boundary value problem unsolved with respect to the derivative was constructed by reducing the problem to the First-order critical case. As an example of application of the constructed iterative scheme, approximations to the solutions of a periodic boundary value problem for the autonomous Lotka–Volterra equation were determined. [ABSTRACT FROM AUTHOR]

In this paper, we address the stabilization problem for nonlinear systems in a critical case. Namely, we study the class of canonical nonlinear systems. Canonical nonlinear systems or chain of power integrators is an important subject of research. Studying such systems is complicated by the fact that they cannot be mapped onto linear systems. Moreover, they have the uncontrollable first approximation. Previous results on smooth stabilization of such systems were obtained under the assumption that the powers in the right-hand side are strictly decreasing. In this work, we consider a case of non-increasing powers in the right-hand side for a three-dimensional system. A popular approach for studying such systems is the backstepping method, which is a method of step-wise stabilization. This method requires a sequential investigation of lower-dimensional subsystems. Backstepping enables the study of a wide range of nonlinear triangular systems but requires technically complex and cumbersome computations. Therefore, a natural question arises about constructing stabilizing controls of a simple form. Polynomial controls can serve as an example of such controls. In the paper, we demonstrate that linear controls can be considered as stabilizing controls. We derive sufficient conditions for the coefficients of the linear control that ensure the asymptotic stability of the zero equilibrium point of the corresponding closed-loop system. The asymptotic stability is proven using the Lyapunov function method, which is found as the sum of squares. The negative definiteness of the Lyapunov function derivative in a neighborhood of the origin guarantees asymptotic stability. In contrast to the case of strictly decreasing powers, additional conditions on the control coefficients, apart from their negativity, emerge. The obtained result extends to a broader class of nonlinear systems through stabilization by nonlinear approximation. This allows the consideration of systems with higher-order terms in the right-hand side. The effectiveness of the applied approach is illustrated by several model examples. The method used in this work to investigate the case of non-increasing powers can be applied to systems of higher dimensions.

V.F. Butuzov and N.N. Nefedov proposed an algorithm for constructing asymptotics with boundary functions of two types for solving a discrete initial value problem with a small step and a nonlinear term of order in the critical case, i.e., when the degenerate equation with is not solvable uniquely for the unknown variable. In this paper, an asymptotic solution of the same problem is constructed by applying a new approach based on orthogonal projectors onto and , where is the matrix premultiplying the unknown variable in the linear part of the equation, is the identity matrix of suitable size, and the prime denotes transposition. This approach considerably simplifies the understanding of the asymptotics-constructing algorithm and makes it possible to represent the problems of finding asymptotic terms of any order in explicit form, which is convenient for researchers applying asymptotic methods for real-world problems. [ABSTRACT FROM AUTHOR]

This paper discusses a class of nonlinear singular perturbation problems with Robin boundary values in critical cases. By using the boundary layer function method and successive approximation theory, the corresponding asymptotic expansions of small parameters are constructed, and the existence of uniformly efficient smooth solutions is proved. Meanwhile, we give a concrete example to prove the validity of our results.

We consider a linear–quadratic control problem in which there is the second power of a small parameter at the derivative of the state variable and the first power of the parameter both in the control term of the state equation and at the quadratic form with respect to the control variable in the performance index; moreover, the state equation represents a critical case of singular perturbation theory. A zero-order asymptotic expansion of the solution is constructed using the so-called direct scheme method in which a postulated asymptotic expansion of the solution is substituted directly into the problem statement and problems for finding the asymptotic terms are stated. [ABSTRACT FROM AUTHOR]

In this paper, we prove some blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime by using an iteration argument, and we derive upper bound estimates for the lifespan. In particular, we will focus on the critical cases which require the employment of a slicing procedure in the iterative mechanism. Furthermore, in order to deal with the main critical case, we will introduce a non-autonomous and parameter-dependent Cauchy problem for a linear ODE of second-order, whose explicit solution will be determined by applying the theory of special functions. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in computing the minimal positive solution of an algebraic Riccati equation arising in transport theory. The coefficient matrices of this equation have two parameters α and c. When 0 < α < 1 , 0 < c < 1 , the existing numerical algorithms can solve its minimal positive solution very quickly and efficiently. However, these algorithms do not converge or converge slowly to the minimal positive solution for α = 0 and c = 1 . In this case, we propose two methods to improve the effectiveness of these algorithms. The first method is a modified double shift technique to transform the original algebraic Riccati equation into a new one, and the two equations have the same minimal positive solution. The second method is a deflating technique to transform the original equation into a new low-order algebraic Riccati equation, and we give the relationship between the minimal positive solutions of these two equations. The minimal positive solutions of two new equations both can be effectively solved by the existing algorithms. We show that two new equations both maintain the symmetry. Moreover, two new equations are M-matrix algebraic Riccati equations. Numerical experiments are given to illustrate the results. [ABSTRACT FROM AUTHOR]

