Jos茅 Manuel Palacios (University of Toronto)
Title: Local Energy control in the presence of a zero-energy resonance
Abstract: We consider the problem of stability and local energy decay for co-dimension one perturbations of the soliton of the cubic Klein-Gordon equation in 1+1 dimensions. Our main result gives a weighted time-averaged control of the local energy over a time interval which is exponentially long in the size of the initial (total) energy. A major difficulty is the presence of a zero-energy resonance in the linearized operator, which is a well-known obstruction to improved local decay properties. We address this issue by using virial estimates that are frequency-localized in a time-dependent way and introducing a "singular virial functional" with time-dependent weights to control the mass of the perturbation projected away from small frequencies.
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