We consider here a spherically symmetric but inhomogeneous universe filled with a massless scalar field. The model obeys two constraints. The first one is that the gradient of the scalar field is timelike everywhere. The second constraint is that the radial coordinate basis vector is a unit vector field in the comoving coordinate system. We find that the resultant dynamical solutions compose a one-parameter family of self-similar models which is known as the Roberts solution. The solutions are divided into three classes. The first class consists of solutions with only one spacelike singularity in the synchronous-comoving chart. The second class consists of solutions with two singularities which are null and spacelike, respectively. The third class consists of solutions with two spacelike singularities which correspond to the big bang and big crunch, respectively. We see that, in the first case, a comoving volume exponentially expands as in an inflationary period; the fluid elements are accelerated outwards form the symmetry center, even though the strong energy condition is satisfied. This behavior is very different from that observed in the homogeneous and isotropic universe in which the fluid elements would move outwards with deceleration, if the strong energy conditions are satisfied. We are thus able to achieve the accelerated expansion of the universe for the models considered here, without a need to violate the energy conditions. The cosmological features of the models are examined in some detail.
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