, Volume 71, Issue 3, pp 253-268

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

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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 u 0 e δ are given functions satisfying suitable conditions.