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We consider a inhomogeneous semilinear wave equation on a noncompact complete Riemannian manifold (Formula presented.) of dimension (Formula presented.), 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 (Formula presented.), our result improves the existing literature. Namely, our main result is valid without assuming that the initial values are compactly supported.

Let (Z n) n≥0 with Z n = (Z n (i, j)) 1≤i,j≤p be a p multi-type critical branching process in random environment, and let M n be the expectation of Z n given a fixed environment. We prove theorems on convergence in distribution of sequences of branching processes Zn |Mn| /|Z n | > 0 and ln Zn √ n /|Z n | > 0. These theorems extend similar results for single-type critical branching process in random environment.; Soit (Z n) n≥0 , avec Z n = (Z n (i, j)) 1≤i,j≤p, un processus de branchement multi-type et critique en environnement aléatoire. Soit M n la matrice moyenne de Z n à environnement fixé. Nous démontrons des théorèmes de convergence en distribution des suites (Zn/ |Mn| ) et ( Zn /√ n) conditionnellement à la non extinction (|Z n | > 0). Ces théorèmes étendent dans le cadre multi-type des résultats antérieurs connus pour des processus uni-type.

We establish weak well-posedness for critical symmetric stable driven SDEs in R d with additive noise Z, d $\ge$ 1. Namely, we study the case where the stable index of the driving process Z is $\alpha$ = 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 Zt = (Z 1 t ,. .. , Z d t) 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. 1. Statement of the problem and main results We are interested in proving well-posedness for the martingale problem associated with the following SDE: (1.1) X t = x + t 0 b(X s)ds + Z t , where (Z s) s$\ge$0 stands for a symmetric d-dimensional stable process of order $\alpha$ = 1 defined on some filtered probability space ($\Omega$, F, (F t) t$\ge$0 , P) (cf. [2] and the references therein) under the sole assumptions of continuity and boundedness on the vector valued coefficient b: (C) The drift b : R d $\rightarrow$ R d is continuous and bounded. 1 Above, the generator L of Z writes: L$\Phi$(x) = p.v. R d \{0} [$\Phi$(x + z) -- $\Phi$(x)]$\nu$(dz), x $\in$ R d , $\Phi$ $\in$ C 2 b (R d), $\nu$(dz) = d$\rho$ $\rho$ 2$\mu$ (d$\theta$), z = $\rho$$\theta$, ($\rho$, $\theta$) $\in$ R * + x S d--1. (1.2) (here $\times$, $\times$ (or $\times$) and | $\times$ | denote respectively the inner product and the norm in R d). In the above equation, $\nu$ is the L{\'e}vy intensity measure of Z, S d--1 is the unit sphere of R d and$\mu$ is a spherical measure on S d--1. It is well know, see e.g. [20] that the L{\'e}vy exponent $\Phi$ of Z writes as: (1.3) $\Phi$($\lambda$) = E[exp(i $\lambda$, Z 1)] = exp -- S d--1 | $\lambda$, $\theta$ |$\mu$(d$\theta$) , $\lambda$ $\in$ R d , where $\mu$ = c 1$\mu$ , for a positive constant c 1 , is the so-called spectral measure of Z. We will assume some non-degeneracy conditions on $\mu$. Namely we introduce assumption (ND) There exists $\kappa$ $\ge$ 1 s.t. (1.4) $\forall$$\lambda$ $\in$ R d , $\kappa$ --1 |$\lambda$| $\le$ S d--1 | $\lambda$, $\theta$ |$\mu$(d$\theta$) $\le$ $\kappa$|$\lambda$|. 1 The boundedness of b is here assumed for technical simplicity. Our methodology could apply, up to suitable localization arguments, to a drift b having linear growth.

In [8], the existence of the solution is proved for a scalar linearly growingbackward stochastic differential equation (BSDE) when the terminal value is$L\exp (\mu \sqrt{2\log (1+L)} )$-integrable for a positive parameter $\mu >\mu _{0}$ with a critical value $\mu _{0}$, and a counterexample is provided to show that the preceding integrability for $\mu \mu _{0}$) is also given in [3] for the preceding BSDE under the uniformly Lipschitz condition of the generator. In this note, we prove that these two results still hold for the critical case: $\mu =\mu _{0}$.

