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We consider a scalar parabolic equation in one spatial dimension. The equation is constituted by a convective term, a reaction term with one or two equilibria, and a positive diffusivity which can however vanish. We prove the existence and several properties of traveling-wave solutions to such an equation. In particular, we provide a sharp estimate for the minimal speed of the profiles and improve previous results about the regularity of wavefronts. Moreover, we show the existence of an infinite number of semi-wavefronts with the same speed., 35 pages, 10 figures; submitted version. Revision with exposition changes, typos fixed and assumption (6.3) added to Propositions 6.1 and 8.2

We study a nonlinear reaction-convection equation with a degenerate diffusion of Perona-Malik's type and a monostable reaction term. Under quite general assumptions, we show the presence of wavefront solutions and prove their main properties. In particular, such wavefronts exist for every speed in a closed half-line and we give estimates of the threshold speed. The wavefront profiles are also strictly monotone and their slopes are uniformly bounded by the critical values of the diffusion.

The paper deals with the exact controllability of partial differential equations by linear controls. The discussion takes place in infinite dimensional state spaces since these equations are considered in their abstract formulation as semilinear equations. The linear parts are densely defined and generate strongly continuous semigroups. The nonlinear terms may also include a nonlocal part. The solutions satisfy nonlocal properties, which are possibly nonlinear. The states belong to Banach spaces with a Schauder basis and the results exploit topological methods. The novelty of this investigation is in the use of an approximation solvability method which involves a sequence of controllability problems in finite-dimensional spaces. The exact controllability of nonlocal solutions can be proved, with controls in \begin{document}$ L^p $\end{document} spaces, \begin{document}$ 1. The results apply to the study of the exact controllability for the transport equation in arbitrary Euclidean spaces and for the equation of the nonlinear wave equation.

The paper deals with second order parabolic equations on bounded domains with Dirichlet conditions in arbitrary Euclidean spaces. Their interest comes from being models for describing reaction–diffusion processes in several frameworks. A linear diffusion term in divergence form is included which generates a strongly elliptic differential operator. A further linear part, of integral type, is present which accounts of nonlocal diffusion behaviours. The main result provides a unifying method for studying the existence and localization of solutions satisfying nonlocal associated boundary conditions. The Cauchy multipoint and the mean value conditions are included in this investigation. The problem is transformed into its abstract setting and the proofs are based on the homotopic invariance of the Leray–Schauder topological degree. A bounding function (i.e. Lyapunov-like function) theory is developed, which is new in this infinite dimensional context. It allows that the associated vector fields have no fixed points on the boundary of their domains and then it makes possible the use of a degree argument.

We consider a degenerate scalar parabolic equation, in one spatial dimension, of flux-saturated type. The equation also contains a convective term. We study the existence and regularity of traveling-wave solutions; in particular we show that they can be discontinuous. Uniqueness is recovered by requiring an entropy condition, and entropic solutions turn out to be the vanishing-diffusion limits of traveling-wave solutions to the equation with an additional non-degenerate diffusion. Applications to crowds dynamics, which motivated the present research, are also provided.

In this paper we give an overview of some recent results concerning partial differential equations modeling collective movements, namely, vehicular traffic flows and crowds dynamics. The focus is on traveling-wave solutions to degenerate parabolic equations in one space dimension, even if we briefly discuss models based on different equations. The case of networks is also taken into consideration. The parabolic degeneration opens the possibilities of several different behaviors of the traveling-wave solutions, which are investigated in details.

We consider in this paper a diffusion-convection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or twice; then, we deal with a forward-backward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviours at the points where the diffusivity degenerates. In particular, if such points are interior points, then the sharp behaviours are new and unusual., Comment: 25 pages, 9 figures. Changes in the exposition; final version

The paper deals with semilinear evolution equations in Banach spaces. By means of linear control terms, the controllability problem is investigated and the solutions satisfy suitable nonlocal conditions. The Cauchy multi-point condition and the mean value condition are included in the present discussion. The final configuration is always achieved with a control with minimum norm. The results make use of fixed point techniques; two different approaches are proposed, depending on the use of norm or weak topology in the state space. The discussion is completed with some applications to dynamics of diffusion processes.

