  In this article, we are interested in the large time behavior of solutions of
the Dirichlet problem for subquadratic viscous Hamilton-Jacobi Equations. In
the superquadratic case, the third author has proved that these solutions can
have only two different behaviors: either the solution of the evolution
equation converges to the solution of the associated stationary generalized
Dirichlet problem (provided that it exists) or it behaves like $-ct+\varphi
(x)$ where $c\geq0$ is a constant, often called the "ergodic constant" and
$\varphi$ is a solution of the so-called "ergodic problem". In the present
subquadratic case, we show that the situation is slightly more complicated: if
the gradient-growth in the equation is like $|Du|^m$ with $m>3/2,$ then
analogous results hold as in the superquadratic case, at least if $c>0.$ But,
on the contrary, if $m\leq 3/2$ or $c=0,$ then another different behavior
appears since $u(x,t) + ct$ can be unbounded from below where $u$ is the
solution of the subquadratic viscous Hamilton-Jacobi Equations.