Abstract We investigate the existence of solutions of weakly nonlinear periodic boundary value problems for systems of ordinary differential equations with switchings and the construction of these solutions. We consider the critical case where the equation for the generating constants of a weakly nonlinear periodic boundary-value problem with switchings does not turn into an identity. We improve the classification of critical and noncritical cases and construct an iterative algorithm for finding solutions of weakly nonlinear periodic boundary value problems with switchings in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary value problem for the mathematical model of nonisothermal chemical reactions. To check the accuracy of the proposed approximations, we evaluate discrepancies in the original equation.

ObjectiveTo analyze influence factors of severe/critical conditions among elderly patients ( ≥ 60 years) infected with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) Omicron variant. MethodsMedical records of all elderly patients (≥ 60 years) with SARS-CoV-2 Omicron variant infection during April 6 – June 12, 2022 were extracted from a designated hospital for coronavirus disease 2019 (COVID-19) in Shanghai city and the patients were assigned into different severity groups according to their clinical conditions. Univariate and multivariate logistic regression analysis were adopted in analysis on factors associated with clinical severity in the patents. ResultsOf the 625 patients included in the study, 455, 125, and 45 were mild, common, and severe/critical cases. In the patients, the proportion of severe/critical cases increased significantly by age, with the proportions of 2.99% (7/234), 7.34% (13/177), 9.03% (13/144), and 17.14% (12/70) for the patients aged 60 – 69, 70 – 79, 80 – 89, ≥ 90 years (χ2trend = 8.288, P = 0.004); the proportion also significantly increased inversely with the dose number of COVID-19 vaccination, with the proportions of 2.33% (1/43), 1.35% (1/74), 6.98% (3/43), and 8.60% (40/465) for the patients having 3, 2, 1, and zero dose vaccine, respectively (χ2trend = 10.201, P = 0.001). The results of multivariate logistic regression analysis showed that at older age (odds ratio [OR] = 1.606, 95% confidence interval [95% CI]: 1.177 – 2.192) and suffering from diabetes (OR = 1.916, 95% CI: 1.030 – 3.565) were independent risk factors for severe/critical conditions. ConclusionFor SARS-CoV-2 Omicron variant infected patients over 60 years old, advanced age and having diabetes are risk factors for severe/critical conditions and vaccination could effectively reduce that risk.

We investigate the existence of solutions of weakly nonlinear periodic boundary value problems for systems of ordinary differential equations with switchings and the construction of these solutions. We consider the critical case where the equation for the generating constants of a weakly nonlinear periodic boundary-value problem with switchings does not turn into an identity. We improve the classification of critical and noncritical cases and construct an iterative algorithm for finding solutions of weakly nonlinear periodic boundary value problems with switchings in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary value problem for the mathematical model of nonisothermal chemical reactions. To check the accuracy of the proposed approximations, we evaluate discrepancies in the original equation. [ABSTRACT FROM AUTHOR]

We consider a inhomogeneous semilinear wave equation on a noncompact complete Riemannian manifold (M , g) of dimension N ≥ 3 , without boundary. The reaction exhibits the combined effects of a critical term and of a forcing term. Using a rescaled test function argument together with appropriate estimates, we show that the equation admits no global solution. Moreover, in the special case when M = R 3 , our result improves the existing literature. Namely, our main result is valid without assuming that the initial values are compactly supported. [ABSTRACT FROM AUTHOR]

