Shoshana Kamin

Fundamental Solutions and Asymptotic Behaviour for the

We establish the uniqueness of fundamental solutions to the p-Laplacian equation ut = div (|Du|p-2 Du), p > 2, defined for x I RN, 0 < t < T. We derive from this result the asymptotic behavoir of nonnegative solutions with finite mass, i.e., such that u(*,t) I L1(RN). Our methods also apply to the porous medium equation ut = ?(um), m > 1, giving new and simpler proofs of known results. We finally introduce yet another method of proving asymptotic results based on the idea of asymptotic radial symmetry. This method can be useful in dealing with more general equations.

p

The quasi-linear parabolic equation ∂tu = a∂xxuα + b∂xuβ − cuγ exhibits a wide variety of wave phenomena, some of which are studied in this work; and some solvable cases are presented. The motion of the wave front is characterized in terms of α, β and γ. Among the interesting phenomena we note the effect of fast absorption (b  0, 0 < γ < 1) that causes extinction within a finite time, may break the evolving pulse into several sub-pulses and causes the expanding front to reverse its direction. In the convecting case (c  0, b ≠ 0) propagation has many features in common with Burgers equation, α = 1; particularly, if 0 < a ≪ 1, a shock-like transit layer is formed.

-Laplacian Equation

It is proved that solutions of the porous medium equation

Thermal waves in an absorbing and convecting medium

u_t = \Delta (| u |^{m - 1} u),m > 1

Asymptotic behaviour of solutions of the porous medium equation with changing sign

, defined in

Existence and uniqueness of the very singular solution of the porous media equation with absorption

Q = \mathbb{R}^N \times (0,\infty )

Metastability in a flame front evolution equation

with initial data

Flat core properties associated to the p-Laplace operator

u(x,0)

Singular solutions of some nonlinear parabolic equations

integrable, compactly supported, and with changing sign, become nonnegative in finite time if

Nonlinear thermal evolution in an inhomogeneous medium

\int {u_0 (x)} dx > 0