Local Existence Research Articles

Existence and energy decay rate of the solutions for the wave equation with a nonlinear distributed delay

<abstract><p>This paper is concerned with the wave equation having a nonlinear distributed delay. First, we prove the local existence of the solutions by using the semigroup theory, where the source term is globally Lipschitz. Next, we establish the global existence of solutions and the energy decay result under the local Lipschitz source and suitable conditions on the initial data.</p></abstract>

Hyperbolic Navier-Stokes equations in three space dimensions

We consider in this paper a hyperbolic quasilinear version of the Navier-Stokes equations in three space dimensions, obtained by using Cattaneo type law instead of a Fourier law. In our earlier work [2], we proved the global existence and uniqueness of solutions for initial data small enough in the space H4(R3)3 ? H3(R3)3. In this paper, we refine our previous result in [2], we establish the existence under a significantly lower regularity. We first prove the local existence and uniqueness of solution, for initial data in the space H5 2 +?(R3)3 ?H32 +?(R3)3, ? > 0. Under weaker smallness assumptions on the initial data and the forcing term, we prove the global existence of solutions. Finally, we show that if ? is close to 0, then the solution of the perturbed equation is close to the solution of the classical Navier-Stokes equations.

Population Projection of Mozambique

In this paper, the evolution of a dynamical system representing the population growth of Mozambique is analyzed. The correspondence between the boundary value problem and the integral equation is leveraged to address the issue of the local existence of solutions to the boundary value problem. The conclusion that the function converges to a function that is the unique solution to the boundary value problem is arrived at by way of constructing a sequence of approximations using Picard’s method of successive approximations and contraction mapping. The exponential function is globally Lipschitz, hence uniformly continuous; however, its solution does not converge to a fixed point implying that the population will grow without bounds as t→∞. The logistic model solves T (ϕ) = ϕ, whence T has a unique fixed point ϕ that is a continuous solution to the integral equation and consequently to the boundary value problem. Therefore, population growth is bounded. In addition, this function is locally Lipschitz and, therefore, not uniformly continuous.

Local Solution of Three-Dimensional Axisymmetric Supersonic Flow in a Nozzle

In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.

Stability analysis and Hopf bifurcation of a delayed prey–predator model with Hattaf–Yousfi functional response and Allee effect

The Allee effect is an important phenomena in the context of ecology characterized by a correlation between population density and the mean individual fitness of a population. In this work, we examine the influences of Allee effect on the dynamics of a delayed prey–predator model with Hattaf–Yousfi functional response. We first prove that the proposed model with Allee effect is mathematically and ecologically well-posed. Moreover, we study the stability of equilibriums and discuss the local existence of Hopf bifurcation.

Asymptotic expansion of solutions for the Robin-Dirichlet problem of Kirchhoff-Carrier type with Balakrishnan-Taylor damping

In this paper, we consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type with Balakrishnan-Taylor damping. First, under suitable conditions on the initial data, the local existence and uniqueness of a weak solution are proved. Next, an asymptotic expansion of solutions in a small parameter with high order is established. The used main tools are the linearization method for nonlinear terms together with the Faedo-Galerkin method, and the key lemmas of the expansion of high-order polynomials and the Taylor expansion for multi-variable functions.

On a nonlinear system of plate equations with variable exponent nonlinearity and logarithmic source terms: Existence and stability results

<abstract><p>In this paper, we consider a coupling non-linear system of two plate equations with logarithmic source terms. First, we study the local existence of solutions of the system using the Faedo-Galerkin method and Banach fixed point theorem. Second, we prove the global existence of solutions of the system by using the potential wells. Finally, using the multiplier method, we establish an exponential decay result for the energy of solutions of the system. Some conditions on the variable exponents that appear in the coupling functions and the involved constants that appear in the source terms are determined to ensure the existence and stability of solutions of the system. A series of lemmas and theorems have been proved and used to overcome the difficulties caused by the variable exponent and the logarithmic nonlinearities. Our result generalizes some earlier related results in the literature from the case of only constant exponent of the nonlinear internal forcing terms to the case of variable exponent and logarithmic source terms, which is more useful from the physical point of view and needed in several applications.</p></abstract>

Half-harmonic gradient flow: aspects of a non-local geometric PDE

<abstract><p>The goal of this paper is to discuss some of the results in the author's previous papers and expand upon the work there by proving two new results: a global weak existence result as well as a first bubbling analysis for the half-harmonic gradient flow in finite time. In addition, an alternative local existence proof to the one provided in <sup>[<xref ref-type="bibr" rid="b47">47</xref>]</sup> is presented based on a fixed-point argument. This preliminary bubbling analysis leads to two potential outcomes for the possibility of finite-time bubbling until a conjecture by Sire, Wei and Zheng, see <sup>[<xref ref-type="bibr" rid="b40">40</xref>]</sup>, is settled: Either there always exists a global smooth solution to the half-harmonic gradient flow without concentration of energy in finite-time, which still allows for the formation of half-harmonic bubbles as $ t \to +\infty $, or finite-time bubbling may occur in a similar way as for the harmonic gradient flow due to energy concentration in finitely many points. In the first part of the introduction to this paper, we provide a survey of the theory of harmonic and fractional harmonic maps and the associated gradient flows. For clarity's sake, we restrict our attention to the case of spherical target manifolds $ S^{n-1} $, but our discussion extends to the general case after taking care of technicalities associated with arbitrary closed target manifolds $ N $ (cf. <sup>[<xref ref-type="bibr" rid="b48">48</xref>]</sup>).</p></abstract>

Local well-posedness for 2D stochastic tropical climate model

In this paper, we consider a class of 2D tropical climate model under random perturbations, which is used to describe the interaction of large-scale flow fields and precipitation in the tropical atmosphere. Under some appropriate assumptions, we establish the local existence and uniqueness of the 2D stochastic tropical climate model. Moreover, we also discuss the blow-up criterion of the solution.