Formal asymptotic expansions of the solution to the Cauchy problem for a singularly perturbed operator differential transport equation with weak diffusion and small nonlinearity are constructed in the critical case. Under certain conditions imposed on the data of the problem, an asymptotic expansion of the solution is constructed in the form of series in powers of a small parameter with coefficients depending on stretched variables. Problems for determining all terms of the asymptotic expansion are obtained. It is shown that the leading term of the solution asymptotics is determined by solving Cauchy problems for a parabolic Burgers-type equation and, under certain conditions, for a Korteweg–de Vries–Burgers type equation. The remainder terms are estimated with respect to the residual. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of singularly perturbed equations with discontinuous right-hand side in the critical case. An asymptotic expansion of a contrast structure solution for the system is contrasted. Moreover, results for existence of the solution and estimations of remainders are presented. Finally, an example is provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper starts from several applications described by initial/boundary value problems for \(1D\) (time and one space variable) hyperbolic partial differential equations whose basic properties and stability of equilibria are studied throughout the same properties for certain associated neutral functional differential equations. It is a common fact that asymptotic stability for neutral functional differential equations is normally obtained under the assumption of asymptotic stability of the difference operator associated to the aforementioned neutral functional differential equations. However the physically meaningful applications presented in the paper have the associated difference operator(s) in critical cases (their stability is, generally speaking, non-asymptotic). Consequently the stability of the considered application models is either non-asymptotic or fragile (in a sense introduced in the paper). The models represent an overview gathered from various fields, processed here in order to emphasize the associated neutral functional differential equations which, consequently, are a challenge to the usual approaches. In the concluding part there are suggested possible ways to overcome these difficulties.

We found constructive conditions of solvability and the scheme for constructing solutions of the nonlinear autonomous boundary-value problem in the case of multiple solutions of the equation for generating constants. The convergent iterative scheme was constructed for finding approximations to solutions of the nonlinear autonomous boundary-value problem for the system of ordinary differential equations in the case of multiple solutions of the equation for generating constants. We found the approximations to solutions of the periodic boundary-value problem for autonomous equation of Duffing type as an example of applying the constructed iterative scheme. We applied residuals in the original equation for controlling the accuracy of the found approximations to solutions of the periodic boundary-value problem for the autonomous equation of Duffing type. [ABSTRACT FROM AUTHOR]

This paper starts from several applications described by initial/boundary value problems for 1D (time and one space variable) hyperbolic partial differential equations whose basic properties and stability of equilibria are studied throughout the same properties for certain associated neutral functional differential equations. It is a common fact that asymptotic stability for neutral functional differential equations is normally obtained under the assumption of asymptotic stability of the difference operator associated to the aforementioned neutral functional differential equations. However the physically meaningful applications presented in the paper have the associated difference operator(s) in critical cases (their stability is, generally speaking, non-asymptotic). Consequently the stability of the considered application models is either non-asymptotic or fragile (in a sense introduced in the paper). The models represent an overview gathered from various fields, processed here in order to emphasize the associated neutral functional differential equations which, consequently, are a challenge to the usual approaches. In the concluding part there are suggested possible ways to overcome these difficulties. [ABSTRACT FROM AUTHOR]

We mainly study the existence and uniqueness of solutions to one-dimensional backward stochastic differential equations with terminal values satisfying a critical integrability and infinite time horizon. The main result is established under the assumption that the generator f exhibits a time-varying monotonicity in y and uniform continuity in z and the terminal value is required to satisfy the critical L exp (μ 0 2 log (1 + L) ) -integrability for a μ 0 > 0 , a condition stronger than L log L -integrability but weaker than L p (p > 1) -integrability. [ABSTRACT FROM AUTHOR]

Our aim is to study the large time asymptotics of solutions to the fourth-order nonlinear Schrödinger equation in two space dimensions i ∂ t u + 1 4 Δ 2 u = λ | u | 2 u , t > 0 , x ∈ ℝ 2 , u (0 , x) = u 0 (x) , x ∈ ℝ 2 , where λ > 0. We show that the nonlinearity has a dissipative character, so the solutions obtain more rapid time decay rate comparing with the corresponding linear case, if we assume the nonzero total mass condition ∫ ℝ u 0 (x) d x ≠ 0. We continue to develop the factorization techniques. The crucial points of our approach presented here are the L 2 — estimates of the pseudodifferential operators and the application of the Kato–Ponce commutator estimates. [ABSTRACT FROM AUTHOR]

The paper studies the asymptotic behavior of the optimal control for the Poisson type boundary value problem in a domain perforated by holes of an arbitrary shape with Robin-type boundary conditions on the internal boundaries. The cost functional is assumed to be dependent on the gradient of the state and on the usual L2-norm of the control. We consider the so-called "critical" relation between the problem parameters and the period of the structure . Two "strange" terms arise in the limit. The paper extends, by first time in the literature, previous papers devoted to the homogenization of the control problem which always assumed the symmetry of the periodic holes. [ABSTRACT FROM AUTHOR]

