aequationes mathematicae

, Volume 71, Issue 3, pp 253–268

On a type of evolution of self-referred and hereditary phenomena

Original Paper

Summary.

In this note we establish some results of local existence and uniqueness of solutions of the equations
$$ u\left( {x,t} \right) = u_{0}\left( x \right) + \int_0^t {u\left( {\int_0^\tau {u\left( {x,s} \right)ds,\tau } } \right)d\tau ,\quad t \geqslant 0,x \in \mathbb{R},} $$
$$ u\left( {x,t} \right) = u_{0}\left( x \right) + \int_0^t {u\left( {\frac{1} {\tau }\int_0^\tau {u\left( {x,s} \right)ds,\tau } } \right)d\tau ,\quad t \geqslant 0,x \in \mathbb{R},} $$
and
$$ u\left( {x,t} \right) = u_{0}\left( x \right) + \int_0^t {u\left( {\int_0^\tau {\frac{1} {{2\delta \left( s \right)}}\int_{x - \delta \left( s \right)}^{x + \delta \left( s \right)} {u\left( {\varepsilon ,s} \right)d\varepsilon ds,\tau } } } \right)d\tau ,\quad t \geqslant 0,x \in \mathbb{R},} $$
or, equivalently, for the initial value problem, respectively:
$$ \left\{ {\begin{array}{*{20}l} {\frac{\partial } {{\partial t}}u\left( {x,t} \right) = u\left( {\int_0^t {u\left( {x,s} \right)ds,t} } \right),\quad t \geqslant 0,x \in \mathbb{R}} \hfill\\ {u\left( {x,0} \right) = u_0 \left( x \right),\quad x \in \mathbb{R}} \hfill\\ \end{array} } \right. $$
$$ \left\{ {\begin{array}{*{20}l} {\frac{\partial } {{\partial t}}u\left( {x,t} \right) = u\left( {\frac{1} {t}\int_0^t {u\left( {x,s} \right)ds,t} } \right),\quad t \geqslant 0,x \in \mathbb{R}} \hfill\\ {u\left( {x,0} \right) = u_0 \left( x \right),\quad x \in \mathbb{R}} \hfill\\ \end{array} } \right. $$
and
$$ \left\{ {\begin{array}{*{20}l} {\frac{\partial } {{\partial t}}u\left( {x,t} \right) = u\left( {\int_0^t {\frac{1} {{2\delta \left( s \right)}}\int_{x - \delta \left( s \right)}^{x + \delta \left( s \right)} {u\left( {\xi ,\tau } \right)d\xi d\tau ,t} } } \right),\quad t \geqslant 0,x \in \mathbb{R}} \hfill\\ {u\left( {x,0} \right) = u_0 \left( x \right),\quad x \in \mathbb{R}} \hfill\\ \end{array} } \right. $$
where u0 e δ are given functions satisfying suitable conditions.

Mathematics Subject Classification (2000).

Primary 47J35 45G10 

Keywords.

Evolution equations hereditary equations functional differential equations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Dipartimento di Matematica “Ennio De Giorgi"Università di LecceLecceItaly

Personalised recommendations