%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The abstract must be written in LaTeX2e               %%
%% Make sure that your abstract can be compiled before   %%
%% you send it to the organizer.                         %%
%% Do not use personal macros in your TeX file, and,     %%
%% for the sake of making your abstract understandable   %%
%% to a large audience and to avoid difficulties with    %%
%% the handling of your file, please limit the use of    %%
%% formulas as much as possible.                         %%
%% We suggest that you do not include references in      %%
%% your abstract. In case that you must have references, %%
%% then do not use \cite commands.                       %%
%%                                                       %%
%% For labels, try to write : \label{yourname01},        %%
%% \label{yourname02}...                                 %%
%% Your abstract should be no more than one page long.   %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass{article}
\usepackage{amsthm,amsmath,amssymb,latexsym}
\pagestyle{empty}
\textwidth=12.8cm
\textheight=21.7cm
\hoffset=-0.3in
\voffset=-0.6in
\parskip=6pt
\lineskip=18pt
%------------------------------------------------------------
% MACRO PER INSIEME DEI NUMERI REALI COMPLESSI ETC ETC ETC
\makeatletter\newif\ifmsbmloaded@
\def\loadmsbm{\msbmloaded@true
  \font\tenmsb=msbm10 scaled 1\@ptsize00
  \font\sevenmsb=msbm7 scaled 1\@ptsize00
  \font\fivemsb=msbm5 scaled 1\@ptsize00
  \alloc@8\fam\chardef\sixt@@n\msbfam
  \textfont\msbfam=\tenmsb
  \scriptfont\msbfam=\sevenmsb
  \scriptscriptfont\msbfam=\fivemsb}
\def\Bbb{\relax\ifmmode\expandafter\Bbb@\else
  \expandafter\nonmatherr@\expandafter\Bbb\fi}
\def\Bbb@#1{{\Bbb@@{#1}}}\def\Bbb@@#1{\fam\msbfam\relax#1}
\loadmsbm
\def\R{\Bbb R}\def\N{\Bbb N}\def\Z{\Bbb Z}\def\C{\Bbb C}
\def\E{\Bbb E}\def\P{\Bbb P}\def\H{\Bbb H}
% FINE DELLA MACRO

\begin{document}
\baselineskip=15pt

\begin{center}
{\bf First Joint Italian-American Meeting UMI-AMS}\\
{Department of Mathematics, 
University of Pisa, June 12- 16, 2002}\medskip

{Special session on}\medskip

{\Large\bf ``NONLINEAR EVOLUTION EQUATIONS''}\medskip
\end{center}\bigskip

\centerline{\large \bf  Multispike solutions to nonlinear 
elliptic equations}\bigskip

\centerline{P. Bates}
\centerline{\it Brigham Young University (Provo)}
\centerline{e-mail {\tt bates@nemo.mth.msu.edu}}
%%%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
An abstract framework is outlined to study
the existence and Morse index of spike-layer solutions to a class
of singular perturbation problems. We then show how the abstract
results apply in the following situation. Let $\epsilon>0$ be a small
positive parameter and $\Omega$, a smooth bounded
domain in ${\mathbf R}^n$ for $n\ge 2$. Consider
\begin{equation}
\label{1}
\begin{cases}
\epsilon^2\Delta v -av+f(v) =0,& \qquad x\in \Omega, \\
%\ds 
\frac{\partial v}{\partial n}=0,& \qquad x\in \partial
\Omega,
\end{cases}
\end{equation}
where $a>0$, $f(v)$ is a smooth positive function, superlinear as
$v \to \infty,$ and such that $f(0)=f'(0)=0$. We seek solutions
$u_\varepsilon$ which tend to zero as $\varepsilon\to 0$ uniformly on 
compact subsets of $\bar \Omega/P_N$ Here $P_N$ is a collection of $N$
points in $\bar\Omega$ which is characterized in terms of the
geometry, and $u_\varepsilon(p)$ does not approach zero with 
$\varepsilon$ for each $p\in P_N$.
\bigskip

\centerline{\large \bf To be communicated}\bigskip

\centerline{ G.I. Barenblatt}
\centerline{\it University of California (Berkeley)}
%\centerline{e-mail {\tt Author 1}}
\bigskip

