When a Mach number dependent parameter σ ≠ 0 is accounted forin an expansive free interaction analogous to that of Stewartson [7], the solution terminates in a singularity with a finite pressure, but with a pressure gradient varying as the inverse three-fifths power of the distance from the singularity, apart from a subtle logarithmic modification.

Let the coefficient q be real-valued on the half-line [0, ∞) and let q′ be locally absolutely continuous on [0, ∞). The ordinary symmetric differential expressions M and M2 are determined by

It has been shown in a previous paper by the authors that if for non-negative numbers k and X the coefficient q satisfies the condition

then M is limit-point and M2 is limit–2 at ∞.

This paper is concerned with showing that for powers of the independent variable x the condition (*) is best possible in order that both M and M2 should have the classification at ∞ given above.

A widely accepted model of nerve conduction in the squid axon is the systemof four non-linear partial differential equations developed by Hodgkin and Huxley. Under space clamp and current clamp conditions these equations are reduced to a system of ordinary differential equations.

We find that under appropriate assumptions on the functions and parameters in the resulting fourth order Hodgkin-Huxley equations there occurs a bifurcation of periodic solutions from the steady state. This bifurcation takes place as the current parameter, I, passes through a critical value.

A fundamental solution for equations of pseudoparabolic type is constructed by a method analogous to Hadamard's construction for elliptic equations. By the use of this fundamental solution we obtain a regularity theorem and integral representations for the solution to the more common initial boundary value problems that are associated with these equations.