Approximation properties of the qq-sine bases

For $q>12/11$ the eigenfunctions of the non-linear eigenvalue problem associated to the one-dimensional $q$-Laplacian are known to form a Riesz basis of $L^2(0,1)$. We examine in this paper the approximation properties of this family of functions and its dual, in order to establish non-orthogonal spectral methods for the $p$-Poisson boundary value problem and its corresponding parabolic time evolution initial value problem. The principal objective of our analysis is the determination of optimal values of $q$ for which the best approximation is achieved for a given $p$ problem.Comment: 20 pages, 11 figures and 2 tables. We have fixed a number of typos and added references. Changed the title to better reflect the conten

Long time behavior of a parabolic p-Laplacian equation coupled to a compartmental ODE system with an induction threshold phenomenon.

We study the existence and the longterm behavior of solutions of a parabolic equation governed by the p-Laplacian with nonlinear growth terms that are coupled with the solutions of a system of ordinary differential equations. The existence and the uniqueness are shown by using a fixed point argument and the longterm behavior of solutions is discussed by using energy estimates together with the nonlinear peculiarity of the p-Laplacian. Numerical simulations are carried out by using a Finite Volume Method for spatial treatment. For time integration of the p-Laplacian, an implicit Euler method is used, and direct integration for the ODE system

Analysis of a PDE model of the swelling of mitochondria accounting for spatial movement.

We analyze existence and asymptotic behavior of a system of semilinear diffusion-reaction equations that arises in the modeling of the mitochondrial swelling process. The model itself expands previous work in which the mitochondria were assumed to be stationary, whereas now their movement is modeled by linear diffusion. While in the previous model certain formal structural conditions were required for the rate functions describing the swelling process, we show that these are not required in the extended model. Numerical simulations are included to visualize the solutions of the new model and to compare them with the solutions of the previous model

A note on positive eigenfunctions and hidden convexity

We give a simple convexity-based proof of the following fact: the only eigenfunction of the p-Laplacian that does not change sign is the first one. The method of proof covers also more general nonlinear eigenvalue problems. Copyright 2012 Springer Basel