\par
%   Text of the abstract
\bigskip
\newpage

\centerline{\large \bf Global Existence of a Strongly-Coupled Quasilinear}

\centerline{\large \bf Parabolic System Arising from Electrochemistry}
\bigskip

\centerline{Y.S. Choi}
\centerline{\it Department of Mathematics, University of Connecticut, Storrs}
%\centerline{e-mail {\tt Author 1}}
\bigskip

\par
We considered a strongly coupled quasilinear parabolic system
\[
\frac{\partial v_i}{\partial t} =
\sum_{j=1}^{m} \frac{\partial}{\partial x}
\left(a^{ij}({\bf v})\frac{\partial v_j}{\partial x}\right)
\,,\;\;\;\;\;i=1,\cdots,m\;
\]
modeling an electrochemical problem.
Its coefficient $a_{ij}$ may become discontinuous.
First we proved the global existence
of weak solutions of a strongly-coupled parabolic system with continuous
coefficent. Using it to approximate the original problem, we established
the global existence of the weak solution to the electrochemistry problem.
Global stability of
the steady-state solutions were also shown.
\bigskip

\centerline{\large \bf Operators of Thin-Film Type:}

\centerline{\large \bf Qualitative Properties and Open Problems}

\bigskip

\centerline{Roberta Dal Passo}
\centerline{\it Universit\`a di Roma '' Tor Vergata'', Italy}
\centerline{e-mail {\tt dalpasso@mat.uniroma2.it}}
\bigskip

\par
This class of (nonlinear degenerate higher order) operators has recently
attracted an enormous interest both for its relevance in a number of
applications (from fluid dynamics to material sciences) and for its
mathematical significance as relatively uncharted territory in the theory
of PDE's.
The mathematical investigation has revealed  a great richness of
structure, but still  several challenging problems remain open.
\bigskip
\newpage

\centerline{\large \bf Geometric Estimates for the Vanishing Behavior}

\centerline{\large \bf of Solutions to the Logarithmic Fast 
Diffusion Equation}
\bigskip

\centerline{P. Daskalopoulos}
\centerline{\it Columbia University (New York)}
%\centerline{e-mail {\tt Author 1}}
\bigskip

\par
We study the vanishing behavior of maximal solutions to the
logarithmic fast diffusion equation $u_t=\Delta \log u$ on the plane.
This equation describes the evolution of a metric $g$, conformal to the
standard metric under the Ricci flow.
We derive   upper and lower bounds on the geometric width of
the solution and on the maximum curvature. Using these geometric
estimates we  deduce upper and lower pointwise bounds on
the solution  near its vanishing time. More precise  bounds on
the solution  are obtained  in the rotationally symmetric case, using
comparison methods.
\bigskip

\centerline{\large \bf Repelling Spikes in the Dynamics}

\centerline{\large \bf of a System of Reaction-Diffusion Equations}
\bigskip

\centerline{G. Bellettini}
\centerline{\it Dipartimento di Matematica,
Universit\`a di Roma ``Tor Vergata''}
%\centerline{e-mail {\tt Author 1}}

%%% in case of several authors:
\smallskip
\centerline{G.Fusco}
\centerline{\it Dipartimento di Matematica,
Universit\`a de L'Aquila}
%\centerline{e-mail {\tt Author 2}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
We consider an activator-inhibitor system of the form
$$
 \left\{\begin{array}{ll}
  u_t=\epsilon^2\Delta u+F(u)+\sigma(u-v),\qquad      &x\in\Omega,\\
  \tau v_t=\epsilon^2\Delta v+F(v)+\sigma(u-v),\qquad &x\in\Omega,\\
  \displaystyle{{\frac{\partial u}{\partial n}}=
  {\frac{\partial v}{\partial n}}=0,}\qquad &x\in\partial\Omega,
        \end{array}\right.
\leqno{(1)}
$$
where: $F(s)=s^2-s$, $\epsilon>0$ is a small parameter,
$\tau,\sigma$ positive constants and $\tau<1, \sigma >\bar\sigma$ 
for some $\bar\sigma$.

