Acta Applicandae Mathematicae最新文献

In this paper, we consider the Cauchy problem for the spatially inhomogeneous relativistic Boltzmann equation where the collision kernel is generated by Israel particle. Unique global (in time) classical solution is obtained in a suitable weighted space by considering small initial data and the Bianchi type I space-time as background.

A multiple degenerate parabolic equation related to the (p(x,t))-Laplacian is considered. Since it is with multiple degeneracy, how to obtain the (L^{infty })-estimate becomes difficult, and the usual Dirichlet boundary value condition may be invalid or overdetermined. By adding some restrictions on the growth order, using the maximum value principle, the corresponding (L^{infty })-estimate of the weak solution is obtained first time. Since the solution is so weak that its trace on the boundary cannot be defined in the conventional manner. By employing the weak characteristic function method introduced in (Zhan and Feng in J. Differ. Equ. 268:389–413, 2020)), the classical trace in (W_{0}^{1,p(cdot )}(Omega )) is generalized to the function space (W_{loc}^{1, p(cdot )}(Omega )bigcap L^{infty }(Omega )). Through this framework, the partial boundary value condition is imposed on a submanifold of (partial Omega times (0,T)), thereby establishing the stability of weak solutions.

A three-dimensional mathematical model of a viscous incompressible fluid with two stiff particles is investigated in the near-contact regime. When one of the particles approaches the other motionless particle with prescribed translational and angular velocities, there always appears blow-up of hydrodynamic force exerted on the moving particle. In this paper, we construct explicit singular functions corresponding to the fluid velocity and pressure to establish precise asymptotic formulas for hydrodynamic force with respect to small interparticle distance, which show that its largest singularity is determined by squeeze motion between two particles. Finally, the primal-dual variational principle is employed to give a complete justification for these asymptotics.

This article studies a (2times 2) hyperbolic system of conservation laws with general source term whose Riemann problem is solved with the use of the variable substitution. Four kinds of solutions involving delta-shock (delta standing wave) are constructed. We clarify the generalized Rankine-Hugoniot relation and entropy condition which are used to determine the position, propagation speed and strength of the delta-shock. The solutions are non-self-similar under the influence of source term. Compared with the homogeneous case, only the strength of delta-shock has changed, while the position and propagation speed of the delta-shock remain unchanged. Additionally, we propose a time-dependent viscous system to show the stability of the solutions including delta-shocks by adopting the vanishing viscosity method.

This study examines the propagation of Love waves in two layer model, consisting of an inhomogeneous isotropic upper layer with damping, overlying a homogeneous lower layer and a homogeneous half-space. To provide a more flexible measure of damping, fractional damping is introduced in the upper layer using Caputo derivative. The Laplace transform and Green’s function approach are employed to derive the displacement in the upper layer in transformed plane. The inverse Laplace transform is computed numerically using Stehfest’s algorithm and results are presented through numerical simulations and graphical illustrations.

In this article we derive the equations that constitute the nonlinear mathematical model of extensible Timoshenko microbeam with memories based on the modified couple stress theory. The nonlinear governing equations are derived by applying the Hamilton principle to full von Kármán equations together with Boltzmann theory for viscoelastic materials. The model takes into account the effects of extensibility, where the dissipation is entirely contributed by memories. Based on semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. By using a resolvent criterion, developed by Borichev and Tomilov, we prove the optimality of the polynomial decay rate of the derived equations without extensibility when the viscoelastic law acts only on the shear force under the condition (4.10). In particular, we show that the considered problem is not exponentially stable. Moreover, by following a result due to Arendt-Batty, we show that the derived problem (without extensibility) is strongly stable.

The age-structured approach plays a crucial role in epidemiological modelling as it accounts for age-specific variations in susceptibility, transmission and disease progressions, providing a more accurate description of disease dynamics. In this paper, we create an age-structured epidemic model that incorporates age-dependent susceptibility and latency, as well as a relapse phase, with the objective of investigating the global dynamics of this model under the impact of that combination. The very important threshold parameter (mathcal{R}_{0}) was introduced, and it has shown that it completely controls the stability of each equilibrium of the model. Based on the Lyapunov functional approach, we show that the disease-free equilibrium is globally asymptotically stable when (mathcal{R}_{0}<1), while the positive endemic equilibrium is globally asymptotically stable whenever (mathcal{R}_{0}>1). Our results suggest that early diagnostic of latency individuals, reduction in transmission rate and improvements in treatment and heath-care of infected individuals may effectively control the spread of the disease.

Exact solutions describing trapped modes in a plane quantum waveguide with a small rigid obstacle are constructed in the form of convergent series in powers of the small parameter characterizing the smallness of the obstacle. The terms of this series are expressed through the solution of the exterior Neumann problem for the Laplace equation describing the flow of unbounded fluid past the inflated obstacle. The exact solutions obtained describe discrete eigenvalues of the problem under certain geometric conditions, and, when the obstacle is symmetric, these solutions describe embedded eigenvalues. For obstacles symmetric with respect to the centerline of the waveguide, the existence of embedded trapped modes is known (due to the decomposition trick of the domain of the corresponding differential operator) even without the smallness assumption. We construct these solutions in an explicit form for small obstacles. For obstacles symmetric with respect to the vertical axis, we find embedded trapped modes for a specific vertical displacement of the obstacle.

In this paper we obtain the local regularity estimates in Besov spaces of weak solutions for a class of elliptic obstacle problems with variable exponents (p(x)). We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality in the following form

under some proper assumptions on the function (p(x)), (A), (varphi ) and (F). Moreover, we would like to point out that our results improve the known results for such problems.