arXiv subjects

Luis F. López Ríos

Publications and source records attributed to Luis F. López Ríos.

3 recordsLinked to original sources

arXiv2025

Stability of stationary reaction diffusion-degenerate Nagumo fronts I: spectral analysis

This paper establishes the spectral stability of monotone, stationary front solutions for reaction-diffusion equations where the reaction function is of Nagumo (or bistable) type and with diffusion coefficients which are density dependent and degenerate at zero (one of the equilibrium points of the reaction). These stationary profiles connect the non-degenerate equilibrium point with the degenerate state at zero, they are monotone, and arrive to the degenerate state at a finite point. They are neither sharp nor smooth. The degeneracy of the diffusion precludes the application of standard techniques to locate the essential spectrum of the linearized operator around the wave in the energy space $L^2$. This difficulty is overcome with a suitable partition of the spectrum, the analysis of singular sequences, a generalized convergence of operators technique and refined energy estimates. It is shown that the $L^2$-spectrum of the linearized operator around the front is real and with a spectral gap, that is, a positive distance between the imaginary axis and the rest of the spectrum, with the exception of the origin. Moreover, the origin is a simple isolated eigenvalue, associated to the derivative of the profile as eigenfunction (the translation eigenvalue). Finally, it is shown that the linearization generates an analytic semigroup that decays exponentially outside a one-dimensional eigenspace associated to the zero eigenvalue.

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arXiv2022

Spectral stability of monotone traveling fronts for reaction diffusion-degenerate Nagumo equations

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 that 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. Three different types of monotone waves are studied: (i) stationary diffusion-degenerate fronts, connecting the two stable equilibria of the reaction; (ii) traveling diffusion-degenerate fronts connecting zero with the unstable equilibrium; and, (iii) non-degenerate fronts. In the first two cases, the degeneracy 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 $L^2$ energy space.

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arXiv2020

Long-time behavior of solutions to the generalized Allen-Cahn model with degenerate diffusivity

The generalized Allen-Cahn equation, \[ u_t=\varepsilon^2(D(u)u_x)_x-\frac{\varepsilon^2}2D'(u)u_x^2-F'(u), \] with nonlinear diffusion, $D = D(u)$, and potential, $F = F(u)$, of the form \[ D(u) = |1-u^2|^{m}, \quad \text{or} \quad D(u) = |1-u|^{m}, \quad m >1, \] and \[ F(u)=\frac{1}{2n}|1-u^2|^{n}, \qquad n\geq2, \] respectively, is studied. These choices correspond to a reaction function that can be derived from a double well potential, and to a generalized degenerate diffusivity coefficient depending on the density $u$ that vanishes at one or at the two wells, $u = \pm 1$. It is shown that interface layer solutions that are equal to $\pm 1$ except at a finite number of thin transitions of width $\varepsilon$ persist for an either exponentially or algebraically long time, depending upon the interplay between the exponents $n$ and $m$. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg-Landau type are derived.

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