The paper deals with the exact controllability of a semilinear system in a separable Hilbert space. A bounded linear part is considered and a linear control introduced. The state space is compactly embedded in a Banach space and the nonlinear term is continuous in its state variable in the norm of the Banach space. An infinite sequence of finite dimensional controllability problems is introduced and the solution is obtained by a limiting procedure. To the best of our knowledge, the method is new in controllability theory. An application to an integro-differential system in euclidean spaces completes the discussion.

We prove a global existence result for bounded solutions to a class of abstract semilinear delay evolution equations with measures subjected to nonlocal initial data of the form $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \mathrm{d}u(t)=\{Au(t)+f(t,u_t)\}\mathrm{d}t+\mathrm{d}g(t),&{}\quad t\in \mathbf{R}_+,\\ u(t)=h(u)(t),&{}\quad t\in [\,-\tau ,0\,], \end{array}\right. \end{aligned}$$ where \(\tau \ge 0\), \(A:D(A)\subseteq X\rightarrow X\) is the infinitesimal generator of a \(C_0\)-semigroup, \(f:\mathbf{R}_+\times \mathcal {R} ([\,-\tau ,0\,];X)\rightarrow X\) is continuous, \(g\in BV_{\mathrm{loc}}(\mathbf{R}_+;X)\), and \(h:\mathcal {R} _b(\mathbf{R}_+;X)\rightarrow \mathcal {R} ([\,-\tau ,0\,];X)\) is nonexpansive.

In this paper we consider an advection-diffusion equation, in one space dimension, whose diffusivity can be negative. Such equations arise in particular in the modeling of vehicular traffic flows or crowds dynamics, where a negative diffusivity simulates aggregation phenomena. We focus on traveling-wave solutions that connect two states whose diffusivity has different signs; under some geometric conditions we prove the existence, uniqueness (in a suitable class of solutions avoiding plateaus) and sharpness of the corresponding profiles. Such results are then extended to the case of end states where the diffusivity is positive but it becomes negative in some interval between them. Also the vanishing-viscosity limit is considered. At last, we provide and discuss several examples of diffusivities that change sign and show that our conditions are satisfied for a large class of them in correspondence of real data., Comment: 27 pages

An existence theorem for differential inclusions in Hilbert spaces with nonlocal conditions is proved. Periodic, anti-periodic, mean value and multipoint conditions are included in this study. The investigation is based on a combination of the approximation solvability method with Hartman-type inequalities. A feedback control problem associated to a first order partial differential equation completes this discussion.

We consider a parabolic partial differential equation that can be understood as a simple model for crowds flows. Our main assumption is that the diffusivity and the source/sink term vanish at the same point; the nonhomogeneous term is different from zero at any other point and so the equation is not monostable. We investigate the existence, regularity and monotone properties of semi-wavefront solutions as well as their convergence to wavefront solutions., 29 pages

The paper deals with a second order integro-partial differential equation in R n with a nonlocal, degenerate diffusion term. Nonlocal conditions, such as the Cauchy multipoint and the weighted mean value problem, are investigated. The existence of periodic solutions is also studied. The dynamic is transformed into an abstract setting and the results come from an approximation solvability method. It combines a Schauder degree argument with an Hartman-type inequality and it involves a Scorza-Dragoni type result. The compact embedding of a suitable Sobolev space in the corresponding Lebesgue space is the unique amount of compactness which is needed in this discussion. The solutions are located in bounded sets and they are limits of functions with values in finitely dimensional spaces.

