  We develop a new geometric framework suitable for dealing with Hamiltonian
field theories with dissipation. To this end we define the notions of
$k$-contact structure and $k$-contact Hamiltonian system. This is a
generalization of both the contact Hamiltonian systems in mechanics and the
$k$-symplectic Hamiltonian systems in field theory. The concepts of symmetries
and dissipation laws are introduced and developed. Two relevant examples are
analyzed in detail: the damped vibrating string and Burgers' equation.
