Rate of Type II blowup for a semilinear heat equation

A solution u of a Cauchy problem for a semilinear heat equation

On Universality of Critical Behavior in the Focusing Nonlinear Schrödinger Equation, Elliptic Umbilic Catastrophe and the Tritronquée Solution to the Painlevé-I Equation

We argue that the critical behavior near the point of “gradient catastrophe” of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation $i\epsilon \varPsi _{t}+\frac{\epsilon^{2}}{2}\varPsi _{xx}+|\varPsi |^{2}\varPsi =0We argue that the critical behavior near the point of “gradient catastrophe” of the solution to the Cauchy problem for the focusing nonlinear Schr?dinger equation , ε ≪1, with analytic initial data of the form is approximately described by a particular solution to the Painlevé-I equation.      

Infinite-order differential equations with analytic solutions

An infinite-order linear differential equation with constant coefficients and characteristic equation of the class [1, 0] is investigated, and a class of solutions is introduced. It is shown that, if the zeros _k = _k+ i _k of the characteristic function satisfy the condition , then all solutions of the class under consideration are analytic functions.Translated from Matematicheskie Zametki, Vol. 10, No. 3, pp. 269–278, September, 1971.     

Blow-up for semilinear parabolic equations with nonlinear memory

In this paper, we consider the semilinear parabolic equation with homogeneous Dirichlet boundary conditions, where p, q are nonnegative constants. The blowup criteria and the blowup rate are obtained.     

New Quasi-Newton Equation and Related Methods for Unconstrained Optimization

In unconstrained optimization, the usual quasi-Newton equation is B _k+1 s _k=y _k, where y _k is the difference of the gradients at the last two iterates. In this paper, we propose a new quasi-Newton equation, , in which is based on both the function values and gradients at the last two iterates. The new equation is superior to the old equation in the sense that better approximates ² f(x _k+1)s _k than y _k. Modified quasi-Newton methods based on the new quasi-Newton equation are locally and superlinearly convergent. Extensive numerical experiments have been conducted which show that the new quasi-Newton methods are encouraging.     

Hilbert metrics and Minkowski norms

It is shown that the Hilbert geometry (D, h_D) associated to a bounded convex domain is isometric to a normed vector space if and only if D is an open n-simplex. One further result on the asymptotic geometry of Hilbert’s metric is obtained with corollaries for the behavior of geodesics. Finally we prove that every geodesic ray in a Hilbert geometry converges to a point of the boundary.     

On a More General Characterisation of Steiner Systems

We introduce [k,d]-sparse geometries of cardinality n, which are natural generalizations of partial Steiner systems PS(t,k;n), with d=2(k−t+1). We will verify whether Steiner systems are characterised in the following way. (*) Let be a [k,2(k−t+1)]-sparse geometry of cardinality n, with k \> t \> 1$$" align="middle" border="0"> . If , then Γ is a S(t,k;n). If (*) holds for fixed parameters t, k and n, then we say S(t,k;n) satisfies, or has, characterisation (*). We could not prove (*) in general, but we prove the Theorems 1, 2, 3 and 4, which state conditions under which (*) is satisfied. Moreover, we verify characterisation (*) for every Steiner system appearing in list of the sporadic Steiner systems of small cardinality, and the list of infinite series of Steiner systems, both mentioned in the latest edition of the book ‘Design Theory’ by T. Beth, D. Jungnickel and H. Lenz. As an interesting application, one can use these results to build (almost) maximal binary codes in the following way. Every [k,d]-sparse geometry is associated with a [k,d]-sparse binary code of the same size (let and link every block with the code word where c_i=1 if and only if the point p_i is a member of B), so one can construct maximal [k,d]-sparse binary codes using (partial) Steiner systems. These [k,d]-sparse codes can then be used as building bricks for binary codes having a bigger variety of weights (the weight of a code word is the sum of its entries).     

Positive solution to semilinear parabolic equation associated with critical Sobolev exponent

We study the semilinear parabolic equation ${u_{t}- \Delta u = u^{p}, u \geq 0}$ on the whole space R ^N , ${N \geq 3}$ associated with the critical Sobolev exponent p = (N + 2)/(N ? 2). Similarly to the bounded domain case, there is threshold blowup modulus concerning the blowup in finite time. Furthermore, global in time behavior of the threshold solution is prescribed in connection with the energy level, blowup rate, and symmetry.     

Nullstellensatz effectif et Conjecture de Serre (Théorème de Quillen-Suslin) pour le Calcul Formel

Let k be an arbitrary field, X₁,….,X_n indeterminates over k and F₁…, F₃ ε ∈ k[X₁…,X_n] polynomials of maximal degree $ d: = \mathop {\max }\limits_{1 \le i \le a} \deg $ (F_i). We give an elementary proof of the following effective Nullstellensatz: Assume that F₁,…,F have no common zero in the algebraic closure of k. Then there exist polynomials P₁…, P₃ ε ∈ k[X₁…,X_n] such that $ 1: = \mathop \Sigma \limits_{1 \le i \le a} $ P_iF_i and This result has many applications in Computer Algebra. To exemplify this, we give an effective quantitative and algorithmic version of the Quillen-Suslin Theorem baaed on our effective Nullstellensatz.     

