Uniqueness and growth of weak solutions to certain linear differential equations in Hilbert space

AbstractIn this paper, we study, by means of a modification of the weighted energy method, the questions of uniqueness and growth of weak solutions to evolutionary equations of the form utt = Mu where M is a symmetric operator and u takes values in a Hilbert space. We show that if the initial energy is negative, then the kinetic and potential energies have exponential growth. This is also the case when the initial energy is nonnegative provided it is not too large and the cosine of the angle between the initial displacement and initial velocity is sufficiently close to one.We also derive a continuous dependence result