We address the quantitative uniqueness properties of the solutions of the parabolic equationwhere v and w are bounded. We prove that for solutions u, the order of vanishing is bounded by C( v 2/3 L ∞ + w 2 L ∞ ) matching the upper bound previously established in the elliptic case.
We consider the wave equation with an energy-supercritical focusing nonlinearity in dimension seven. We prove that any radial solution that remains bounded in the critical Sobolev space is global and scatters to a linear solution.
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