Local Existence Research Articles

Local Existence of Solutions to the Non-Resistive 3D MHD Equations with Power-law Type

Local Existence of Solutions to the Non-Resistive 3D MHD Equations with Power-law Type

Existence and blow up time estimate for a nonlinear Cauchy problem with variable exponents: theory and numerics

A nonlinear Cauchy problem of a wave equation with variable exponents is considered. The damping term is subjected to a coefficient of both variables space and time. We prove the local existence of solutions using the Galerkin approach while the blow-up result is obtained via the perturbed energy method. We finally end our paper with some numerical illustrations. Problems that are frequently used in the literature are special cases.

Stability and Hopf bifurcation in delayed age-structured SVIR epidemic model with vaccination and incubation

An age-structured SVIR epidemic model with vaccination and incubation in two delays effects is investigated. In the model, one delay expresses the incubation period of disease and another delay expresses the minimum infection age of which the infected individuals possesses the infection ability. For the stringency of analysis, the model is firstly transformed into a non-densely defined abstract Cauchy problem. The nonnegativity and boundedness of solutions, basic reproduction number R0 and the existence of equilibria are established. The characteristic equations of linearized systems at equilibria are calculated from the corresponding abstract Cauchy problem. When R0≤1, the global stability of disease-free equilibrium is proved, and hence the disease will be deracinated. When R0>1, the local stability and existence of Hopf bifurcation at the endemic equilibrium are established. Particularly, two different cases of Hopf bifurcation are discussed in detail. These results show that the transmission of disease with the incubation period and the minimum infection age have very complex dynamical behavior at the endemic equilibrium. Finally, the numerical examples and simulations are presented to verify the theoretical results and further exhibit the existence of stability switch.

Impeded thermal transport in aperiodic BN/C nanotube superlattices due to phonon Anderson localization

Anderson localization of phonons is a kind of phonon wave effect, which has been proved to occur in many structures with disorders. In this work, we introduced aperiodicity to boron nitride/carbon nanotube superlattices (BN/C NT SLs), and used molecular dynamics to calculate the thermal conductivity and the phonon transmission spectrum of the models. The existence of phonon Anderson localization was proved in this quasi one-dimensional structure by analyzing the phonon transmission spectra. Moreover, we introduced interfacial mixing to the aperiodic BN/C NT SLs and found that the coexistence of the two disorder entities (aperiodicity and interfacial mixing) can further decrease the thermal conductivity. In addition, we also showed that anharmonicity can destroy phonon localization at high temperatures. This work provides a reference for designing thermoelectric materials with low thermal conductivity by taking advantage of phonon localization.

Local existence of weak solutions to kinetic Cucker‐Smale‐Fokker‐Planck equation with singular commutation weights

In this paper, we investigate the Cauchy problem of the dimensional kinetic Cucker‐Smale‐Fokker‐Planck equation with singular communication weight behaving like as . First, local existence of weak solutions with finite kinetic energy is established for and . Second, for , we show that any initial datum with finite velocity moment of order launches a weak solution that propagates the initial velocity moment.

Local existence for the d‐dimensional magneto‐micropolar equations with fractional dissipation in Besov spaces

In this paper, we consider the Cauchy problem of the d‐dimensional magneto‐micropolar equations ( or ) with general fractional dissipation. The aim of this paper is to obtain the existence and uniqueness of solutions in the weakest possible inhomogeneous Besov spaces. Using the technical tools of Litttlewood–Paley decomposition and Besov spaces theory, we obtain the local existence in the functional setting of inhomogeneous Besov spaces. Furthermore, such solutions are unique only in 2D case.

Isotopes of nitrate and gadolinium fingerprints to assay human inputs in Guarani Aquifer System

