Coefficient determination via asymptotic spreading speeds
Abstract
In this paper, we analyze the inverse problem of determining the reaction term f (x, u) in reaction-diffusion equations of the form ∂tu − D∂xxu = f (x, u), where f is assumed to be periodic with respect to x ∈ R. Starting from a family of exponentially decaying initial conditions u0,λ, we show that the solutions uλ of this equation propagate with constant asymptotic spreading speeds wλ. Our main result shows that the linearization of f around the steady state 0, ∂u f (x, 0), is uniquely determined (up to a symmetry) among a subset of piecewise linear functions, by the observation of the asymptotic spreading speeds wλ.
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