Ricerche Di Matematica 

Abstract: Two mathematical models for phase segregation and diffusion of an order parameter are derived, within one and the same continuum mechanical framework. These models are, respectively, of the Allen-Cahn type and of the Cahn-Hilliard type. Our framework is similar to that used in [1], in that a postulated balance of microforces plays a central role in both deductive paths, but differs from it, mainly in three ways: imbalance of entropy replaces for a dissipation inequality, whose form depends on the case, restricting the growth of free energy; balance of energy replaces for the mass balance introduced in [1] to arrive at (a generalization of) the C-H equation; and chemical potential is given the same role played by coldness in the deduction of the heat equation. When appropriate constitutive prescriptions are made, different in the cases of segregation and diffusion but consistent with the entropy imbalance, it is found that standard A-C and C-H processes are solutions of constant chemical potential of the corresponding generalized equations; in particular, the stationary solutions are the same. Keywords: Phase segregation, Diffusion, Allen-Cahn equation, Cahn-Hilliard equation, Phase-field methods Mathematics Subject Classification (2000): 74N25, 74A50, 35K60

Abstract: In this paper we consider the following elliptic system in $${\mathbb{R}^3}$$ $$\qquad\left\{\begin{array}{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\end{array}\right.$$ where λ is a real parameter, $${p\in (1, 5)}$$ if λ < 0 while $${p\in (3, 5)}$$ if λ > 0 and K(x), a(x) are non-negative real functions defined on $${\mathbb{R}^3}$$ . Assuming that $${\lim_{|x|\rightarrow+\infty}K(x)=K_{\infty} >0 }$$ and $${\lim_{|x|\rightarrow+\infty}a(x)=a_{\infty} >0 }$$ and satisfying suitable assumptions, but not requiring any symmetry property on them, we prove the existence of positive ground states, namely the existence of positive solutions with minimal energy.

Abstract: An SEIR epidemic model with a nonlinear incidence rate is studied The incidence is assumed to be a convex function with respect to the infective class of a host population A bifurcation analysis is performed and conditions ensuring that the system exhibits backward bifurcation are provided The global dynamics is also studied, through a geometric approach to stability Numerical simulations are presented to illustrate the results obtained analytically This research is discussed in the framework of the recent literature on the subject

Abstract: In this paper we study a model which describes the relation of the matter and the electromagnetic field from a unitarian standpoint in the spirit of the ideas of Born and Infeld. This model, introduced in [1], is based on a semilinear perturbation of the Maxwell equation (SME). The particles are described by the finite energy solitary waves of SME whose existence is due to the presence of the nonlinearity. In the magnetostatic case (i.e. when the electric field ${\bf E}=0$ and the magnetic field ${\bf H}$ does not depend on time) the semilinear Maxwell equations reduce to semilinear equation where “ $ abla\times $ ” is the curl operator, f′ is the gradient of a smooth function $f:{\mathbb{R}}^3\to{\mathbb{R}}$ and ${\bf A}:{\mathbb{R}}^3\to{\mathbb{R}}^3$ is the gauge potential related to the magnetic field ${\bf H}$ ( ${\bf H}= abla\times {\bf A}$ ). The presence of the curl operator causes (1) to be a strongly degenerate elliptic equation. The existence of a nontrivial finite energy solution of (1) having a kind of cylindrical symmetry is proved. The proof is carried out by using a variational approach based on two main ingredients: the Principle of symmetric criticality of Palais, which allows to avoid the difficulties due to the curl operator, and the concentration-compactness argument combined with a suitable minimization argument. Keywords: Maxwell equations, Natural constraint, Minimizing sequence Mathematics Subject Classification (2000): 35B40, 35B45, 92C15

Abstract: Volterra’s model for population growth in a closed system includes an integral term to indicate accumulated toxicity in addition to the usual terms of the logistic equation, that occurs in ecology. In this paper, a new numerical approximation is introduced for solving this model of arbitrary (integer or fractional) order. The proposed numerical approach is based on the generalized fractional order Chebyshev orthogonal functions of the first kind and the collocation method. Accordingly, we employ a collocation approach, by computing through Volterra’s population model in the integro-differential form. This method reduces the solution of a problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show that the new method is efficient and applicable.