Semilinear multivalued equations are considered, in separable Banach spaces with the Radon-Nikodym property. An effective criterion for the existence of solutions to the associated Floquet boundary value problem is showed. Its proof is obtained combining a continuation principle with a Liapunov-like technique and a Scorza-Dragoni type theorem. A strictly localized transversality condition is assumed. The employed method enables to localize the solution values in a not necessarily invariant set; it allows also to introduce nonlinearities with superlinear growth in the state variable.

Aim of the paper is to provide a method to analyze the behavior of $T$-periodic solutions $x_\eps, \eps>0$, of a perturbed planar Hamiltonian system near a cycle $x_0$, of smallest period $T$, of the unperturbed system. The perturbation is represented by a $T$-periodic multivalued map which vanishes as $\eps\to0$. In several problems from nonsmooth mechanical systems this multivalued perturbation comes from the Filippov regularization of a nonlinear discontinuous $T$-periodic term. \noindent Through the paper, assuming the existence of a $T$-periodic solution $x_\eps$ for $\eps>0$ small, under the condition that $x_0$ is a nondegenerate cycle of the linearized unperturbed Hamiltonian system we provide a formula for the distance between any point $x_0(t)$ and the trajectories $x_\eps([0,T])$ along a transversal direction to $x_0(t).$

The paper deals with the multivalued initial value problem x ′ ∈ A ( t , x ) x + F ( t , x ) for a.a. t ∈ [ a , b ] , x ( a ) = x 0 in a separable, reflexive Banach space E . The nonlinearity F is weakly upper semicontinuous in x and the investigation includes the case when both A and F have a superlinear growth in x . We prove the existence of local and global classical solutions in the Sobolev space W 1 , p ( [ a , b ] , E ) with 1 p ∞ . Introducing a suitably defined Lyapunov-like function, we are able to investigate the topological structure of the solution set. Our main tool is a continuation principle in Frechet spaces and we prove the required pushing condition in two different ways. Some examples complete the discussion.

The solvability of the second-order Floquet problem in a given set is established by means of C 2 -bounding functions for vector upper-Caratheodory systems. The applied Scorza–Dragoni type technique allows us to impose related conditions strictly on the boundaries of bound sets. An illustrating example is supplied for a dry friction problem.

This paper deals with the semilinear multivalued evolution equation $$ x'(t) + A(t)x(t) \in \Sigma (t,x(t)),t \in [a,b]andx \in E, $$ in an arbitrary Banach space E.

This paper deals with a nonhomogeneous scalar parabolic equation with possibly degenerate diffusion term; the process has only one stationary state. The equation can be interpreted as modeling collective movements (crowd dynamics, for instance). We first prove the existence of semi-wavefront solutions for every wave speed; their properties are investigated. Then, a family of travelling wave solutions is constructed by a suitable combination of the previous semi-wavefront solutions. Proofs exploit comparison-type techniques and are carried out in the case of one spatial variable; the extension to the general case is straightforward., 37 pages

The paper deals with a nonlocal diffusion equation which is a model for biological invasion and disease spread. A nonsmooth feedback control term is included and the existence of controlled dynamics is proved, satisfying different kinds of nonlocal condition. Jump discontinuities appear in the process. The existence of optimal control strategies is also shown, under suitably regular control functionals. The investigation makes use of techniques of multivalued analysis and is based on the degree theory for condensing operators in Hilbert spaces.

The classical Hartman's Theorem in [18] for the solvability of the vector Dirichlet problem will be generalized and extended in several directions. We will consider its multivalued versions for Marchaud and upper-Caratheodory right-hand sides with only certain amount of compactness in Banach spaces. Advanced topological methods are combined with a bound sets technique. Besides the existence, the localization of solutions can be obtained in this way.

