
Abstract
Let $A$ be a closed linear operator defined
on a dense set in the Hilbert space $H$. Fractional evolution
equations of the form $\frac{d^\alpha u(t)}{dt^\alpha} = Au(t), 0
< \alpha \leq 1$, are studied in $H$, for a wide class of the
operators $A$. Some properties of the solutions of the Cauchy
problem for the considered equation are studied under suitable
conditions . It is proved also that there exists a dense set $S$
in $H$, such that if the initial condition $u(0)$ is an element of
$S$, then there exists a solution $u(t)$ of the considered Cauchy
problem. Applications to general partial differential equations of
the form
$$
\frac{\partial^\alpha u(x,t)}{\partial t^\alpha} = \sum_{q \leq m} a_q(x) D^q u(x,t)
$$
are given without any restrictions on the characteristic form
$\sum_{q=m} a_\alpha(x) \xi^q$, where $D^q = D_1^{q_1} ...
D_n^{q_n}, x = (x_1, ..., x_n), D_j= \frac{\partial}{\partial
x_j}, \xi^q = \xi_1^{q_1}, ..., \xi_n^{q_n}, q = q_1 + ... +
q_n$, and $q = (q_1, ..., q_n)$ is a multi index.\par}
Mathematics Subject Classification: 47 D 09, 34 G 10, 34 G 99, 35 K 90.
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