We prove two stability results for the scale invariant solutions of the nonlinear heat equation ∂ t u = ∆u−|u| p−1 u with 1 < p < 1+ 2 n , n being the spatial dimension. The first result is that a small perturbation of a scale invariant solution vanishes as t → ∞. The second result is global, with a positivity condition on the initial data.

We address a well known problem of nonlinear image diffusion techniques, namely the loss of texture information. We do so by first determining that it is due to unaccounted correlation structure in the image which we subsequently mitigate by proposing a wavelet frame-based technique. This, by the same token establishes a theoretical bridge between the scale space methodology and the multiscale ...

We give conditions under which the trivial solution of a first-order nonlinear variable-delay dynamic equation is asymptotically stable, for arbitrary time scales that are unbounded above. In an example, we apply our techniques to a logistic dynamic equation on isolated, unbounded time scales. c © 2006 Elsevier Ltd. All rights reserved. MSC: 39A10; 34B10

We establish some new oscillation criteria for the second-order quasilinear neutral delay dynamic equations r t zΔ t γ Δ q1 t x τ1 t q2 t x τ2 t 0 on a time scale T, where z t x t p t x τ0 t , 0 < α < γ < β. Our results generalize and improve some known results for oscillation of second-order nonlinear delay dynamic equations on time scales. Some examples are considered to illustrate our main r...