The use of environmental tracers brings comprehensive benefits to the management of water resources since it helps to prevent their pollution, minimize public health risks, and thus reduce the impact of urbanization. In Brazil, the Guarani Aquifer System (GAS) has strategic and environmental importance, making its preservation and sustainable exploitation mandatory. The present study aimed at evaluating sources of contamination in the GAS using the combination of geochemical data and two environmental tracers: nitrate isotopes (15NNO3 and 18ONO3) and one rare earth element (Gadolinium-Gd). For that, five wells-four exploiting the GAS and one the Bauru Aquifer System (BAS)-were selected to discuss the human inputs in groundwater used for public supply in an urban area. Traditional physicochemical analyses were conducted for six campaign samplings and nitrate monitoring for this period was evaluated on a time scale, also considering the accumulated rainfall. Besides that, the double isotopic method (DIM), e.g., δ18ONO3 e δ15NNO3, was applied to identify the fractionation and enable the distinction of the nitrate contamination source. In addition, the determination of anomalies of Gd, a wastewater-derived contaminant, was also performed to verify recent human inputs in groundwater. The results show that the local existence of nitrate in the GAS and BAS-even at low concentrations (values from 0.26 to 6.68mg L-1)-originated from anthropogenic inputs (septic waste), as indicates the typical isotopic signals ratio in the isotopic approach. Associated with that, the evaluation of Gd permitted the separation of groundwater samples into older or more recent leakages. The use of environmental tracers to assess anthropogenic inputs in groundwater reiterates the importance of adopting more effective protection strategies for water resources management systems, in order to prevent contamination.

Incompressible Euler equations with stochastic forcing: A geometric approach

We consider a stochastic version of Euler equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden (1970). For the Euler equations on a compact manifold (possibly with smooth boundary) we establish local existence and uniqueness of a strong solution in spaces of Sobolev mappings (of high enough regularity). Our approach combines techniques from stochastic analysis and infinite-dimensional geometry and provides a novel toolbox to establish local well-posedness of stochastic non-linear partial differential equations.

Analytical solution for a cylinder glaciation model with variable latent heat and thermal diffusivity

We consider a Stefan type problem that models the glaciation of the outer surface of a cylindrical gas pipeline in sea water. An equivalent integral formulation is obtained, considering that the latent heat and the thermal diffusivity depend on the radial distance. When the radial component of the heat flux vector from sea water to glaciation front is inversely proportional to the radial distance, local existence and uniqueness of solution is shown. In the general case, existence is ensured but uniqueness of solution is not guaranteed.

Strong solutions to a nonlinear stochastic aggregation-diffusion equation

It is well-known that solutions to deterministic nonlocal aggregation-diffusion models may blow up in two or higher dimensions. Various mechanisms hence have been proposed to “regularize” the deterministic aggregation-diffusion equations in a manner that allows pattern formation without blow-up. However, stochastic effect has not been ever considered among other things. In this work, we consider a nonlocal aggregation-diffusion model with multiplicative noise and establish the local existence and uniqueness of strong solutions on [Formula: see text]. If the noise is non-autonomous and linear, we establish the global existence and large-time behavior of strong solutions with decay properties by combining the Moser-Alikakos iteration technique and some decay estimates of Girsanov type processes. If the noise is nonlinear and strong enough, we show that blow-up can be prevented. As such, our results assert that certain multiplicative noise can also regularize the aggregation-diffusion model.

Decay property of solutions to the wave equation with space‐dependent damping, absorbing nonlinearity, and polynomially decaying data

We study the large time behavior of solutions to the semilinear wave equation with space‐dependent damping and absorbing nonlinearity in the whole space or exterior domains. Our result shows how the amplitude of the damping coefficient, the power of the nonlinearity, and the decay rate of the initial data at the spatial infinity determine the decay rates of the energy and the ‐norm of the solution. In the appendix, we also give a survey of basic results on the local and global existence of solutions and the properties of weight functions used in the energy method.

Local Existence and Uniqueness of Heat Conductive Compressible Navier–Stokes Equations in the Presence of Vacuum Without Initial Compatibility Conditions

Local Existence and Uniqueness of Heat Conductive Compressible Navier–Stokes Equations in the Presence of Vacuum Without Initial Compatibility Conditions

Asymptotic stability of a quasi-linear viscoelastic Kirchhoff plate equation with logarithmic source and time delay

<abstract><p>In this paper, a quasi-linear viscoelastic Kirchhoff plate equation with logarithmic source and time delay involving free boundary conditions in a bounded domain is considered. The local existence and global existence are proved, respectively. Under the assumptions on a more general type of relaxation functions and suitable conditions on the coefficients between damping term and delay term, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions. As the considered assumption here on the kernel is more general than earlier papers, our result improves and generalizes earlier result in the literature.</p></abstract>

BLOW-UP OF SOLUTIONS TO THE SEMILINEAR WAVE EQUATIONS WITH FRICTIONAL AND VISCOELASTIC DAMPING TERMS

Our starting point in this paper is to investigate the weakly coupled system of semilinear wave equations with two types of damping terms and combined nonlinearities $|v_{t}|^{p_{1}}+|v|^{q_{1}}$, $|u_{t}|^{p_{2}}+|u|^{q_{2}}$ on exterior domain in $n\,(n\geq1)$ dimensions. Local existence and uniqueness of mild solutions to the problem are established. Moreover, non-existence of global solutions to the problem with power nonlinearities $|u|^{p},\,|v|^{q}$ in the case of coupled system and the problem with power nonlinearity $|u|^{p}$ in the case of single equation are verified. The proofs are based on the test function technique. It is worth noticing that the frictional damping $u_{t}$ is more dominant than the viscoelastic damping $\Delta u_{t}$ when the time trend to infinity. To the best of our knowledge, the blow-up results in Theorems 1.2--1.3 are new.

