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Two-sided Gaussian estimates for the fundamental solution of a second order linear parabolic differential equation are upper and lower bounds in terms of the fundamental solution of the classical heat conduction equation. In his seminal 1958 paper Nash stated, without proof, two-sided non-Gaussian bounds for the fundamental solution of a uniformly parabolic divergence structure equation assuming only boundedness of the coefficients. In his 1967-1968 papers Aronson derived truly Gaussian estimates for the fundamental solutions of a large class of linear parabolic equations (including the divergence structure equation) under minimal non-regularity assumptions on the coefficients. Subsequently in 1986 Fabes & Stroock derived Gaussian estimates for the divergence structure equation directly from the ideas of Nash and went on to prove Nash's continuity theorem and the Harnack inequality as a consequence of their estimate. In this note I describe these results together with various extensions.

In this note I survey the extensive literature on the dynamics of large series arrays of identical current biased Josephson junctions coupled through various shared loads. The equations describing the dynamics are invariant under permutation of the junctions so that in addition to the usual dynamical systems and numerical methods, group theoretic methods can be applied. In practice it is desirable to operate these circuits at a stable in-phase oscillation. The works summarized here are devoted to the study of where in parameter space the in-phase oscillations are stable and how the stability is lost. In particular, they focus on the variety of states produced through bifurcation as the in-phase oscillation loses stability. These states include discrete rotating waves and semirotors.

James Serrin's fundamental contributions to the theory of quasilinear elliptic equations are well-known and widely appreciated. He also made less well-known contributions to the theory of quasilinear parabolic equations which we dicuss in this note. Jurgen Moser gave greatly simplified proofs of the De Giorgi-Nash regularity results for linear divergence structure elliptic and parabolic differential equations using an original iterative technique. Serrin extended Moser's techniques and applied them to the study of divergence structure quasilinear elliptic equations, and in collaboration with Aronson, to divergence structure quasilinear parabolic equations. Specifically, among other results, they proved a maximum principle, Holder continuity of generalized solutions and derived a Harnack principle for a very broad class of quasilinear parabolic equations. In subsequent work, Aronson applied these results to study non-negative solutions to divergence structure linear equations without regularity assumptions on the coefficients. The results include a two-sided Gaussian estimate for the fundamental solution and a generalization of the Widder Representation Theorem.

We consider a porous mediaum flow in which the gas is initially distributed in the exterior of an empty region (a hole) and study the final stage of the hole-filling process. Hole-filling is asymptotically described by a self-similar solution which depends on a constant determined by the initial configuration. In general, this constant must be found numerically.. Here we give an example of a one-dimensional symmetric flow where the constant is obtained explicitly., Comment: laTex, 5 pages

We consider the equation that models the spreading of thin liquid films of power-law rheology. In particular, we analyze the existence and uniqueness of source-type self-similar solutions in planar and circular symmetries. We find that for shear-thinning fluids there exist a family of such solutions representing both finite and zero contact angle drops and that the solutions with zero contact angle are unique. We also prove the existence of traveling waves in one space dimension and classify them., Comment: 18 pages, 1 Postscript figures

We present a systematic computational approach to the study of self-similar dynamics. The approach, through the use of what can be thought of as a ``dynamic pinning condition" factors out self-similarity, and yields a transformed, non-local evolution equation. The approach, which is capable of treating both first and second kind self-similar solutions, yields as a byproduct the self-similarity exponents of the original dynamics. We illustrate the approach through the porous medium equation, showing how both the Barenblatt (first kind) and the Graveleau (second kind) self-similar solutions arise in this framework. We also discuss certain implications of the dynamics of the transformed equation (which we will name "MN-dynamics"); in particular we discuss the discrete-time implementation of the approach, and connections with time-stepper based methods for the "coarse" integration/bifurcation analysis of microscopic simulators., Comment: 14 pages, 2 figure

In the focusing problem we study a solution of the porous medium equation $u_t=\Delta (u^m)$ whose initial distribution is positive in the exterior of a closed non-circular two dimensional region, and zero inside. We implement a numerical scheme that renormalizes the solution each time that the average size of the empty region reduces by a half. The initial condition is a function with circular level sets distorted with a small sinusoidal perturbation of wave number $k\geq 3$. We find that for nonlinearity exponents m smaller than a critical value which depends on k, the solution tends to a self-similar regime, characterized by rounded polygonal interfaces and similarity exponents that depend on m and on the discrete rotational symmetry number k. For m greater than the critical value, the final form of the interface is circular., Comment: 26 pages, Latex, 13 ps figures