Covid 19 pandemic has placed challenges in front of medical health fraternity in terms of management, prevention and immunity building. Effectiveness of any medication has not conclusively proven; hence there is need for integrative management of Covid 19. We have managed a critical case of Covid-19 having history of thalassemia, hypothyroidism with integrative management of Ayurveda and modern medicine. A male patient (59 years of age) with history of thalassemia had complaints of cough and breathlessness since 4 days. He performed RT PCR because of his exposure to a Covid positive cases in immediate family. He was treated with Favipiravir at home for 5 days. He deteriorated on 6th day with SPO2 dropped to 75%, temp raised to 101 F and respiratory rate (RR) raised to 45/min. He was admitted in Yogeshwari Hospital Daund, Maharashtra; treated with oxygen inhalation, Remdesvir and Ayurveda medicines in intensive care unit (ICU). Ayurveda treatment protocol was advised through telemedicine. Significant improvement in clinical symptoms and normal HRCT was observed at completion of treatment. This case report provides further directions for integrative management in cases of Covid 19. Further clinical research studies in this direction are warranted.

The new type of coronavirus epidemic that started in the People's Republic of China in December 2019 turned into a pandemic in March 2020 and, affected many pediatric patients. 2% of the cases in the Republic of China were pediatric patients. Although the symptoms such as headache, sore throat, myalgia, loss of sense of smell and taste, diarrhea are also seen, the most common symptoms of the infection are fever, cough and dyspnea. The disease may progress in a wide range from asymptomatic cases to severe respiratory failure accompanied by pneumonia and death. The incidence of critically ill children is reported as 11%. The World Health Organization reported that of cases diagnosed with COVID-19, critically ill patients are more likely to have myocarditis, Kawasaki syndrome or multiple systemic inflammatory syndrome (MIS-C). In this article, 4-month-old infant with respiratory failure who presented with respiratory failure on 30 March 2020, was diagnosed with severe COVID-19 pneumonia and myocarditis, and treated with the algorithm recommended by Republic of Turkey Ministry of Health, is presented. [ABSTRACT FROM AUTHOR]

We study a continuous-time branching random walk (BRW) on the lattice ℤd, d ∈ ℕ, with a single source of branching, that is the lattice point where the birth and death of particles can occur. The random walk is assumed to be spatially homogeneous, symmetric and irreducible but, in contrast to the majority of previous investigations, the random walk transition intensities a(x, y) decrease as |y − x|−(d+α) for |y − x| → ∞, where α ∈ (0, 2), that leads to an infinite variance of the random walk jumps. The mechanism of the birth and death of particles at the source is governed by a continuous-time Markov branching process. The source intensity is characterized by a certain parameter β. We calculate the long-time asymptotic behaviour for all integer moments for the number of particles at each lattice point and for the total population size. With respect to the parameter β, a non-trivial critical point βc > 0 is found for every d ≥ 1. In particular, if β > βc the evolutionary operator generated a behaviour of the first moment for the number of particles has a positive eigenvalue. The existence of a positive eigenvalue yields an exponential growth in t of the particle numbers in the case β > βc called supercritical. Classification of the BRW treated as subcritical (β < βc) or critical (β = βc) for the heavy-tailed random walk jumps is more complicated than for a random walk with a finite variance of jumps. We study the asymptotic behaviour of all integer moments of a number of particles at any point y ∈ ℤd and of the particle population on ℤd according to the ratio d/α. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of singular perturbed integral initial values and Robin boundary values in the critical case. Based on the boundary layer function method, we not only construct the asymptotic approximation of the original equation, but also prove the uniform validity of the asymptotic solution by successive approximation. At the same time, we give an example to prove the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

Under study is the problem of asymptotic but exponential stability for a class of linear autonomous neutral functional differential equations. We demonstrate that the asymptotic stability of an equation of the class takes place for all integrable initial functions if the roots of the characteristic equation lie on the left of and approach the imaginary axis. [ABSTRACT FROM AUTHOR]

We establish weak well-posedness for critical symmetric stable driven SDEs in ℝ d with additive noise Z , d ≥ 1. Namely, we study the case where the stable index of the driving process Z is α = 1 which exactly corresponds to the order of the drift term having the coefficient b which is continuous and bounded. In particular, we cover the cylindrical case when Z t = (Z t 1 , ... , Z t d) and Z 1 , ... , Z d are independent one-dimensional Cauchy processes. Our approach relies on L p -estimates for stable operators and uses perturbative arguments. [ABSTRACT FROM AUTHOR]

