Narrow Results

In this paper, we consider the nonlinear cauchy problem

In this paper, we study the local existence of a classical solution to the coupled Einstein and Maxwell–Klein–Gordon system in higher dimensions. This system provides a toy model to study the dynamics of the complex scalar fields under the influence of the interaction of the gravitational and electromagnetic fields. In the starting point, we introduce the metric of the space-time in the Bondi coordinate. Then, we construct the evolution equation in the form of a single first-order nonlinear integro-differential equation. Furthermore, we show that there exists a unique fixed point that is the solution of the main problem based on the contraction mapping arguments. Finally, for a given initial data, we prove the existence of a local classical solution.

In this paper we study the existence and continuation of solution to the general fractional differential equation (FDE) with Riemann–Liouville derivative. If no confusion appears, we call FDE for brevity. We firstly establish a new local existence theorem. Then, we derive the continuation theorems for the general FDE, which can be regarded as a generalization of the continuation theorems of the ordinary differential equation (ODE). Such continuation theorems for FDE which are first obtained are different from those for the classical ODE. With the help of continuation theorems derived in this paper, several global existence results for FDE are constructed. Some illustrative examples are also given to verify the theoretical results.

We propose here a decomposition of the respiratory tree into three stages which correspond to different mechanical models. The resulting system is described by the Navier–Stokes equation coupled with an ODE (a simple spring model) representing the motion of the thoracic cage. We prove that this problem has at least one solution locally in time for any data and, in the special case where the spring stiffness is equal to zero, we obtain an existence result globally in time provided that the data are small enough. The behavior of the global model is illustrated by three-dimensional simulations.

The analysis of criminal behavior with mathematical tools is a fairly new idea, but one which can be used to obtain insight on the dynamics of crime. In a recent work,³⁴ Short et al. developed an agent-based stochastic model for the dynamics of residential burglaries. This model produces the right qualitative behavior, that is, the existence of spatio-temporal collections of criminal activities or "hotspots", which have been observed in residential burglary data. In this paper, we prove local existence and uniqueness of solutions to the continuum version of this model, a coupled system of partial differential equations, as well as a continuation argument. Furthermore, we compare this PDE model with a generalized version of the Keller–Segel model for chemotaxis as a first step to understanding possible conditions for global existence versus blow-up of the solutions in finite time.

We investigate local/global existence, blowup criterion and long-time behavior of classical solutions for a hyperbolic–parabolic system derived from the Keller–Segel model describing chemotaxis. It is shown that local smooth solution blows up if and only if the accumulation of the L^∞ norm of the solution reaches infinity within the lifespan. Our blowup criteria are consistent with the chemotaxis phenomenon that the movement of cells (bacteria) is driven by the gradient of the chemical concentration.

Furthermore, we study the long-time dynamics when the initial data is sufficiently close to a constant positive steady state. By using a new Fourier method adapted to the linear flow, it is shown that the smooth solution exists for all time and converges exponentially to the constant steady state with a frequency-dependent decay rate as time goes to infinity.

In this first part, we study the existence and uniqueness of solutions of a general nonlinear Schrödinger system in the presence of diamagnetic field, local and nonlocal nonlinearities. This kind of systems models many important phenomena in nonlinear optics; multimodal optical fibers, optical pulse propagation, ferromagnetic film and optical pulse propagation in the birefringent fibers. They also govern the interaction of electron and nucleii through Coulombic potential and under the action of external magnetic field in quantum mechanics.

In this paper, we investigate the following Keller–Segel system with flux limitation

This work deals with the proof of local existence theorem of solutions for coupled nonlocal singular viscoelastic system with respect to the nonlinearity of source terms by using the Faedo–Galerkin method together with energy methods. This work makes a new contribution, since most of the previous works did not address the proof of the theorem of the local existence of solutions. It is also a completed study of Boulaaras et al. [Adv. Differ. Equ. 2020 (2020) 310].

