Lower bounds, contractivity, error estimates, and continuity
Lower bounds, contractivity, error estimates, and continuity
This chapter considers a number of bounds that complement the study so far, which was centred on upper bounds. The first section concerns the existence of lower bounds for non-negative solutions. There is a big difference between the ranges m > 1, m = 1, and m ≤ 1. In the first case, the property of finite propagation implies that solutions may travel at a bounded speed so that it will take a certain amount of time for a solution to become positive at points where it was initially zero. It is shown that this does not happen for m ≤ 1. The second section extends the property of contractivity in the L 1 norm into error estimates in different norms.
Keywords: lower bounds, Harnack inequalities, PME, positivity, heat equation, FDE, contractivity
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