Under some conditions, an asymptotic solution containing boundary functions of two types was constructed by V.F. Butuzov and N.N. Nefedov for initial value problems for differential equations involving the second power of a small parameter multiplying the derivative with a right-hand side consisting of a singular matrix times the unknown function (as a linear part of the equation) plus the same small parameter multiplying a nonlinear function. In the present paper, an algorithm for constructing asymptotics using the orthogonal projectors onto and (the prime denotes transposition) is given. This approach can be useful for understanding the algorithm underlying the construction of asymptotics. It allows us to present formulas for finding asymptotic terms of any order in an explicit form. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of the asymptotic behavior of the optimal control for the boundary value problem in an ε-periodically perforated domain with linear Robin-type boundary condition, when the period of the structure tends to zero, and the problem parameters, diameter of perforations and adsorption coefficient, take critical values. [ABSTRACT FROM AUTHOR]

An asymptotic theory is developed for a class of third-order differential equations. We identify a critical case to obtain the asymptotic form of three linearly independent solutions for large x. [ABSTRACT FROM AUTHOR]

We study the blow up profiles associated to the following second order reaction–diffusion equation with non-homogeneous reaction: ∂ t u = ∂ xx (u m) + | x | σ u , with σ > 0 . Through this study, we show that the non-homogeneous coefficient | x | σ has a strong influence on the blow up behavior of the solutions. First of all, it follows that finite time blow up occurs for self-similar solutions u, a feature that does not appear in the well known autonomous case σ = 0 . Moreover, we show that there are three different types of blow up self-similar profiles, depending on whether the exponent σ is closer to zero or not. We also find an explicit blow up profile. The results show in particular that global blow up occurs when σ > 0 is sufficiently small, while for σ > 0 sufficiently large blow up occurs only at infinity, and we give prototypes of these phenomena in form of self-similar solutions with precise behavior. This work is a part of a larger program of understanding the influence of non-homogeneous weights on the blow up sets and rates. [ABSTRACT FROM AUTHOR]

Summary: We consider a linear‐quadratic optimal control problem where the second power of a small parameter stands in front of the derivative and a control in a state equation and in front of a quadratic form with respect to control in a performance index; moreover, in the state equation, the nonhomogeneity has the first order of the power of a small parameter, and a matrix in front of the state variable is singular if the small parameter is equal to zero. Using immediate substituting a postulated asymptotic expansion of a solution, containing a regular series and four boundary layer functions series, into the problem condition, we obtain problems for finding asymptotic terms of the zero order for the optimal control and the first order for the optimal trajectory. An illustrative example is given. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the blowup phenomena for initial–boundary value problem (0.1) τ ∂ t 2 u (x , t) − Δ u (x , t) + a (x) ∂ t u (x , t) = λ | u (x , t) | p , (x , t) ∈ C Σ × (0 , T) , u (x , t) = 0 , (x , t) ∈ ∂ C Σ × (0 , T) , u (x , 0) = ε f (x) , x ∈ C Σ , τ ∂ t u (x , 0) = τ ε g (x) , x ∈ C Σ , where C Σ is a cone-like domain in R N (N ≥ 2) defined as C Σ = int r ω ∈ R N ; r ≥ 0 , ω ∈ Σ with a connected open set Σ in S N − 1 with smooth boundary ∂ Σ. If N = 1 , then we only consider two cases C Σ = (0 , ∞) and C Σ = R. Here a (x) is a non-zero coefficient of ∂ t u 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. [ABSTRACT FROM AUTHOR]

Abstract We obtain a blow-up result for solutions to a semi-linear wave equation with scale-invariant dissipation and mass and power non-linearity, in the case in which the model has a "wave like" behavior. We perform a change of variables that transforms our starting equation in a strictly hyperbolic semi-linear wave equation with time-dependent speed of propagation. Applying Kato's lemma we prove a blow-up result for solutions to the transformed equation under some assumptions on the initial data. The limit case, that is, when the exponent p is exactly equal to the upper bound of the range of admissible values of p yielding blow-up needs special considerations. In this critical case an explicit integral representation formula for solutions of the corresponding linear Cauchy problem in 1d is derived. Finally, carrying out the inverse change of variables we get a non-existence result for global (in time) solutions to the original model. [ABSTRACT FROM AUTHOR]

For the critical branching random walk on the lattice , in the case of an arbitrary total number of produced offspring spreading on the lattice from the parental particle, the existence of a limit distribution (which corresponds to a steady state (or statistical equilibrium)) of the population is proved. If the second factorial moment of the total number of offspring is much larger than the square of the first factorial moment, then the limit particle field displays strong deviations from the uniformity: this is intermittency. [ABSTRACT FROM AUTHOR]