A singular Cahn--Hilliard--Oono phase-field system with hereditary memory

We consider a phase-field system modeling phase transition phenomena, where the Cahn--Hilliard--Oono equation for the order parameter is coupled with the Coleman--Gurtin heat law for the temperature. The former suitably describes both local and nonlocal (long-ranged) interactions in the material undergoing phase-separation, while the latter takes into account thermal memory effects. We study the well-posedness and longtime behavior of the corresponding dynamical system in the history space setting, for a class of physically relevant and singular potentials. Besides, we investigate the regularization properties of the solutions and, for sufficiently smooth data, we establish the strict separation property from the pure phases

Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions

This paper is devoted to the study of the well-posedness and the long time behavior of the Caginalpphase-field model with singular potentials and dynamic boundary conditions.Thanks to a suitable definition of solutions, coinciding with the strong ones underproper assumptions on the bulk and surfacepotentials, we are able to get dissipative estimates, leading tothe existence of the global attractor with finite fractal dimension,as well as of an exponential attractor

Mathematical analysis of a phase-field model of brain cancers with chemotherapy and antiangiogenic therapy effects

Our aim in this paper is to study a mathematical model for brain cancers with chemotherapy and antiangiogenic therapy effects. We prove the existence and uniqueness of biologically relevant (nonnegative) solutions. We then address the important question of optimal treatment. More precisely, we study the problem of finding the controls that provide the optimal cytotoxic and antiangiogenic effects to treat the cancer

Decadimento uniforme per equazioni integro-differenziali lineari di Volterra

This talk is devoted to some recent results concerning the exponential and the polynomial decays of the energy associated with a linear Volterra integro-differential equation of hyperbolic type in a Hilbert space, which is an abstract version of the equation describing the motion of a linearly viscoelastic solid occupying a (bounded) volume at rest.We provide sufficient conditions for the decay to hold, without invoking differential inequalities involving the convolution kernel. A similar analysis is carried on in the whole N-dimensional real space, although both the polynomial and the exponential decay of the memory kernel lead to a polynomial decay of the energy, with a rate influenced by the space dimension N. These results are contained in two joint papers with Monica Conti and Vittorino Pata (Politecnico di Milano)

Attractors for semilinear equations of viscoelasticity with very low disspation

We analyze a differential system arising in the theory of isothermal viscoelasticity. This system is equivalent to an integrodifferential equation of hyperbolic type with a cubic nonlinearity, where the dissipation mechanism is contained only in the convolution integral, accounting for the past history of the displacement. In particular, we consider here a convolution kernel which entails an extremely weak dissipation. In spite of that, we show that the related dynamical system possesses a global attractor of optimal regularity

A construction of a robust family of exponential attractors

Given a dissipative strongly continuous semigroup depending on some parameters, we construct a family of exponential attractors which is robust, in the sense of the symmetric Hausdorff distance, with respect to (even singular) perturbations

Attractors for semilinear equations of viscoelasticity with very low dissipation

We analyze a differential system arising in the theory of isothermal viscoelasticity. This system is equivalent to an integrodifferential equation of hyperbolic type with a cubic nonlinearity, where the dissipation mechanism is contained only in the convolution integral, accounting for the past history of the displacement. In particular, we consider here a convolution kernel which entails an extremely weak dissipation. In spite of that, we show that the related dynamical system possesses a global attractor of optimal regularity

Intracellular heavy metal nanoparticle storage: progressive accumulation within lymph nodes with transformation from chronic inflammation to malignancy

A 25-year-old man had complained of sudden fever spikes for two years and his blood tests were within the normal range. In 1993, a surgical biopsy of swollen left inguinal lymph nodes was negative for malignancy, but showed reactive lymphadenitis and widespread sinus histiocytosis. A concomitant needle biopsy of the periaortic lymph nodes and a bone marrow aspirate were also negative. In 1994, after an emergency hospital admission because of a sport-related thoracic trauma, a right inguinal lymph node biopsy demonstrated Hodgkin’s lymphoma Stage IVB (scleronodular mixed cell subtype). Although it was improved by chemotherapy, the disease suddenly relapsed, and a further lymph node biopsy was performed in 1998 confirming the same diagnosis. Despite further treatment, the patient died of septic shock in 2004, at the age of 38 years. Retrospective analysis of the various specimens showed intracellular heavy metal nanoparticles within lymph node, bone marrow, and liver samples by field emission gun environmental scanning electron microscopy and energy dispersive spectroscopy. Heavy metals from environmental pollution may accumulate in sites far from the entry route and, in genetically conditioned individuals with tissue specificity, may act as cofactors for chronic inflammation or even malignant transformation. The present anecdotal report highlights the need for further pathologic ultrastructural investigations using serial samples and the possible role of intracellular nanoparticles in human disease