We show that,given an integer $N\geq 1$, there is
$\epsilon_N>0$ such that for $\epsilon<\epsilon_N$
system (1) has a solution $t\rightarrow u^\epsilon(t)$
which exhibits $N$ spikes at   points
$\xi^\epsilon_1(t),\dots,\xi^\epsilon_N(t)\in \Omega$.
We also derive a precise asymptotic formula for the speed of the N spikes.
We show that, for small $\epsilon$, the points  $\xi^\epsilon_i(t)$
{\bf repel} each other and are also {\bf repelled} by the
boundary $\partial\Omega$ of $\Omega$.
A consequence of this fact is that the solution
$t\rightarrow u^\epsilon(t)$ exists globally in time
keeping for ever its spike structure. 
The repelling character of the {\bf spike interaction} has also interesting
consequences on the structure of the set of stationary spike solutions. 
For instance one can deduce the existence of unstable spike stationary 
solutions in a $\epsilon^{1\over2}-$neighborhood of stable ones.

%%% in case of several authors:
{\bf  Presented by G. Fusco}
%%%
\bigskip

\centerline{\large \bf Intermediate Scaling Laws for Spreading Droplets:}

\centerline{\large \bf PDE Methods and Asymptotic Analysis}
\bigskip

\centerline{Lorenzo Giacomelli}
\centerline{\it Universit\`a ``La Sapienza'' (Roma)}
\centerline{e-mail {\tt giacomelli@dmmm.uniroma1.it}}

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par

The spreading of a thin droplet of viscous liquid on a plane surface
driven by capillarity is modeled --- in the standard lubrication
approximation --- by a fourth order nonlinear degenerate parabolic
equation for the droplet height $h$.  If the evolution is limited by
the no--slip boundary condition at the liquid--solid interface, then
the
problem is ill--posed, since solutions in fact don't spread due to a
singularity at the moving contact line (in the mythological words of
Huh and Scriven, ``... not even Herakles could sink a solid"). Ad-hoc
relaxations of the no--slip boundary conditions, such as positive
slippage models or shear-thinning rheologies, remove this paradox but
introduce new physical, microscopic length scales $b$.

Here we show that these microscopic length scales only enter
logarithmically in the effective (that is, macroscopic) spreading
behavior. For positive slippage models, we prove a scaling law in time
for both the total energy  and the diameter of the apparent (that is,
macroscopic) support of the droplet. This is an intermediate scaling
law: It takes an initial layer to ``forget'' the initial droplet shape
--- whereas after a long time, the droplet is so thin that its
spreading is governed by the physics on the scale $b$. Our approach
mimics a simple heuristic argument based on the gradient flow
structure, and works by deriving suitable estimates for physically
relevant integral quantities: the free energy, the length of the
apparent support and their respective rates of change. For
shear-thinning models we present formal arguments, based on the
analysis of a class of qusi--selfsimilar solutions, which suggest
analogous logarithmic corrections on time scales which in this case
may not be intermediate. Results are based on joint works with Felix
Otto and Lidia Ansini, respectively.
\bigskip

\centerline{\large \bf Stability in a Mathematical Model 
in Combustion Theory}
\bigskip

\centerline{Alessandra Lunardi}
\centerline{\it Dipartimento di Matematica, Universit\`a di Parma, Italia}
\centerline{e-mail {\tt lunardi@prmat.math.unipr.it}}

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
I will discuss the problem of the stability of the planar travelling wave
solution to a well known free boundary parabolic system modelling the
propagation of near-equidiffusional premixed
flames in the whole plane or in a two-dimensional strip.

I will give stability and instability results, according to the value of
the reduced Lewis number.
\bigskip


\centerline{\large \bf One-dimensional stability of relaxation shocks}
\bigskip

\centerline{Corrado Mascia}
\centerline{\it Dipartimento di Matematica ``G. Castelnuovo''
Universit\`a ``La Sapienza'' (Roma)}
\centerline{e-mail {\tt mascia@mat.uniroma1.it}}

%%% in case of several authors:
\smallskip
\centerline{Kevin Zumbrun}
\centerline{\it Mathematics Department, Indiana University (Bloomington)}
%\centerline{e-mail {\tt Author 2}}
%%%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
Consider a general {\it hyperbolic  relaxation model} of form
$$
  \left\{\begin{array}{l}
  u_t + f(u,v)_x=0\\
  v_t + g(u,v)_x=q(u,v),
  \end{array}\right.
\leqno(1)
$$
with $u$, $f\in \R^n$, $v$, $g$, $q \in \R^r$, under the assumption
$Re\, \sigma\big(q_v(u,v^*(u))\big)<0$ along a smooth 
{\it equilibrium manifold} defined by  $q(u,v^*(u))\equiv 0$.

