We study the formation of singularities for the problem {u(t) = [phi(u)](xx) + epsilon[psi(u)](txx) in Omega x (0, T) phi(u) + epsilon[psi(u)](t) = 0 in partial derivative Omega x(0, T) u = u(0) >= 0 in Omega x {0}, where epsilon and Tare positive constants, Omega a bounded interval, u(0) a nonnegative Radon measure on Omega, phi a nonmonotone and nonnegative function with phi(0) = phi(infinity) = 0, and psi an increasing bounded function. We show that if u(0) is a bounded or continuous function, singularities may appear spontaneously. The class of singularities which can arise in finite time is remarkably large, and includes infinitely many Dirac masses and singular continuous measures.

We consider a mathematical model describing population dynamics of normal and abnormal cell densities with contact inhibition of cell growth from a theoretical point of view. In the first part of this paper, we discuss the global existence of a solution satisfying the segregation property in one space dimension for general initial data. Here, the term segregation property means that the different types of cells keep spatially segregated when the initial densities are segregated. The second part is devoted to singular limit problems for solutions of the PDE system and traveling wave solutions, respectively. Actually, the contact inhibition model considered in this paper possesses quite similar properties to those of the Fisher-KPP equation. In particular, the limit problems reveal a relation between the contact inhibition model and the Fisher-KPP equation.

We prove existence and uniqueness for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous and bounded. The solution class is determined by an additional condition which is needed to prove uniqueness.

We consider a nonlinear system of partial differential equations which describes the dynamics of two types of cell densities with contact inhibition. After a change of variables the system turns out to be parabolic-hyperbolic and admits travelling wave solutions which solve a 3D dynamical system. Compared to the scalar Fisher-KPP equation, the structure of the travelling wave solutions is surprisingly rich and to unravel part of it is the aim of the present paper. In particular, we consider a parameter regime where the minimal wave velocity of the travelling wave solutions is negative. We show that there exists a branch of travelling wave solutions for any nonnegative wave velocity, which is not connected to the travelling wave solution with minimal wave velocity. The travelling wave solutions with nonnegative wave velocity are strictly positive, while the solution with minimal one is segregated in the sense that the product uv vanishes.

We introduce and analyze a new, nonlinear fourth-order regularization of forwardbackward parabolic equations. In one space dimension, under general assumptions on the potentials, which include those of Perona-Malik type, we prove existence of Radon measure-valued solutions under both natural and essential boundary conditions. If the decay at infinity of the nonlinearities is sufficiently fast, we also exhibit examples of local solutions whose atomic part arises and/or persists (in contrast to the linear fourth-order regularization) and even disappears within finite time (in contrast to pseudoparabolic regularizations).

-- We prove existence and uniqueness of Radon measure-valued solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension, the initial data being a finite superposition of Dirac masses and the flux being Lipschitz continuous, bounded and suciently smooth. The novelty of the paper is the introduction of a compatibility condition which, combined with standard entropy conditions, guarantees uniqueness.

We study nonnegative solutions of the Cauchy problem

We study the initial-boundary value problem [Formula presented]where [Formula presented] is an interval and [Formula presented] is a nonnegative Radon measure on [Formula presented]. The map [Formula presented] is increasing in [Formula presented] and decreasing in [Formula presented] for some [Formula presented], and satisfies [Formula presented]. The regularizing map [Formula presented] is increasing and bounded. We prove existence of suitably defined nonnegative Radon measure-valued solutions. The solution class is natural since smooth initial data may generate solutions which become measure-valued after finite time.

In the first part of this paper we review a mathematical model for the onset and progression of Alzheimer's disease (AD) that was developed in subsequent steps over several years. The model is meant to describe the evolution of AD in vivo. In Achdou et al (2013 J. Math. Biol. 67 1369-92) we treated the problem at a microscopic scale, where the typical length scale is a multiple of the size of the soma of a single neuron. Subsequently, in Bertsch et al (2017 Math. Med. Biol. 34 193-214) we concentrated on the macroscopic scale, where brain neurons are regarded as a continuous medium, structured by their degree of malfunctioning. In the second part of the paper we consider the relation between the microscopic and the macroscopic models. In particular we show under which assumptions the kinetic transport equation, which in the macroscopic model governs the evolution of the probability measure for the degree of malfunctioning of neurons, can be derived from a particle-based setting. The models are based on aggregation and diffusion equations for ?-Amyloid (A? from now on), a protein fragment that healthy brains regularly produce and eliminate. In case of dementia A? monomers are no longer properly washed out and begin to coalesce forming eventually plaques. Two different mechanisms are assumed to be relevant for the temporal evolution of the disease: (i) diffusion and agglomeration of soluble polymers of amyloid, produced by damaged neurons; (ii) neuron-to-neuron prion-like transmission. In the microscopic model we consider mechanism (i), modelling it by a system of Smoluchowski equations for the amyloid concentration (describing the agglomeration phenomenon), with the addition of a diffusion term as well as of a source term on the neuronal membrane. At the macroscopic level instead we model processes (i) and (ii) by a system of Smoluchowski equations for the amyloid concentration, coupled to a kinetic-type transport equation for the distribution function of the degree of malfunctioning of the neurons. The transport equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain

In this article we propose a mathematical model for the onset and progression of Alzheimer's disease based on transport and diffusion equations. We regard brain neurons as a continuous medium and structure them by their degree of malfunctioning. Two different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble polymers of amyloid, produced by damaged neurons and ii) neuron-to-neuron prion-like transmission. We model these two processes by a system of Smoluchowski equations for the amyloid concentration, coupled to a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. The second equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain. Our numerical simulations are in good qualitative agreement with clinical images of the disease distribution in the brain which vary from early to advanced stages.

We study the initial-boundary value problem (Formula presented.) with measure-valued initial data. Here ? is a bounded open interval, ?(0)=?(?)=0, ? is increasing in (0,?) and decreasing in (?,?), and the regularising term ? is increasing but bounded. It is natural to study measure-valued solutions since singularities may appear spontaneously in finite time. Nonnegative Radon measure-valued solutions are known to exist and their construction is based on an approximation procedure. Until now nothing was known about their uniqueness. In this note we construct some nontrivial examples of solutions which do not satisfy all properties of the constructed solutions, whence uniqueness fails. In addition, we classify the steady state solutions.

We study a quasilinear parabolic equation of forward-backward type, under assumptions on the nonlinearity which hold for a wide class of mathematical models, using a pseudo-parabolic regularization of power type.We prove existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures. It is shown that these solutions satisfy suitable entropy inequalities. We also study their qualitative properties, in particular proving that the singular part of the solution with respect to the Lebesgue measure is constant in time.

We give a necessary and sufficient condition for the validity of the strong maximum principle in one space dimension.

We consider an initial-boundary value problem for a degenerate pseudoparabolic regularization of a nonlinear forward-backward-forward parabolic equation, with a bounded nonlinearity which is increasing at infinity. We prove existence of suitably defined nonnegative solutions of the problem in a space of Radon measures. Solutions satisfy several monotonicity and regularization properties; in particular, their singular part is nonincreasing and may disappear in finite time. The problem is of intrinsic mathematical interest, but also arises naturally when studying, by time reversal, the spontaneous appearance of singularities in a specific application.

We consider a cell growth model involving a nonlinear system of partial differential equations which describes the growth of two types of cell populations with contact inhibition. Numerical experiments show that there is a parameter regime where, for a large class of initial data, the large time behaviour of the solutions is described by a segregated travelling wave solution with positive wave speed c. Here, the word segregated expresses the fact that the different types of cells are spatially segregated, and that the single densities are discontinuous at the moving interface which separates the two populations. In this paper, we show that, for each wave speed c > c, there exists an overlapping travelling wave solution, whose profile is continuous and no longer segregated. We also show that, for a large class of initial functions, the overlapping travelling wave solutions cannot represent the large time profile of the solutions of the system of partial differential equations. The structure of the travelling wave solutions strongly resembles that of the scalar Fisher-KPP equation, for which the special role played by the travelling wave solution with minimal speed has been extensively studied.

In [Comm. Appl. Math. Comput. Sci., 4 (2009), pp. 153-175], Barenblatt presents a model for partial laminarization and acceleration of shear flows by the presence of suspended particles of different sizes, and provides a formal asymptotic analysis of the resulting velocity equation. In the present paper we revisit the model. In particular we allow for a continuum of particle sizes, rewrite the velocity equation in a form which involves the Laplace transform of a given function or measure, and provide several rigorous asymptotic expansions for the velocity. The model contributes to a better insight to the extreme velocities in hurricanes, fire storms, and dust storms, and the analysis confirms Barenblatt's conclusion that often the smallest suspended particles are responsible for the extreme flow acceleration at large altitudes.

In this paper we introduce the Mathematical Desk for Italian Industry, a project based on applied and industrial mathematics developed by a team of researchers from the Italian National Research Council in collaboration with two major Italian associations for applied mathematics, SIMAI and AIRO. The scope of this paper is to clarify the motivations for this project and to present an overview on the activities, context and organization of the Mathematical Desk, whose mission is to build a concrete bridge of common interests between the Italian scientific community of applied mathematics and the world of the Italian enterprises. Some final considerations on the strategy for the future development of the Mathematical Desk project complete the paper.

We consider a simplified 1-dimensional PDE-model describing the effect of contact inhibition in growth processes of normal and abnormal cells. Varying the value of a significant parameter, numerical tests suggest two different types of contact inhibition between the cell populations: the two populations move with constant velocity and exhibit spatial segregation, or they stop to move and regions of coexistence are formed. In order to understand the different mechanisms, we prove that there exists a segregated traveling wave solution for a unique wave speed, and we present numerical results on the ``stability" of the segregated waves. We conjecture the existence of a non-segregated standing wave for certain parameter values.