Spectral Stability of Monotone Traveling Fronts for Reaction Diffusion-Degenerate Nagumo Equations
Por:
Leyva J.F., Lopez Rios, Luis F., Plaza, Ramon G.
Publicada:
1 ene 2022
Categoría:
Mathematics (miscellaneous)
Resumen:
This paper establishes the spectral stability of monotone traveling
front solutions for reaction-diffusion equations where the reaction
function is of Nagumo (or bistable) type and with diffusivities which
are density dependent and degenerate at zero (one of the equilibrium
points of the reaction). Spectral stability is understood as the
property in which the spectrum of the linearized operator around the
wave, acting on an exponentially weighted space, is contained in the
complex half plane with non-positive real part. The degenerate fronts
studied in this paper travel with positive speed above a threshold value
and connect the (diffusion-degenerate) zero state with the unstable
equilibrium point of the reaction function. In this case, the degeneracy
of the diffusion coefficient is responsible of the loss of hyperbolicity
of the asymptotic coefficient matrices of the spectral problem at one of
the end points, precluding the application of standard techniques to
locate the essential spectrum. This difficulty is overcome with a
suitable partition of the spectrum, a generalized convergence of
operators technique, the analysis of singular (or Weyl) sequences, and
the use of energy estimates. The monotonicity of the fronts, as well as
detailed descriptions of the decay structure of eigenfunctions on a
case-by-case basis, are key ingredients to show that all traveling
fronts under consideration are spectrally stable in a suitably chosen
exponentially weighted L2 energy space.
Filiaciones:
Leyva J.F.:
Facultad de Ciencias de la Computación, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 14 Sur, Ciudad Universitaria, Puebla, Puebla, C.P. 72570, Mexico
Lopez Rios, Luis F.:
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Circuito Escolar s/n, Ciudad Universitaria, Ciudad de México, C.P. 04510, Mexico
Plaza, Ramon G.:
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Circuito Escolar s/n, Ciudad Universitaria, Ciudad de México, C.P. 04510, Mexico
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