We study global behavior of the nonlinear Klein-Gordon equation with a focusing cubic power in three dimensions, in the energy space under the restriction of radial symmetry and an energy upper bound slightly above that of the ground state. We give a complete classification of the solutions into 9 non-empty sets according to whether they blow-up, scatter to 0, or scatter to the ground states, in the forward and backward time directions, and the splitting is given in terms of the stable and the unstable manifoldsof the ground states. This […]

We present a variational model for quasistatic evolutions of brittle cracks in hyperelastic bodies, in the context of finite elasticity.All existence results on this subject that can be found in the mathematical literature were obtained using energy densities with polynomial growth. This is not compatible with the standard assumption in finite elasticity that the strain energy diverges as the determinant of the deformation gradient tends to zero. On the contrary, we consider a wide class of energy densities satisfying this property