A mixed nonlinear time-fractional Rayleigh-Stokes equation

This paper investigates a nonlinear time-fractional Rayleigh-Stokes equation with mixed nonlinearity containing a power-type function, a logarithmic function and an inverse time-forcing term. Applying Lagrange's mean value theorem and the compactness of the Sobolev embeddings, we estimate the complex Lipschitz property of mixed nonlinearity. We investigate the local well-posed results (local existence, regularity estimate, continuation) of the solutions in Hilbert scales space. Moreover, the global existence theory affiliated to the finite-time blow-up is considered.

Sub-diffusion equations with Mittag-Leffler nonlinearity

<p style='text-indent:20px;'>The paper is devoted to the study of subdiffusion equations with Mittag-Leffler nonlinearity. The comparison principle is proved in a bounded domain. The results on the local existence, global existence, and blow-up of solutions to the initial-boundary value problem are obtained. In addition, the results of local existence and blow-up of solutions to the initial problem on the whole Euclidean space are proven.</p>

BLOW-UP PROBLEMS FOR FUJITA-TYPE PARABOLIC SYSTEM INVOLVING TIME-DEPENDENT COEFFICIENTS ON GRAPHS

In this paper, we deal with the blow-up problems for Fujita-type parabolic system involving time-dependent coefficients on graphs. Under appropriate conditions, we prove that the nonnegative solution of the parabolic system blows up in a finite time on finite graphs and locally finite graphs, respectively. The results obtained extend some previous results of [Y. Lin and Y. Wu, Blow-up problems for nonlinear parabolic equations on locally finite graphs, Acta Math. Sci. Ser. B 38(3) (2018) 843–856; Y. Wu, Local existence and blow-up of solutions to Fujita-type equations involving general absorption term on finite graphs, Fractals 30(2) (2022) 2240053].

A study of the time fractional Navier-Stokes equations for vertical flow

<abstract><p>Navier-Stokes (NS) equations dealing with gravitational force with time-fractional derivatives are discussed in this paper. These equations can be used to predict fluid velocity and pressure for a given geometry. This paper investigates the local and global existence and uniqueness of mild solutions to NS equations for the time fractional differential operator. We also work on the regularity effects of such types of equations were caused by orthogonal flow.</p></abstract>

Under nonlinear prey-harvesting, effect of strong Allee effect on the dynamics of a modified Leslie-Gower predator-prey model.

In the present study, the effects of the strong Allee effect on the dynamics of the modified Leslie-Gower predator-prey model, in the presence of nonlinear prey-harvesting, have been investigated. In our findings, it is seen that the behaviors of the described mathematical model are positive and bounded for all future times. The conditions for the local stability and existence for various distinct equilibrium points have been determined. The present research concludes that system dynamics are vulnerable to initial conditions. In addition, the presence of several types of bifurcations (e.g., saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, homoclinic bifurcation) has been investigated. The first Lyapunov coefficient has been evaluated to study the stability of the limit cycle that results from Hopf bifurcation. The presence of a homoclinic loop has been demonstrated by numerical simulation. Finally, possible phase drawings and parametric figures have been depicted to validate the outcomes.

A local Palais-Smale condition and existence of solitary waves for a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations

<abstract><p>This paper is concerned with a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \bigg\{ \begin{array}{rl} & u_t + (|u|^{p-2}u)_x + u_{xxx} +h_x(x-\tau t, y) +\beta \nabla_y v = 0, \\ & v_x = \nabla_y u.\end{array} $\end{document} </tex-math></disp-formula></p> <p>By proving a local Palais-Smale condition, we manage to prove the existence of solitary waves with the help of a variational characterization on the smallest positive constant of an anisotropic Sobolev inequality (Huang and Rocha, J. Inequal. Appl., 2018,163). The novelty is to give an <bold>explicit estimate</bold> on the sufficient condition of $ h $ to get the existence of solitary waves.</p></abstract>