1,720,963 research outputs found
On an initial value problem modeling evolution and selection in living systems
No full textThis 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
Decay and stability for some nonlinear quasi-variational systems
No full textthe paper deals with the decay and stability problem for a nonlinear quasi-variational syste
Global Nonexistence for Nonlinear Kirchhoff Systems
No full textIn 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(x)=2 and even for problems linearly damped
Incremental equations for pre-stressed compressible viscoelastic materials
No full textIn 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
Preface to this special issue - Recent contributions in nonlinear analysis
No full textThis special issue of Discrete and Dynamical Systems - Series S is a tribute to
Patrizia Pucci on the occasion of her 65th birthda
Mathematical tools of the kinetic theory of active particles with some reasoning on the modelling progression and heterogeneity.
No full textOn the use of universal relations in modeling nonlinear electro-elastic materials
No full textIn 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 Pipki
Asymptotic stability for nonlinear Kirchhoff systems
No full textWe 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
Asymptotic stability for anisotropic Kirchhoff systems
No full textAbstractWe 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
Existence and uniqueness of solutions to a Cauchy problem modeling the dynamics of socio-political conflicts
No full textThis 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
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