Zeitschrift für angewandte Mathematik und Physik最新文献

In the paper, we consider a system of thermoelasticity of type I with viscoelastic damping and nonlinear sources. By using the Galerkin method and the Banach fixed point theorem, we first prove the local existence and uniqueness of weak solution. Secondly, by extending the potential well method, we prove that the local solution exists globally if its initial position starts inside a family of “potential wells.” In particular, we also establish an explicit and optimal decay rate of energy driven by the decay rate of the relaxation function which includes exponential, algebraic, and logarithmic decay rates. Finally, by using the continuation theorem and the concavity arguments due to Levine (Trans Am Math Soc 192:1–21, 1974), we show that the local solution blows up at finite time in the sense of Ball (Q J Math Oxf 28(4): 473–486, 1977) if its initial position starts outside the “potential wells.” Further, an upper bound for the blow-up time is also given explicitly. Notice that our results imply a sharp result on the global existence and blow-up of the local weak solution and they also allow a relatively large class of relaxation functions that generalize the existing results in the literature.

This work is concerned with the fractional-in-time parabolic–parabolic Keller–Segel system in a bounded domain (Omega subset mathbb {R}^{d}) ((dge 2)), for distinct fractional diffusions of the cells and chemoattractant. We prove results on existence, uniqueness, continuous dependence on the initial data and its robustness, continuation, and a blow-up alternative of solutions in Lebesgue spaces. Then, we use those results to show the existence of global solutions to the problem, when the chemoattractant diffusion is not slower than the cell diffusion.

In this article, we propose the fractional weak adversarial networks (f-WANs) for the stationary fractional advection dispersion equations based on their weak formulas. This enables us to handle less regular solutions for the fractional equations. To handle the non-local property of the fractional derivatives, convolutional layers and special loss functions are introduced in this neural network. Numerical experiments for both smooth and less regular solutions show the validity of f-WANs.

We prove the existence of weak solutions for a class of second-order traffic models with relaxation, without requiring the sub-characteristic stability condition to hold. With the help of numerical simulations, we show how, in this unstable setting, large but bounded oscillations may arise from small perturbations of equilibria, thus reproducing the formation of stop-and-go waves commonly observed in traffic dynamics. An analysis of the corresponding traveling waves completes the study.

This paper is concerned with the bifurcation properties of the standing wave solutions for the Schrödinger–Poisson system with doubly critical case

The study of system (({mathcal {P}})) is motivated by its important applications in many physical models, such as the quantum mechanical systems under external influences. Here, (3le Nle 6),(0<alpha <N), (lambda in {mathbb {R}}), g is a nonnegative weight function, and (2_alpha ^sharp ) and (2_alpha ^*) are the lower and upper Hardy–Littlewood–Sobolev critical exponents, respectively. Moreover, when (N=6) and (0<alpha <2) existence of the (weak) solutions of the system under consideration is also proved via the global bifurcation theorem due to Rabinowitz.

For a given material, controllable deformations are those deformations that can be maintained in the absence of body forces and by applying only boundary tractions. For a given class of materials, universal deformations are those deformations that are controllable for any material within the class. In this paper, we characterize the universal deformations in compressible isotropic implicit elasticity defined by solids whose constitutive equations, in terms of the Cauchy stress (varvec{sigma }) and the left Cauchy-Green strain (textbf{b}), have the implicit form (varvec{textsf{f}}(varvec{sigma },textbf{b})=textbf{0}). We prove that universal deformations are homogeneous. However, an important observation is that, unlike Cauchy (and Green) elasticity, not every homogeneous deformation is constitutively admissible for a given implicit-elastic solid. In other words, the set of universal deformations is material-dependent, yet it remains a subset of homogeneous deformations.

We consider the Cahn–Hilliard/Navier–Stokes system with non–degenerate mobility in the space–periodic case, describing the flow of two viscous immiscible and incompressible Newtonian fluids with matched densities. We identify sufficient conditions on the velocity field for weak solutions to satisfy an energy identity, improving previous results on the literature.

In this article, a reaction–diffusion model of HIV immunity with chemotaxis and absorption effect is constructed. The paper proves the existence and boundedness of the global classical solution of this mode when the chemotactic coefficient is kept in a suitable range. Five equilibrium points are established based on the different ranges of the basic regeneration number, two immune reproduction numbers and the immune competitive reproduction number. The global asymptotic stability of each equilibrium point is established by constructing an appropriate Lyapunov function when the chemotactic coefficient is kept in a small interval.

We consider semiclassically scaled, weakly nonlinear Schrödinger equations with external confining potentials and additional angular-momentum rotation term. This type of model arises in the Gross–Pitaevskii theory of trapped, rotating quantum gases. We construct asymptotic solutions in the form of semiclassical wave packets, which are concentrated in both space and in frequency around an classical Hamiltonian phase-space flow. The rotation term is thereby seen to alter this flow, but not the corresponding classical action.

In this paper, an extension of the Timoshenko model for plane beams is outlined, with the aim of describing, under the assumption of small displacements and strains, a class of dissipative mechanisms observed in cementitious materials. In the spirit of micromorphic continua, the modified beam model includes a novel kinematic descriptor, conceived as an average sliding relevant to a density of micro-cracks not varying along time. For the pairs of rough surfaces, in which such a distribution of micro-cracks is articulated, both an elastic deformation and a frictional dissipation are considered, similarly to what occurs for the fingers of the joints having a tooth saw profile. The system of governing differential equations, of the second order, is provided by a variational approach, endowed by standard boundary conditions. To this purpose, a generalized version of the principle of virtual work is used, in the spirit of Hamilton–Rayleigh approach, including as contributions: (i) the variation of the inner elastic energy, generated by the linear elasticity of the sound material and, in a nonlinear way, by the mutual, reversible deformation of the asperities inside the micro-cracks; (ii) the virtual work of the external actions consistent with the beam model, i.e., the distributed transversal forces and the moments per unit lengths; besides these two contributions, constituting the conservative part of the system, (iii) the dissipation due to friction specified through a smooth Rayleigh potential, entering a nonlinear dependence of viscous and Coulomb type on the sliding rate. Through a COMSOL Multiphysics implementation, 1D finite element analyses are carried out to simulate structural elements subjected to three- and four-point bending tests with alternating loading cycles. The dissipation of energy is investigated at varying the model parameters, and the predictions turn out to be in agreement with preliminary data from an experimental campaign. The present approach is expected to provide a valuable tool for the quantitative and comparative assessment of the hysteresis cycles, favoring the robust design of cementitious materials.