Milan Journal of Mathematics 

About: Milan Journal of Mathematics is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Nonlinear system & Type (model theory). It has an ISSN identifier of 1424-9286. Over the lifetime, 362 publications have been published receiving 6640 citations. The journal is also known as: Issued by the Seminario Matematico e Fisico di Milano & MJM.

Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions

Abstract: We consider two simple conservative systems of parabolic-elliptic and parabolic-degenerate type arising in modeling chemotaxis and angiogenesis. Both systems share the same property that when the % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaCa % aaleqabaWaaSaaaeaacaWGKbaabaGaaGOmaaaaaaaaaa!38A1! $$L^{\frac{d} {2}} $$ norm of initial data is small enough, where d ≥ 2 is the space dimension, then there is a global (in time) weak solution that stays in all the L p spaces with max % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaada % GacaqaaiaaigdacaGG7aWaaSaaaeaacaWGKbaabaGaaGOmaaaacqGH % sislcaaIXaaacaGL9baacqGHKjYOcaWGWbGaeyipaWJaeyOhIuQaai % OlaaGaay5Eaaaaaa!42CD! $$\left\{ {\left. {1;\frac{d} {2} - 1} \right\} \leq p < \infty .} \right.$$ This result is already known for the parabolic-elliptic system of chemotaxis, moreover blow-up can occur in finite time for large initial data and Dirac concentrations can occur. For the parabolic-degenerate system of angiogenesis in two dimensions, we also prove that weak solutions (which are equi-integrable in L1) exist even for large initial data. But break-down of regularity or propagation of smoothness is an open problem.

On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics

Abstract: The standard central limit theorem plays a fundamental role in Boltzmann-Gibbs statistical mechanics. This important physical theory has been generalized [1] in 1988 by using the entropy \(S_{q} = \frac{1-\sum_{i} p^{q}_{i}}{q-1}\)\(({\rm with}\,q\,\in {{{\mathcal{R}}}})\) instead of its particular BG case \(S_{1} = S_{BG} = - \sum_{i} p_{i}\,{\rm ln}\,p_{i}\). The theory which emerges is usually referred to as nonextensive statistical mechanics and recovers the standard theory for q = 1. During the last two decades, this q-generalized statistical mechanics has been successfully applied to a considerable amount of physically interesting complex phenomena. A conjecture[2] and numerical indications available in the literature have been, for a few years, suggesting the possibility of q-versions of the standard central limit theorem by allowing the random variables that are being summed to be strongly correlated in some special manner, the case q= 1 corresponding to standard probabilistic independence. This is what we prove in the present paper for \(1{\leqslant}\,q 0\), and normalizing constant Cq. These distributions, sometimes referred to as q-Gaussians, are known to make, under appropriate constraints, extremal the functional Sq (in its continuous version). Their q = 1 and q = 2 particular cases recover respectively Gaussian and Cauchy distributions.

Global Existence in Reaction-Diffusion Systems with Control of Mass: a Survey

Abstract: The goal of this paper is to describe the state of the art on the question of global existence of solutions to reaction-diffusion systems for which two main properties hold: on one hand, the positivity of the solutions is preserved for all time; on the other hand, the total mass of the components is uniformly controlled in time. This uniform control on the mass (or – in mathematical terms- on the L1-norm of the solution) suggests that no blow up should occur in finite time. It turns out that the situation is not so simple. This explains why so many partial results in different directions are found in the literature on this topic, and why also the general question of global existence is still open, while lots of systems arise in applications with these two natural properties. We recall here the main positive and negative results on global existence, together with many references, a description of the still open problems and a few new results as well.

Periodic Nonlinear Schrödinger Equation with Application to Photonic Crystals

Abstract: We present some results on existence of nontrivial solutions of periodic stationary nonlinear Schrodinger equations. We also sketch an application to nonlinear optics and discuss some open problems.

On Schrödinger-Poisson Systems

Abstract: We discuss some recent results dealing with the existence of bound states of the nonlinear Schrodinger-Poisson system $$\left\{ \begin{gathered} - \Delta u + V(x)u + \lambda K(x)\phi (x)u = |u|^{{p - 1}} u, \hfill \\ - \Delta \phi = K(x)u^{2}, \hfill \\ \end{gathered} \right.$$ as well as of the corresponding semiclassical limits. The proofs are based upon Critical Point theory and Perturbation Methods.