Semilinear parabolic equations of the form u1 = ∇2u + δf(u), where f is positive and is finite, are known to exhibit the phenomenon of blow-up, i.e. for sufficiently large S, u becomes infinite after a finite time t*. We consider one-dimensional problems in the semi-infinite region x>0 and find the time to blow-up (t*). Also, the limiting behaviour of u as t→t*- and x→∞ is determined; in particular, it is seen that u blows up at infinity, i.e. for any given finite x, u is bounded as t→t*. The results are extended to problems with convection.

The modified equation xu, = uxx +f(u) is discussed. This shows the possibility of blow-up at x =0 even if u(0, f) = 0. The manner of blow-up is estimated.

Finally, bounds on the time to blow-up for problems in finite regions are obtained by comparing u with upper and lower solutions.

The global topological structure of the space of configurations of a non-rotating elastic string under compression and tension is studied. The part of the string under tension is specified by a measurable subset of the interval. The set of such intervals, with the Hausdorff topology, is considered a parameter space for the equation satisfied by the string, and the solutions are shown to form an infinitedimensional continuum over this parameter space. A new global topological theorem is needed, since the parameter space is not Euclidean. The topological theorem is based on the fixed-point transfer.

We give a definition of “quasispectral maximalč subspaces for a quasinilpotent, but not nilpotent, bounded Banach space operator. The definition applies to a class of operators, close to the Volterra operator.

We examine the case of plane, time-harmonic acoustic waves in two dimensions, scattered by an obstacle on the surface of which an impedance boundary condition is imposed. A stable method is developed for solving the inverse problem ofdetermining both the shape of the scatterer and the surface impedance from measurements of the asymptotic behaviour of the scattered waves at low frequencies. We accomplish this by minimizing an appropriate functional over a compact set of admissible boundary curves and admissible impedances.

In this paper, a new variational formulation of the Signorini problem with friction is given in terms of the contact stresses. The method corresponds to the direct integral equation approach in classical elastostatic problems. First the displacement and mixed problems are briefly described together with some numerical results. Next the displacements are eliminated by the use of Green's function, and a constrained minimum problem with respect to the normal and tangential tractions on the contact boundary is derived. Then the resulting approximation procedure is studied and certain convergence results are proved. Finally, some remarks on the Signorini problem with Coulomb friction are presented. Numerical results illustrate the theory.

We prove the following: suppose that a and b are complex numbers, with a non-zero, and that n is an integer not less than 5. Then, if F is a family of functions meromorphic in a plane domain Dsuch that, for each f in F, the equation

has no solutions in D, then F is normal in D.