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In this article we employ the nonlinear constitutive framework of isotropic electro-elasticity to derive universal relations. These are connections between the components of the total stress, the electric field and the deformation and, for a class of materials, are independent of the specific free energy function. Universal relations are derived by investigating the coaxiality of the total stress tensor and the corresponding deformation, but only the universal manifold method gives the general set of universal relations for a given material class. Universal relations must hold independently of the constitutive law for a given family of materials and can be used by the experimentalist to determine if a particular material should be included in such a family, i.e. the universal relations must be satisfied by the experimental data. To inform the experimentalist, we illustrate the universal relations for the full constitutive relation and show the consequences if the number of constitutive functions is reduced. In particular, we consider the homogeneous deformation known as simple shear and a non-homogeneous deformation of a cylindrical solid with circular cross-sectional area. The latter is one of the controllable states proposed by Singh and Pipkin [25].

In this paper, we face the question of describing the incremental motion of pre-stressed isotropic homogeneous compressible viscoelastic materials of differential type. We obtain a set of linear evolution equations which generalizes the previous mathematical description of the problem. Well-posedeness of the associated Cauchy problems and dissipation properties are established as well. In the final part, a reduction to a unique equation of the sixth order is derived, and a physical example is exhibited.

The paper presents a model of virus mutations and evolution of epidemics in a system of interacting individuals, where the intensity of the pathology, described by a real discrete positive variable, is heterogeneously distributed, and the virus is in competition with the immune system or therapeutical actions. The model is developed within the framework of the Kinetic Theory of Active Particles. The paper also presents a qualitative analysis developed to study the well-posedness of the mathematical problem associated to the general framework. Finally, simulations show the ability of the model to predict some interesting emerging phenomena, such as the mutation to a subsequent virus stage, the heterogeneous evolution of the pathology with the co-presence of individual carriers of the virus at different levels of progression, and the presence of oscillating time phases with either virus prevalence or immune system control.

We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative anisotropic Kirchhoff systems, involving the p(x)-Laplacian operator, governed by time-dependent nonlinear damping forces and strongly nonlinear power-like variable potential energies. This problem had been considered earlier for potential energies which arise from restoring forces, whereas here we allow also the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. The results are further extended to the more delicate problem involving higher order damping terms.

A free boundary problem for a nonlinear diffusion–convection equation is considered. The problem is reduced to a nonlinear integral equation in time which is shown to admit a unique solution for small time. An explicit traveling wave solution is also obtained, which travels with the same velocity as that of the free boundary.

This paper deals with decay properties for the solutions of a large class of ordinary differential systems, with time dependent restoring potential, which include the system $$\left({\left|{u\prime}\right|^{\mu - 2} u\prime}\right)\prime+\beta_1 t^{\theta_{\text{1}}}\left|{u\prime}\right|^{\mu-2}u\prime+\beta_2 t^{\theta_{\text{2}}}\left|{u\prime}\right|^{m-2}u\prime+ct^v\left|u\right|^{p-2}u=0,$$ t e [T, ∞), u : [T, ∞) → \(\mathbb{R}^N\), 1 < µ < m, ν ≥ 0, c0, β10, β2 ≥ 0, -1 ≤ θ1 < µ+ν-1, θ2 < m + ν - 1, and the nonlinear system $$Lu+T\left( t \right)u\prime+V\left( {t,u\prime } \right)+t^\nu Su=e\left(t\right),$$ where L and S are positive definite matrices, T is a skew-symmetric matrix continuous function, V is a quasilinear replacement of a linear resistive term R(t)u′ and e is continuous.

This paper is devoted to the qualitative analysis of a new broad class of nonlinear initial value problems that model evolution and selection in living systems derived by the mathematical tools of the kinetic theory of active particles. The paper is divided into two parts. The first shows how to obtain the nonlinear equations with proliferative/distructive nonlinear terms. The latter presents a detailed analysis of the related initial value problem. In particular, it is proved that the corresponding initial value problem admits a unique non--negative maximal solution. However, the solution cannot be in general global in time, due to the possibility of blow--up. The blow--up occurs when the biological life system is globally proliferative, see Theorem 3.3.

This paper presents the analysis of a system of ordinary differential equations modeling a socio-political competition toward a possible onset of extreme conflicts. The dependent variable is a probability density distribution, while the equations are characterized by quadratic type nonlinearities. The model was derived within the framework of the kinetic theory for active particles, where interactions are modeled according to game-theoretical tools. Global existence and uniqueness of the solutions to the initial value problem related to the model are proved in a special case. The main tool used is the Banach–Caccioppoli fixed point theorem.

