We study the blow-up of positive solutions of the Cauchy problem for the semilinear parabolic equation

where u is a scalar function of the spatial variable x ∈ ℝN and time t > 0, a ∈ ℝV, a ≠ 0, 1 < p and 1 ≦ q. We show that: (a) if p > 1 and 1 ≦ q ≦ p, there always exist solutions which blow up in finite time; (b) if 1 < q ≦ p ≦ min {1 + 2/N, 1 + 2q/(N + 1)} or if q = 1 and 1 < p ≦ l + 2/N, then all positive solutions blow up in finite time; (c) if q > 1 and p > min {1 + 2/N, 1 + 2q/N + 1)}, then global solutions exist; (d) if q = 1 and p > 1 + 2/N, then global solutions exist.

In this paper we introduce a new tool in geometric measure theory and then we apply it to study the rank properties of the derivatives of vector functions with bounded variation.

It is shown that for each v ∊ {0, 1, 2}, the class of v-primitive near-rings is not axiomatisable.

The nonuniqueness of solutions of Riemann problems for a system of conservation laws of mixed type which are admissible according to kinetic relation criteria or travelling wave criteria is proved. The above-mentioned admissibility criteria all consist of local restrictions on points of discontinuity of weak solutions. An example is given which has more than one solution admissible by these local admissibility criteria, but has only one solution, the one-phase solution, satisfying the vanishing viscosity criterion. The entropy rate criterion, however, prefers the two-phase solution.

We construct the Conley index for maps. We do not assume any compactness of map or space. We prove the Ważewski property, additivity property and continuation property.

We investigate the existence of positive principal eigenvalues of the problem - ∆u(x) = λg(x)u(x) for x ∈ ℝk where the weight function g changes sign in ℝk and is negative for |x| sufficiently large.

A fundamental comparison theorem is established for general semilinear parabolic systems via the notions of sectorial operators, analytic semigroups and the application of the Tychonoff Fixed Point Theorem. Based on this result, we establish a maximum principle for systems of general parabolic operators and general comparison theorems for parabolic systems with quasimonotone or mixed quasimonotone nonlinearities. These results cover and extend most currently used forms of maximum principles and comparison theorems. A global existence theorem for parabolic systems is derived as an application which, in particular, gives rise to some global existence results for Fujita type systems and certain generalisations.

We consider the diffusion-advection equation ut = uxx + (ε/(1 − u)β)x(ε >0, β >0), 0 < x < 1, t >0, under the boundary conditions ux + ε/(1 − u)β = 0. We prove that there is a critical number ε(β) such that when ε < ε(β) for certain initial data a global solution exists and converges to the corresponding stationary solution; any solution must quench (u reaches one in finite or infinite time) if ε ≧ε(β). We also show that quenching can only occur at x = 0, and that for each ε > 0 there exist initial data for which the solution quenches in finite time.

In the first part of the paper a variational characterisation of the periodic eigenvalues (the so-called Fučik spectrum) of a semilinear, positive homogeneous Sturm–Liouville equation is given. The proof relies on the S1-invariance of the equation.

In the second part a nonlinear Sturm–Liouville equation with, typically, an exponential nonlinearity is considered. It is proved that under certain conditions this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values in the Fučik spectrum.

We study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular. We prove that the positive solution is in general Hölder-continuous up to the boundary and has even better regularity in some special cases.

We use a direct, geometric approach to study the free surface boundary conditions for stationary flows of viscous liquids. The free surface problem is characterised by a mapping on smooth variations of a given configuration; this mapping has a simple structure, which we determine by computing its differential, and studying it in terms of the space dimension and the surface tension coefficient. Applications are given to problems of existence, uniqueness and regularity in free surface flows.

Let ℐN denote the set of (N + 1)-tuples of C1 ω-periodic functions; the point P = (pN,…, p1p0) ϵ ℐN is identified with the differential equation

We examine the way in which the total number of ω-periodic solutions can vary as P traverses a path in ℐN.

The Cauchy problem of ut, = ∆uα + uβ, where 0 < α < l and α>1, is studied. It is proved that if 1< β<α + 2/n then every nontrivial non-negative solution is not global in time. But if β>α+ 2/n there exist both blow-up solutions and global positive solutions which decay to zero as t–1/(β–1) when t →∞. Thus the famous Fujita result on ut = ∆u + up is generalised to the present fast diffusion equation. Furthermore, regarding the equation as an infinite dimensional dynamical system on Sobolev space W1,s (W2.s) with S > 1, a non-uniqueness result is established which shows that there exists a positive solution u(x, t) with u(., t) → 0 in W1.s (W2.s) as t → 0.

In this paper we establish bounds constraining the effective conductivity tensor of composites made of an arbitrary number n of possibly anisotropic phases in prescribed volume fractions. The bounds are valid in any spatial dimension d≧2. The bounds have a very simple and concise form and include those previously obtained by Hashin and Shtrikman, Murat and Tartar, Lurie and Cherkaev, Kohn and Milton, Avellaneda, Cherkaev, Lurie and Milton and Dell'Antonio and Nesi.

A basic problem in population dynamics is that of finding criteria for the long-term coexistence of interacting species. An important aspect of the problem is determining how coexistence is affected by spatial dispersal and environmental heterogeneity. The object of this paper is to study the problem of coexistence for two interacting species dispersing through a spatially heterogeneous region. We model the population dynamics of the species with a system of two reaction–diffusion equations which we interpret as a semi-dynamical system. We say that the system is permanent if any state with all components positive initially must ultimately enter and remain within a fixed set of positive states that are strictly bounded away from zero in each component. Our analysis produces conditions that can be interpreted in a natural way in terms of environmental conditions and parameters, by combining the dynamic idea of permanence with the static idea of studying geometric problems via eigenvalue estimation.

An upper bound for the “sausage catastrophe” of dense sphere packings in 4-space is given.

A basic problem in the theory of finite packing is to determine, for a given positive integer k, the minimal volume of all convex bodies into which k translates of the unit ball Bd of the Euclidean d-dimensional space Ed can be packed ([5]). For d = 2 this problem was solved by Groemer ([6]).