System (1) supports smooth traveling front solutions,
known as {\it relaxation shocks}, i.e. solutions of form
$(u,v)(x,t)= (\bar u, \bar v)(x-st)$ such that
$\lim\limits_{z\to \pm \infty} (\bar u, \bar v)= (u_\pm, v^*(u_\pm))$.
Such relaxation shocks are the counterpart of shock solutions to
the associated {\it equilibrium}, or ``relaxed'' system of conservation laws
$u_t + f(u,v^*(u))_x=0$, hence these solutions are expected to be stable.

Under the weak assumption of spectral stability, or stable point spectrum 
of the linearized operator about the wave, we establish sharp pointwise 
Green's function bounds and consequent linear and nonlinear orbital 
stability for relaxation shocks.
A consequence is stability of small-amplitude profiles of
Broadwell and Jin-Xin models for each of which spectral stability has 
been verified in other works.
These are the first complete stability results for relaxation models with
nonscalar equilibrium equations.

%%% in case of several authors:
{\bf  Presented by Corrado Mascia}
%%%
\bigskip


\centerline{\large \bf Diffusion and Cross-Diffusion in Pattern Formation:}

\centerline{\large \bf from Single Equations to Systems}
\bigskip

\centerline{Wei-Ming Ni}
\centerline{\it University of Minnesota (Minneapolis)}
%\centerline{e-mail {\tt Author 1}}

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
In this lecture I plan to explain how diffusions and
cross-diffusions are used in modeling pattern formation, both from a
modeling point of view and from a mathematical point of view. Examples
will be used to illustrate various approaches.
\bigskip

\centerline{\large\bf Diffusive N-Waves and Metastability in 
Burgers Equation}
\medskip

\centerline{Yong Jung Kim}
\centerline{\it Department of  Mathematics, University of 
Minnesota (Minneapolis)}
%\centerline{e-mail {\tt Author 1}}

%%% in case of several authors:
\smallskip
\centerline{Athanasios E. Tzavaras}
\centerline{\it Department of Mathematics, University of 
Wisconsin-Madison (Madison)}
%\centerline{e-mail {\tt Author 2}}
%%%
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
We study the effect of viscosity on the large time behavior of
the viscous Burgers equation by using a transformed version
of Burgers (in self-similar variables) that captures efficiently 
the mechanism of transition to the asymptotic states, and allows to 
estimate the time of evolution from an N-wave to the final stage of 
a diffusion wave.
Then, we construct certain special solutions of diffusive N-waves
with unequal masses.
Finally, using a set of similarity variables
and a variant of the Cole-Hopf transformation,
we obtain an integrated Fokker-Planck equation.
The latter is solvable and provides an explicit solution
of the viscous Burgers in a series of Hermite polynomials.
This format captures the long time - small viscosity interplay,
as the diffusion wave and the diffusive N-waves correspond
respectively to the first two terms in the Hermite polynomial expansion.

%%% in case of several authors:
{\bf  Presented by Athanasios E. Tzavaras}
%%%
\bigskip

\centerline{\large \bf Qualitative Behavior of Solutions}

\centerline{\large \bf of Chemotactic Diffusion Systems}
\smallskip

\centerline{Xuefeng Wang}
\centerline{\it Tulane University (New Orleans)}
%\centerline{e-mail {\tt Author 1}}
\smallskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
Chemotaxis is the oriented movement of cells in response to the
concentration gradient of chemical substances in their environment. 
It is ``anti-diffusion''. 
We are interested in the effects of chemotaxis and diffusion on the 
growth of cells.
This kind of problems leads to new challenges in nonlinear analysis, 
even in special cases.
 We first consider the situation of a single bacterial population which
responds chemotactically to a nutrient diffusing from an adjacent phase 
not accessible to the bacteria. 
The bacteria and the chemical are assumed to be diffusive.
The concentration and density of the substrate and cells (resp.) satisfy a
quasi-linear parabolic system , with nonlinear boundary condition. 
Our first set of results addresses the effects of two important
biological parameters $\lambda >0$ and $\chi >0$ on the steady states, where
$\lambda$ measures the (random) motility of bacteria and $\chi$ the 
magnitude of chemotactic response (or sensitivity) to the chemical.

\noindent I shall also talk about the dynamics of the system:
global boundedness of time-dependent solutions and  stability issues of
 trivial and nontrivial steady states with small amplitudes.

\noindent Finally, I will present some results (joint work with
Yaping Wu) for the situation when two
species of cells compete for the same source of substrate.
\end{document}