This paper is devoted to the study of the behaviors of the solution to Fujita-type equations on finite graphs. Under certain conditions given by absorption term of the equations, we prove respectively local existence and blow-up results of solutions to Fujita-type equations on finite graphs. Our results contain some previous results as special cases. Finally, we provide some numerical experiments to illustrate the applicability of the obtained results.

In this paper, we construct for every $\alpha >0$ and $\lambda \in ℂ$ a class of initial values $u_0$ for which there exists a local solution of the nonlinear Schrödinger equation $i{u}_t+\mathrm{\Delta }u+\lambda |u{|}^{\alpha }u=0$ on ${ℝ}^N$ with the initial condition $u(0,x)=u_0$. Moreover, we construct for every $\alpha >\frac{2}{N}$ a class of (arbitrarily large) initial values for which there exists a global solution that scatters as $t\to \infty$.

The skew mean curvature flow (SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfaces in Euclidean space ${ℝ}^4$. A Sobolev-type embedding theorem for the second fundamental forms of two-dimensional surfaces is also proved, which might be of independent interest.

The existence of classical solutions to the Cauchy problem for the Boltzmann equation without angular cutoff has been extensively studied in the framework when the solution has Maxwellian decay in the velocity variable, cf. [6, 8] and the references therein. In this paper, we prove local existence of solutions with polynomial decay in the velocity variable for the Boltzmann equation with soft potential. In the proof, the singular change of variables between post- and pre-collision velocities plays an important role, as well as the regular one introduced in the celebrated cancelation lemma by Alexandre–Desvillettes–Villani–Wennberg [Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal.  152 (2000) 327–355].

In this paper, we consider the Cauchy problem of a viscous compressible shallow water equations with the Coriolis force term and non-constant viscosities. More precisely, the viscous coefficients are constants multiple of height, the equations are degenerate when vacuum appears. For initial data allowing vacuum, the local existence of strong solution is obtained and a blow-up criterion is established.

In this paper, we study the Cauchy problem of the two-species incompressible viscoelastic fluid of Oldroyd-B system, which involving a reaction effect between two species of polymers. We prove the local existence with initial data in $H^k(k\ge 2)$ in a classical solution framework, and then provide a blow-up criteria. We concentrate on the a priori estimate, by using the energy method. In particular, the variant system in a general formulation is also studied, and the corresponding local well-posedness is established.

In this work we are concerned with a local existence of certain semi-linear wave equations for which the initial data has minimal regularity. Assuming the initial data are in H^1+∊ and H^∊ for any ∊ > 0, we prove a local result by using a fixed point argument, the main ingredient being an a priori estimate for the quadratic nonlinear term uDu. The technique applies to the Yang–Mills equations in the Lorentz gauge.

We construct a metric such that the Cauchy problem for the corresponding nonlinear wave equation admits a nontrivial solution but the solution map is not uniformly continuous.

Global existence of solutions is proved and asymptotic behavior is investigated, in the case of a positive cosmological constant and positive initial velocity of the cosmological expansion factor.

We study the Cauchy problem for multi-dimensional compressible relativistic hydrodynamics in the presence of a radiation field. First, based on the theory of quasilinear symmetric hyperbolic, we establish the local existence of smooth solutions for both non-vacuum and vacuum cases. Next, in the spirit of Sideris’ work [T. Sideris, Formation of singularities of solutions to nonlinear hyperbolic equations, Arch. Ration. Mech. Anal.86 (1984) 369–381; T. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys.101 (1985) 475–485], we show that smooth solutions blow-up in finite time if the initial data is compactly supported and large enough. Compared with the previous work, the main difficulties of the first problem lie in two aspects, we must first deal with the source terms relying on radiative quantities, and we also need to solve out the new coefficients matrices under the Lorentz transformation for vacuum case. The second difficulty arises on how to verifying that the smooth solution has finite propagation speed..

We study the density-dependent incompressible Cahn–Hilliard–Navier–Stokes system, which describes a two-phase flow of two incompressible fluids with different densities. We establish the local existence and uniqueness of strong solutions to the initial value problem in a bounded domain, when the initial density function enjoys a positive lower bound.