We present two new models describing the dynamic behavior of an automotive thermostat, involving delay-differential equations with hysteresis. Existence, uniqueness, and regularity of the solutions for both models are obtained by a continuation argument. We establish sufficient conditions for the models to exhibit intrinsic oscillations. We also present an algorithm for numerical approximations of the solutions and give some representative numerical simulations. These reveal a rather interesting dynamical behavior of the solutions.

The onset of shear-banding in a deforming elastoplastic solid has been linked to change of type of the governing partial differential equations. If uniform material properties are assumed, then (i) deformations prior to shear-banding are uniform, and (ii) the onset of shear-banding occurs simultaneously at all points in the sample. In this paper we study, in the context of a model for anti-plane shearing of a granular material, the effect of a small variation in material properties (e.g. in yield strength) within the sample. Using matched asymptotic expansions, we find that (i) the deformation is extremely non-uniform in a short time period immediately preceding the formation of shear-bands; and (ii) generically, a shear-band forms at a single location in the sample.

Many elastodynamic solutions exist for wave interactions with defects upon or within an elastic half space bounded above by a vacuum. The aim of this paper is to significantly widen the scope of these solutions by showing that they can be directly utilized, when the vacuum is replaced by a fluid, to find the leading order acoustic (or elastic) responses in the light fluid-loading limit. Using the procedure developed here one can take virtually any elastodynamic solution and use it to generate the leading-order solution for a fluid-loaded elastic solid.

We prove that the thin film equation ht+div (hn grad (Δh))=0 in dimension d[ges ]2 has a unique C1 source-type radial self-similar non-negative solution if 0<n<3 and has no solution of this type if n[ges ]3. When 0<n3 the solution h has finite speed of propagation and we obtain the first order asymptotic behaviour of h at the interface or free boundary separating the regions where h>0 and h=0. (The case d=1 was already known [1]).

In this paper we deal with the one-dimensional Stefan problem

ut−uxx =s˙(t)δ(x−s(t)) in ℝ ;× ℝ+, u(x, 0) =u0(x)

with kinetic condition s˙(t)=f(u) on the free boundary F={(x, t), x=s(t)}, where δ(x) is the Dirac function. We proved in [1] that if [mid ]f(u)[mid ][les ]Meγ[mid ]u[mid ] for some M>0 and γ∈(0, 1/4), then there exists a global solution to the above problem; and the solution may blow up in finite time if f(u)[ges ] Ceγ1[mid ]u[mid ] for some γ1 large. In this paper we obtain the optimal exponent, which turns out to be √2πe. That is, the above problem has a global solution if [mid ]f(u)[mid ][les ]Meγ[mid ]u[mid ] for some γ∈(0, √2πe), and the solution may blow up in finite time if f(u)[ges ] Ce√2πe[mid ]u[mid ].