This paper deals with the second order nonlinear difference equation $$ \dd(r_k\dd u_k)+q_kg(u_{k+1})=0, $$ where $ \{r_k\} $ and $ \{q_k\} $ are positive real sequences defined on $\N\cup \{0\}$, and the nonlinearity $g:\R \to \R $ is nonnegative and nontrivial. Sufficient and necessary conditions are given, for the existence of bounded solutions starting from a fixed initial condition $u_0$. The same dynamic, with $f$ instead of $g$ such that $uf(u)>0$ for $u\not=0$, was recently extensively investigated. On the contrary, our nonlinearity $ g $ is of a small appearance in the discrete case. Its introduction is motivated by the analysis of wavefront profiles in biological and chemical models. The paper emphasizes the many different dynamical behaviors caused by such a $g$ with respect to the equation involving function $f$. Some applications in the study of wavefronts complete this work.

An existence result for differential inclusions in a separable Hilbert space is furnished. A wide family of nonlocal boundary value problems is treated, including periodic, anti-periodic, mean value and multipoint conditions. The study is based on an approximation solvability method. Advanced topological methods are used as well as a Scorza Dragoni-type result for multivalued maps. The conclusions are original also in the single-valued setting. An application to a nonlocal dispersal model is given.

We describe in this paper a variance reduction method based on control variates. The technique uses the fact that, if all stochastic assets but one are replaced in the payoff function by their mean, the resulting integral can most often be evaluated in closed form. We exploit this idea by applying the univariate payoff as control variate and develop a general Monte Carlo procedure, called Mean Monte Carlo (MMC). The method is then tested on a variety of multifactor options and compared to other Monte Carlo approaches or numerical techniques. The method is of easy and broad applicability and gives good results especially for low to medium dimension and in high volatility environments.

The paper deals with the boundary value problem on the whole line $u'' - f(u,u') + g(u) = 0 $ $u(\infty,1) = 0$, $ u(+\infty) = 1$ $(P)$ where $g : R \to R$ is a continuous non-negative function with support $[0,1]$, and $f : R^2\to R$ is a continuous function. By means of a new approach, based on a combination of lower and upper-solutions methods and phase-plane techniques, we prove an existence result for $(P)$ when $f$ is superlinear in $u'$; by a similar technique, we also get a non-existence one. As an application, we investigate the attractivity of the singular point $(0,0)$ in the phase-plane $(u,u')$. We refer to a forthcoming paper [13] for a further application in the field of front-type solutions for reaction diffusion equations.

Criteria for the existence and uniqueness of a solution of the boundary value problem are established, where ƒ :]a, b[×R 2 → R satisfies the local Carathéodory conditions, and μ : [a, b] → R is the function of bounded variation. These criteria apply to the case where the function ƒ has nonintegrable singularities in the first argument at the points a and b.

Hartman-type conditions are presented for the solvability of a multivalued Dirichlet problem in a Banach space by means of topological degree arguments, bounding functions, and a Scorza-Dragoni approximation technique. The required transversality conditions are strictly localized on the boundaries of given bound sets. The main existence and localization result is applied to a partial integro-differential equation involving possible discontinuities in state variables. Two illustrative examples are supplied. The comparison with classical single-valued results in this field is also made.

A new approach is developed for the solvability of nonlocal problems in Hilbert spaces associated to nonlinear differential equations. It is based on a joint combination of the degree theory with the approximation solvability method and the bounding functions technique. No compactness or condensivity condition on the nonlinearities is assumed. Some applications of the abstract result to the study of nonlocal problems for integro-differential equations and systems of integro-differential equations are then showed. A generalization of the result by using nonsmooth bounding functions is given.

The existence and localization result is obtained for a multivalued Dirichlet problem in a Banach space. The upper-Caratheodory and Marchaud right-hand sides are treated separately because in the latter case, the transversality conditions derived by means of bounding functions can be strictly localized on the boundaries of bound sets.

The paper deals with a semilinear evolution equation in a reflexive and separable Banach space. The non-linear term is multivalued, upper Caratheodory and it depends on a retarded argument. The existence of global almost exact, i.e. classical, solutions is investigated. The results are based on a continuation principle for condensing multifields and the required transversalities derive from the introduction of suitable generalized guiding functions. As a consequence, the equation has a bounded globally viable set. The results are new also in the lack of retard and in the single valued case. A brief discussion of a non-local diffusion model completes this investigation.

