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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 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.

In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of the unknown function at the node of the graph and, moreover, the diffusivity can be degenerate. Our main result concerns a necessary and sufficient algebraic condition for the existence of traveling waves in the graph. We also study in great detail some examples corresponding to quadratic and logarithmic flux functions, for different diffusivities, to which our results apply.

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 a degenerate reaction-diffusion equation, including aggregative movements and convective terms. The model also incorporates a real parameter causing the change from a purely diffusive to a diffusive-aggregative and to a purely aggregative regime. Existence and qualitative properties of traveling wave solutions are investigated, and estimates of their threshold speeds are furnished. Further, the continuous dependence of the threshold wave speed and of the wave profiles on a real parameter is studied, both when the process maintains its diffusion-aggregation nature and when it switches from it to another regime.

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 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.

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 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.

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.

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.

A bound sets technique is developed for Floquet problems of Carathéodory differential inclusions. It relies on the construction of either continuous or locally ipschitzian Lyapunov-like bounding functions. Proceeding sequentially, the existence of bounded trajectories is then obtained. Nontrivial examples are supplied to illustrate our approach.

In this paper, we define a topological index for compact multivalued maps in convex metrizable subsets of a locally convex topological vector space in order to correct the proofs of Theorems 4.1 and 4.2 in Benedetti et al. (2011) [1] .

This paper investigates the effects of a degenerate diffusion term in reaction–diffusion models ut=[D(u)ux]x+g(u) with Fisher-KPP type g. Both in the case when D(0)=0 and when D(0)=D(1)=0, with D(u)>0 elsewhere, we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c∗ and we show the appearance of a sharp-type profile when c=c∗. These results solve recent conjectures formulated by Sánchez-Garduño and Maini (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339).

✩ This work was supported by the Research Grant of the Italian Ministry of Foreign Affairs (P.A. 31524/02) and carried out while the third author was visiting the University of Modena and Reggio Emilia and the Polytechnic University of Marche. * Corresponding author. Tel.: +39 0522 522616; fax: +39 0522 522609. E-mail addresses: malaguti.luisa@unimore.it (L. Malaguti), marcelli@dipmat.univpm.it (C. Marcelli), ninopa@rmi.acnet.ge (N. Partsvania).

This paper analyses front propagation of the equation $u_\tau=[D(u)v_x]_x +f(v) \;\;\; \tau < 0, x \in \mathbb{R}$ where $f$ is a monostable (ie Fisher-type) nonlinear reaction term and $D(v)$ changes its sign once, from positive to negative values,in the interval $ v \in[0,1]$ where the process is studied. This model equation accounts for simultaneous diffusive and aggregative behaviors of a population dynamic depending on the population density $v$ at time $\tau$ and position $x$. The existence of infinitely many travelling wave solutions is proven. These fronts are parametrized by their wave speed and monotonically connect the stationary states u = 0 and v = 1. In the degenerate case, i.e. when D(0) and/or D(1) = 0, sharp profiles appear, corresponding to the minimum wave speed. They also have new behaviors, in addition to those already observed in diffusive models, since they can be right compactly supported, left compactly supported, or both. The dynamics can exhibit, respectively, the phenomena of finite speed of propagation, finite speed of saturation, or both.