In this paper we consider the problem of non-continuation of solutions of dissipative nonlinear Kirchhoff systems, involving the p(x)-Laplacian operator and governed by nonlinear driving forces f = f (t, x, u), as well as nonlinear external damping terms Q = Q(t, x, u, u t ), both of which could significantly dependent on the time t. The theorems are obtained through the study of the natural energy Eu associated to the solutions u of the systems. Thanks to a new approach of the classical potential well and concavity methods, we show the nonexistence of global solutions, when the initial energy is controlled above by a critical value; that is, when the initial data belong to a specific region in the phase plane. Several consequences, interesting in applications, are given in particular subcases. The results are original also for the scalar standard wave equation when p ≡ 2 and even for problems linearly damped.

The present study is devoted to modelling the onset and the spread of epidemics. The mathematical approach is based on the generalized kinetic theory for active particles. The modelling includes virus mutations and the role of the immune system. Moreover, the heterogeneous distribution of patients is also taken into account. The structure allows the derivation of specific models and of numerical simulations related to real systems.

We study the asymptotic stability for solutions of the nonlinear damped Kirchhoff system, with homogeneous Dirichlet boundary conditions, under fairly natural assumptions on the external force f and the distributed damping Q . Then the results are extended to a more delicate problem involving also an internal dissipation of higher order, the so called strongly damped Kirchhoff system. Finally, the study is further extended to strongly damped Kirchhoff–polyharmonic systems, which model several interesting problems of the Woinowsky–Krieger type.

The mathematical approach proposed in this paper refers to the modelling, and related mathematical problems, of large systems of interacting entities whose microscopic state includes not only geometrical and mechanical variables (typically position and velocity), but also peculiar functions or specific activities. The number of the above entities is sufficiently large for describing the overall state of the system using a suitable probability distribution over the microscopic state. The first part of the paper is devoted to the derivation of suitable mathematical structures which can be properly used to model a variety of models in different fields of applied sciences. Then some research perspectives are analyzed, focussed on applications to biological systems.

We consider the propagation of finite amplitude plane transverse waves in a class of homogeneous isotropic incompressible viscoelastic solids with memory. It is assumed that the Cauchy stress may be written as the sum of an elastic part and a dissipative viscoelastic part. The elastic part is of the form of the stress corresponding to a Mooney–Rivlin material, whereas the dissipative part depends not only on current but also on previous deformations. The body is first subjected to a homogeneous static deformation. It is seen that two finite amplitude transverse plane waves may propagate in every direction in the deformed body. It is also seen that finite amplitude circularly polarized waves may propagate along either n + or n − , where n + , n − are the normals to the planes of the central circular section of the ellipsoid x · B −1 x = 1. Here B is the left Cauchy–Green strain tensor corresponding to the finite static homogeneous deformation.

A one phase Stefan problem in nonlinear conduction is considered. The problem is shown to admit a unique solution for small times. An exact solution is obtained which is a travelling front moving with constant speed.

A two-phase free boundary problem associated with nonlinear heat conduction is considered. The problem is mapped into two one-phase moving boundary problems for the linear heat equation, connected through a constraint on the relative motion of their moving boundaries. Existence and uniqueness of the solution is proved for small times and a particular exact solution is discussed. Free Boundary Problems (FBP) motivated several studies in the past due to their relevance in applications [1 − 4]. From the mathematical point of view FBPare initial/ boundary value problems with a moving boundary [5]. The motion of the boundary is unknown (free boundary) and has to be determined together with the solution of the given partial differential equation. As a consequence the solution of FBPis in most cases equivalent to the solution of a nonlinear system. In recent studies [6 − 13] some free boundary problems for nonlinear evolution equations relevant in applications have been considered. In particular in [12] the solution of a one-phase Stefan Problem in nonlinear conduction is proved to exist and to be unique for short intervals of time. Furthemore a particular travelling wave solution was obtained. On the other hand two-phase Stefan Problems are more complicated than their one-phase counterparts and the theory is more elaborate. It is the aim of this paper to analyse a two–phase Stefan Problem for the nonlinear heat equation considered in [12]. Such an equation arises as a model of heat conduction in solid crystalline hydrogen [14]. It admits an exact linearization into the heat equation and therefore belongs to the class of C-integrable equations [16]. In the following we show that the two-phase Stefan Problem for the nonlinear heat equation admits a linearization into two distinct one-phase moving boundary problems for the linear heat equation. The two linearized problems are connected through a constraint on the relative motion of their moving boundaries. Such a constraint is induced by the free boundary motion of the nonlinear problem. We start our analysis with the following system of nonlinear heat equations