Abstract: The existence of solutions for many systems of
integro-differential equations discovered and generalized in the
process of applying the Galerkin method for some initial-boundary
value problems will be investigated in this paper.
Abstract: In this paper, we mention to the initial-boundary value problem
for the linear wave equation
\begin{equation}\label{1}
\begin{cases}
u_{tt}-u_{xx}+K u+\lambda u_t=f(x,t),\quad 0<x<1,0<t<T,\\
u(0,t)=0, \\
-u_x(1,t)=P(t),\\
u(x,0)=u_0(x),u_t(x,0)=u_1(x),
\end{cases}
\end{equation}
where $K,\lambda $ are given constants and $u_0,u_1,f$ are given
functions, the unknown function $u(x,t)$ and the unknown boundary
value $P(t)$ satisfy the following integral equation
\begin{equation}\label{2}
P(t)=u(1,t)+u_t(1,t)+g(t)-\int_0^tk(t-s)u(1,s)ds,
\end{equation}
where $g,k$ are given functions. The paper consists of two parts.
In part $1,$ for $(u_0, u_1)\in H^1\times L^2,K,\lambda\in
\mathbb{R},f\in L^1(0,T;L^2),g\in L^2(0,T),k\in L^1(0,T),$ we
prove a theorem of existence and uniqueness of a weak solution
$(u,P)$ of problem (\ref{1})-(\ref{2}). In the proof, we use the
Galerkin approximation associated with a Schauder's fixed point
theorem and some compact standards. The second part is devoted to
the study of global existence and the decay of the solution
$\big(u(t),P(t)\big)$ with respect to $t.$ We first study the
decay of the component $u(t)$ of the solution
$\big(u(t),P(t)\big)$ under more restrictive conditions, namely
$0<|K|<(2-\varepsilon_1-3\varepsilon_2)/7,0<\lambda<2-2\varepsilon_2-4|K|,
f\in L^1(\mathbb{R}_+;L^2)\cap L^2(\mathbb{R}_+;L^2),g\in
L^2(\mathbb{R}_+),k,h\in L^1(\mathbb{R}_+)$,
$\int_0^{+\infty}e^{\alpha t}g^2(t)dt<+\infty$, $
\int_0^{+\infty}e^{\alpha
t}\bigg(\int_0^tk(t-s)ds\bigg)^2dt<+\infty$,
$\int_0^{+\infty}e^{\alpha t}\|f(t)\|^2dt<+\infty,$ for some
$\alpha,\varepsilon_1,\varepsilon_2>0,$ where
$h(t)=\int_0^t|k(t-s)|ds.$ Later by modifying some stronger
assumptions, we obtain the decays of both $u(t)$ and $P(t)$ in
which we also base on a theorem of unique existence for problem
(\ref{1})-(\ref{2}) published in \cite{L-U-T}.
Abstract: In this paper we investigate the existence of solutions of a system of self-referred and hereditary differential equations. The initial data are assumed to be lower semi-continuous. We also formulate some open questions.
Abstract: In this note we establish a results of existence of solution for
the equation $$ u(x,t)=u_0(x)+\int _0^t u\left( \int _0^\tau
u(x,s)ds,\tau \right)d\tau,\quad t\geq 0, x\in \mathbb{R}, $$ and
for the initial value problem: $$ \left\{
\begin{array}{l}
{\displaystyle \frac{\partial}{\partial t}u(x,t)=u\left( \int _0^t
u(x,s)ds,t\right)},\quad x\in \mathbb{R}; a.a. t>0 \\ \mbox{ }\\
u(x,0)=u_0(x),\quad x\in \mathbb{R}
\end{array}
\right. $$ with a suitable weak condition on $u_0.$
Abstract: In this paper, the unique solvability of a semi-linear wave equation associated with a linear integral equation at the boundary is proved by a contracted procedure.
Abstract: In this paper the unique existence and regularity of the weak solution
of an initial-boundary value problem relating to a semi-linear wave equation
and two integral equations at the boundaries are given.
Abstract: In this paper, we prove the well-posedness for a mixed nonhomogeneous problem for a semilinear wave equation associated with a linear integral equation at the boundary.
Abstract: In this paper the stability and asymptotic expansion of the weak
solution of an initial-boundary problem relating to a semi-linear wave equation
and two integral equations at the boundaries are given.
Notes: a.a. supported by Faculty of Science, University of Oulu, 2010 Travel Grant; the free hospitality granted by the conference including accommodation, 300 India Rupees per day and free-service transportation.
Abstract:
The purpose of this meeting is to bring together researchers with common interest in the field. There will be opportunities for informal discussions. Graduate students and others beginning their mathematical career are encouraged to participate.
Notes: Professors: Anders Björn (Linköping University, Linköping, Sweden. Newtonian spaces: First-order Sobolev spaces on metric spaces) ; Fernando Cobos (Universidad Complutense, Madrid, Spain. Interpolation theory and compactness); Thierry Coulhon (Cergy-Pontoise University, Cergy-Pontoise, France. Sobolev inequalities on non-compact Riemannian manifolds);
Peter A. Hästö (University of Helsinki, Helsinki, Finland. Muckenhoupt weights and variable-exponent function spaces).
Notes: Basically it is competitive to get a grant from my faculty, since there are many applicants from different departments. I was lucky to be recommended by Peter Hasto, so I got it. I sincerely thank him for his kindness.
Notes: According to Mathematical Reviews, "new reviewers are frequently enlisted on the recommendation of a current reviewer. Your suggestions are very welcome. A potential reviewer should ordinarily have already published reviewable work. Exceptions are made in the case of strongly recommended recent Ph.D.'s, especially if they can offer combinations that we need of expertise in certain fields and languages (our shortage of Russian-reading and Chinese-reading reviewers is chronic)". However, a reviewer can be invited by an editor of Mathematical Reviews. For my case, I was invited by Graeme Fairweather, Executive Editor, in 2009. All in all, I am truly grateful to Dr. Fairweather for his invitation.
Notes: In 2007, I completed two preprints which were then published in Mem. Differential Equations and Math. Phys. and Proc. A. Razmadze Math. Inst.. These helped me to be awarded a Nguyen Thai Hoc scholarship which is worth 750$. This scholarship foundation is founded by a group of Vietnamese American people. I sincerely thank them for their kindness.
Notes: Since I got a very good record from my graduate studies, which is over 8/10, Dr. Giao M. Nguyá»n recommended me to receive an Odon Vallet scholarship in 2005. It is worth 200$. It is noted that Dr. Nguyá»n was an adviser of the graduate program I took. In addition, he was also a member of the representative board of the Odon Vallet Scholarship in South Vietnam. During my graduate time, Dr. Nguyá»n sometimes visited our class, and encouraged us to go further in our studies. He gave us his email address. That was the reason why I had an opportunity to write to him. A funny story is that he asked me to send him my C.V., and I replied him by asking "Would you please tell me what a C.V. is?" (This may have helped him to know how innocent I was). It were a mistake if I would not remind myself of M.Sc. Thuáºn V. Äặng regarding my graduate studies. Without his help, I could not have had a chance to take the graduate entrance examination. All in all, I am truly grateful to them for their kindness and sincerity.