Conditioned on the generating functions of offspring distribution, we study the asymp-totic behaviour of the probability of non-extinction of a critical multi-type Galton-Watson process in i.i.d. random environments by using limits theorems for products of positive random matrices. Under some certain assumptions, the survival probability is proportional to 1/ √ n.

Special issue; International audience; In [3], the authors proved that uniqueness holds among solutions whose exponentials are $L^p$ with $p$ bigger than a constant $\gamma$ ($p>\gamma$). In this paper, we consider the critical case: $p=\gamma$. We prove that the uniqueness holds among solutions whose exponentials are $L^\gamma$ under the additional assumption that the generator is strongly convex.

The periodic quasilinear system of differential equations with small parameter and piecewise constant argument of generalized type [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA, 66 (2007) 367-383, M.U. Akhmet, On the reduction principle for differential equations with piecewise argument of generalized type, J. Math. Anal. Appl. 336 (2007) 646-663] is addressed. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. One of the main auxiliary results of our paper is an analogue of Gronwall-Bellman Lemma for functions with piecewise constant and retarded-advanced type arguments. Dependence of solutions on the parameter is investigated. Appropriate examples are given to show our results. (c) 2008 Elsevier Ltd. All rights reserved.

In this paper, we study the conditions under which the zero solution is stable in the semi- liner case for certain third order differential equation of the form : Where , s = 1,2,3 , , , The characteristic equation of the above differential equation has complex roots of the form : and the other root has the following , . property

In this paper, we consider a kind of neutral functional differential equation as follows: x(t)−cx(t−τ)′=gt,xt−μ(t)+e(t) in the critical case |c|=1. By employing the topological degree theory and some analysis techniques, we obtain some new results on the existence of periodic solutions.

We investigate the statistics and dynamics of failure in a two-dimensional load-bearing network with branching hierarchical structure, and its variants. The variants strengthen the original lattice by using connectivity strategies which add new sites to the maximal cluster in top-to-bottom or bottom-to-top versions. We study the load-bearing capacity and the failure tolerance of all versions, as well as that of the strongest realization of the original lattice, the V lattice. The average number of failures as a function of the test load shows power-law behavior with power 5/2 for the V lattice, but sigmoidal behavior for all other versions. Thus the V lattice turns out to be the critical case of the load-bearing lattices. The distribution of failures is Gaussian for the original lattice, the V lattice, and the bottom-to-top strategy, but is non-Gaussian for the top-to-bottom one. The bottom-to-top strategy leads to stable and strong lattices, and can resist failure even when tested with weights which greatly exceed the capacity of its backbone. We also examine the behavior of asymmetric lattices and discover that the mean failure rates are minimized if the probability of connection p is symmetric with respect to both neighbors. Our results can be of relevance in the context of realistic networks. � 2011 American Physical Society.

In this paper, for the “critical case” with two delays, we establish two relations between any two solutions y ( t ) and y ∗ ( t ) for the Volterra integral equation of non-convolution type y ( t ) = f ( t ) + ∫ t − τ t − δ k ( t , s ) g ( y ( s ) ) d s and a solution z ( t ) of the first order differential equation z ( t ) = β ( t ) [ z ( t − δ ) − z ( t − τ ) ] , and offer a sufficient condition that lim t → + ∞ ( y ( t ) − y ∗ ( t ) ) = 0 .

We obtain coefficient conditions of existence and an iterational algorithm for constructing periodic solutions of weakly perturbed autonomous nonlinear differential systems in critical cases in the presence of multiple roots for generating amplitudes., Отримано коефіцієнтні умови існування та ітераційний алгоритм побудови періодичних розв'язків слабозбурених автономних нелінійних диференціальних систем в критичних випадках при кратних коренях рівняння для породжуючих амплітуд.

A scalar linear differential equation with time-dependent delay x@?(t)=-a(t)x(t-@t(t)) is considered, where t@?I@?[t"0,~), t"0@?R, a:I->R^+@?(0,~) is a continuous function and @t:I->R^+ is a continuous function such that t-@t(t)>t"0-@t(t"0) if t>t"0. The goal of our investigation is to give sufficient conditions for the existence of positive solutions as t->~ in the critical case in terms of inequalities on a and @t. A generalization of one known final (in a certain sense) result is given for the case of @t being not a constant. Analysing this generalization, we show, e.g., that it differs from the original statement with a constant delay since it does not give the best possible result. This is demonstrated on a suitable example.