Local and Global LIPSCHITZian Mappings on Ordered Metric Spaces

We consider the set ?? of nonhomogeneous Markov fields on T = N or T = Z with finite state spaces E_n, n ? T , with fixed local characteristics. For T = N we show that ?? has at most \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop N\nolimits_\infty = \mathop {\lim \inf}\limits_{n \to \infty} \left| {\mathop E\nolimits_n} \right| $\end{document} phases. If T = Z , ?? has at most N_-∞ · N_∞; phases, where \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop N\nolimits_{-\infty} = \mathop {\lim \inf}\limits_{n \to -\infty} \left| {\mathop E\nolimits_n} \right| $\end{document}. We give examples, that for T = N for any number k, 1 ≦ k ≦ N_∞, there are local characteristics with k phases, whereas for T = Z every number l · k, 1 ≦ l ≦ N_-∞, 1 ≦ k ≦ N_∞ occurs. We describe the inner structure of ??, the behaviour at infinity and the connection between the one-sided and the two-sided tail-fields. Simple examples of Markov fields which are no Markov processes are given.     

Refined asymptotics for the blowup of ut — δu = up

This work is concerned with positive, blowing-up solutions of the semilinear heat equation u_t — δu = u^p in Rⁿ. Our main contribution is a sort of center manifold analysis for the equation in similarity variables, leading to refined asymptotics for u in a backward space-time parabola near any blowup point. We also explore a connection between the asymptotics of u and the local geometry of the blowup set.     

Some Extensions and Embeddings of Near Polygons

Let (P, L, *) be a near polygon having s + 1 points per line, s > 1, and suppose k is a field. Let V _k be the k-vector space with basis Then the subspace generated by the vectors , where l L, has codimension at least 2 in V _k.This observation is used in two ways. First we derive the existence of certain diagram geometries with flag transitive automorphism group, and secondly, we show that any finite near polygon with 3 points per line can be embedded in an affine GF(3)-space.     

Orthogonal two-direction multiscaling functions

The concept of a two-direction multiscaling functions is introduced. We investigate the existence of solutions of the two-direction matrix refinable equation

Light paths with an odd number of vertices in polyhedral maps

Let P _k be a path on k vertices. In an earlier paper we have proved that each polyhedral map G on any compact 2-manifold with Euler characteristic contains a path P _k such that each vertex of this path has, in G, degree . Moreover, this bound is attained for k = 1 or k 2, k even. In this paper we prove that for each odd , this bound is the best possible on infinitely many compact 2-manifolds, but on infinitely many other compact 2-manifolds the upper bound can be lowered to .     

Monotone Solutions of a Higher-Order Neutral Difference Equation

A real sequence x _k is said to be (*)-monotone with respect to a sequence p _k and a positive integer if x _k > 0 and . This paper is concerned with the existence of (*)-monotone solutions of a neutral difference equation. Existence criteria are derived by means of a comparison theorem and by establishing explicit existence criteria for positive and/or bounded solutions of a majorant recurrence relation.     

Extended equivariant Picard complexes and homogeneous spaces

Let k be a field of characteristic 0 and let [`(k)] \bar{k} be a fixed algebraic closure of k. Let X be a smooth geometrically integral k-variety; we set [`(X)] = X ×_k[`(k)] \bar{X} = X{ \times_k}\bar{k} and denote by [`(X)] \bar{X} . In [BvH2] we defined the extended Picard complex of X as the complex of Gal( [`(k)]

Solvability and Fredholm Properties of Integral Equations on the Half-Line in Weighted Spaces

The solvability of integral equations of the form and the behaviour of the solution x at infinity are investigated. Conditions on k and on a weight function w are obtained which ensure that the integral operator K with kernel k is bounded as an operator on X_w, where X_w denotes the weighted space of those continuous functions defined on the half-line which are O(w(s)) as We also derive conditions on w and k which imply that the spectrum and essential spectrum of K on X_w are the same as on BC[0,). In particular, the results apply when when the integral equation is of Wiener-Hopf type. In this case we show that our results are particularly sharp.     

The bordism class of manifolds with involution

In this paper, we consider the following question. For some givenn-bordism class and (n–k _i)-bordism classes _n–ki,(11<...<k _i),. under what condition can we find a representationM of and an involutionT onM, such that the bordism class of the fixed point set ofT is . This paper also gives some examples not satisfying the above property.     

Singularities of Steady Axisymmetric Free Surface Flows with Gravity

We consider a steady axisymmetric solution of the Euler equations describing the irrotational flow without swirl of an incompressible inviscid fluid acted on by gravity and with a free surface. We analyze stagnation points as well as points on the axis of symmetry. At points on the axis of symmetry that are not stagnation points, constant velocity motion is the only blowup profile consistent with the invariant scaling of the equation. This suggests the presence of downward‐pointing cusps at those points. At stagnation points on the axis of symmetry, the unique blowup profile consistent with the invariant scaling of the equation is the Garabedian pointed bubble solution with water above air. Thus at stagnation points on the axis of symmetry with no water above the stagnation point, the invariant scaling of the equation cannot be the right scaling. A finer blowup analysis of the velocity field yields that in the case when the surface is described by an injective curve, the velocity field scales almost like and is asymptotically given by with a nonzero constant c₀. The last result relies on a frequency formula in combination with a concentration compactness result for the axially symmetric Euler equations by Delort. While the concentration compactness result alone does not lead to strong convergence in general, we prove the convergence to be strong in our application. © 2014 Wiley Periodicals, Inc.     

Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogeneous density

The following quasilinear parabolic equation with a source term and an inhomogeneous density is considered:

$\rho (x)\frac{{\partial u}}{{\partial t}} = div(u^{m - 1} \left| {Du} \right|^{\lambda - 1} Du) + u^p $

. The conditions on the parameters of the problem are found under which the solution to the Cauchy problem blows up in a finite time. A sharp universal (i.e., independent of the initial function) estimate of the solution near the blowup time is obtained.