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\centerline{\large \bf Global Existence of Smooth Solutions}

\centerline{\large \bf for Partially 
Dissipative Hyperbolic Systems}

\bigskip

\centerline{Roberto Natalini}
\centerline{\it IIAC - CNR, Rome, Italy}
\centerline{e-mail natalini@asterix.iac.rm.cnr.it }
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%   Text of the abstract

We consider the Cauchy problem for a  general 
one dimensional n x n hyperbolic
symmetrizable system of balance laws. It is well known that, in many physical
examples, for instance for the isentropic Euler system with damping, the
dissipation due to the source term may prevent the shock formation, at least
for smooth and small initial data.

Our main goal is to  consider the following problem: to find a set of general
and realistic sufficient conditions  to guarantee the global existence of
smooth solutions, and possibly to investigate their asymptotic behavior.

For systems which are entropy dissipative, a quite natural generalization of
the Kawashima condition for hyperbolic-parabolic systems can be given. In
this talk we  first propose a general framework to set this kind of problems,
by using the so-called entropy variables. Therefore, we pass to prove some
general statements about the global existence of smooth solutions, under
different sets of conditions. Our main tools will be some refined energy
estimates and the use of a suitable version, possibly relaxed, of the
Kawashima condition.        

\bigskip


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