The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise by Arnaud Debussche Michael Högele & Peter Imkeller

Author:Arnaud Debussche, Michael Högele & Peter Imkeller
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

4.1 Uniformly Absorbing Ball for (2.​12)

Proposition 4.1.

There exists a constant r ∗ and a time such that for any x ∈ H and with , the solution v θ (⋅;x) of (2.​12) satisfies

Proof.

We fix such that and write v = v θ . Then (2.​12) can be written in the form

We now reproduce the estimates used in the case θ = 0 (see [EFNT94, Tem92]). We multiply this equation by | v |  k v and integrate with respect to the space variable to obtain after integration by part1:

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