A two-point boundary value problem associated to a semilinear multivalued evolution equation is investigated, in reflexive and separable Banach spaces. To this aim, an original method is proposed based on the use of weak topologies and on a suitable continuation principle in Frechet spaces. Lyapunov-like functions are introduced, for proving the required transversality condition. The linear part can also depend on the state variable x and the discussion comprises the cases of a nonlinearity with sublinear growth in x or of a noncompact valued one. Some applications are given, to the study of periodic and Floquet boundary value problems of partial integro-differential equations and inclusions appearing in dispersal population models. Comparisons are included, with recent related achievements.

The paper deals with the multivalued boundary value problem x 0 2 A(t, x)x+ F(t, x) for a.a. t 2 (a, b), Mx(a) + Nx(b) = 0, in a separable, reflexive Banach space E. The nonlinearity F is weakly upper semicontinuous in x. We prove the existence of global solutions in the Sobolev space W 1,p ((a, b), E) with 1 < p < 1 endowed with the weak topol- ogy. We consider the case of multiple solutions of the associated homogeneous linearized problem. An example completes the discussion.

Continuous dependence of the threshold wave speed and of the travelling wave profiles for reaction-diffusion-convection equations $ u_t + h(u)u_x = (d(u)u_x)_x + f(u)$ is here studied with respect to the diffusion, reaction and convection terms.

The paper deals with the initial value problem for the degenerate reaction-diffusion-convection equation ut + h(u)ux = (um)xx + f(u), x ∈ ℝ, t > 0, where h is continuous, m > 1, and f is of Fisher-type. By means of comparison type techniques, we prove that the equilibrium u ≡ 1 is an attractor for all solutions with a continuous, bounded, non-negative initial condition u0(x) = u(x, 0) ≢ 0. When u0 is also compactly supported and satisfies 0 ≤ u0 ≤ 1, the convergence is such that an asymptotic estimate of the interface can be obtained. The employed techniques involve the theory of travelling-wave solutions that we improve in this context. The assumptions on f and h guarantee that the threshold speed wavefront is not stationary and we show that the asymptotic speed of the interface equals this minimal speed.

Solutions in a given set of the Floquet boundary value problem are investigated for second-order Marchaud systems. The methods used involve a fixed point index technique developed by ourselves earlier with a bound sets approach. Since the related bounding (Liapunov-like) functions are strictly localized on the boundaries of parameter sets of candidate solutions, some trajectories are allowed to escape from these sets. The main existence and localization theorem is illustrated by two examples for periodic and anti-periodic problems.

A continuation principle is established for the solvability of vector second-order boundary value problems associated with upper-Caratheodory differential inclusions. For Floquet second-order problems, this principle is combined with a bound sets approach. The viability result is also obtained in this way.

Front propagation for the aggregation-diffusion-reaction equation v τ = [D(v)vx]x + f(v) is investigated, where f is a bi-stable reaction-term and D(v) is a diffusion coefficient with changing sign, modeling aggregating-diffusing processes. We provide necessary and sufficient conditions for the existence of traveling wave solutions and classify them according to how or if they attain their equilibria at finite times. We also show that the dynamics can exhibit the phenomena of finite speed of propagation and/or finite speed of saturation. © World Scientific Publishing Company.

We study the existence and properties of travelling wave solutions of the Fisher-KPP reaction-diffusion-convection equation ut+ h(u)ux= [D(u)ux]x+ g(u), where the diffusivity D(u) is simply or doubly degenerate. Both the cases when Ḋ(0) and Ḋ(1) are possibly zero real values or infinity, are treated. We discuss the effects, due to the presence of a convective term, concerning the property of finite speed of propagation. Moreover, in the doubly degenerate case we show the appearance of new types of profiles and provide their classification according to sharp relations between the nonlinear terms of the model. An application is also presented, concerning the evolution of a bacterial colony.