AN ALGORITHMIC APPROACH TO LOCAL SOLUTION OF THE NONLINEAR SECOND ORDER ORDINARY HYBRID INTEGRODIFFERENTIAL EQUATIONS

In this paper, we establish a couple of approximation results for local existence and uniqueness of the solution of an IVP of nonlinear second order ordinary hybrid integrodifferential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2022) and Dhage et al. (2022). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Finally, our main abstract results are also illustrated with a couple of numerical examples.

Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials

<abstract><p>In this article, we investigate the initial-boundary value problem for a class of finitely degenerate semilinear parabolic equations with singular potential term. By applying the Galerkin method and Banach fixed theorem we establish the local existence and uniqueness of the weak solution. On the other hand, by constructing a family of potential wells, we prove the global existence, the decay estimate and the finite time blow-up of solutions with subcritical or critical initial energy.</p></abstract>

ON THE NONCHARACTERISTICITY OF A CAUCHY SURFACE AND APPLICATIONS

In mathematical physics, Cauchy surfaces play a crucial role in defining initial data for the evolution of physical systems. The characteristic nature of Cauchy surface means that information about the future evolution of the system can be obtained from the data on the surface itself. However, not all Cauchy surfaces are characteristic. This article aims to investigate the noncharacteristic nature of Cauchy surfaces and their applications in various physical scenarios. We demonstrate that the noncharacteristicity of a Cauchy surface has important implications for the behavior of physical systems, including the existence of singularities and the formulation of initial value problems. We explore the applications of noncharacteristic Cauchy surfaces in classical and quantum field theory, general relativity, and quantum gravity. Our analysis provides a deeper understanding of the mathematical and physical properties of Cauchy surfaces and their significance in the study of fundamental physical phenomena. In this work we show that under the noncharacteristic condition one can prove a local existence result for the Cauchy problem similar to the semi-linear and quasilinear cases. For this, we consider the Eikonal equation and Hamilton-Jacobi equations. Therefore, we solve simultaneously for the characteristics and the solution value, resulting in curves that live in the solution graph.

A priori bounds for positive radial solutions of quasilinear equations of Lane–Emden type

We consider the quasilinear equation $\Delta _p u +K(|x|)u^q=0$, and present the proof of the local existence of positive radial solutions near $0$ under suitable conditions on $K$. Moreover, we provide a priori estimates of positive radial solutions near $\infty $ when $r^{-\ell }K(r)$ for $\ell \ge -p$ is bounded near $\infty $.

Stability of vacuum for the Landau equation with hard potentials

We consider the spatially inhomogeneous Landau equation with Maxwellian and hard potentials (i.e with $\gamma\in[0,1)$) on the whole space $\mathbb{R}^3$. We prove that if the initial data $f_{\text{in}}$ are close to the vacuum solution $f_{\text{vac}}=0$ in an appropriate weighted norm then the solution $f$ exists globally in time. This work builds up on the author's earlier work on local existence of solutions to Landau equation with hard potentials. Our proof uses $L^2$ estimates and exploits the null-structure established by Luk [Stability of vacuum for the Landau equation with moderately soft potentials, Annals of PDE (2019) 5:11]. To be able to close our estimates, we have to couple the weighted energy estimates, which were established by the author in a previous paper [Local existence for the Landau equation with hard potentials, arXiv:1910.11866], with the null-structure and devise new weighted norms that take this into account.

Existence of weak solutions to a convection–diffusion equation in amalgam spaces

We consider the local existence and uniqueness of a weak solution for a convection–diffusion equation in amalgam spaces. We establish the local existence and uniqueness of solution for the initial condition in amalgam spaces. Furthermore, we prove the validity of the Fujita–Weissler critical exponent for local existence and uniqueness of solution in the amalgam function class that is identified by Escobedo and Zuazua (J Funct Anal 100:119–161, 1991).

On explicit abstract neutral differential equations with state-dependent delay

We introduce and study a class of abstract neutral differential equations with state-dependent delay at the temporal derivative. We study the local and global existence and uniqueness of mild solution and the existence of strict solution. Some examples concerning partial neutral integro-differential equations are presented.

Existence and blow up criterion for strong solutions to the compressible biaxial nematic liquid crystal flow

In recent paper, we study the compressible biaxial nematic liquid crystal flow in a simply connected and smooth bounded domain . An initial vacuum may exist in this flow. We firstly obtain a local existence of a unique strong solution by Galerkin's approximation method. Finally, we get a blow up criterion for such a local strong solution in terms of blow up of and for any .

Local and global existence of solutions to a time-fractional wave equation with an exponential growth

A time-fractional wave equation with an exponential growth source function is considered. This model can be regarded as a modified version of the one studied by Nakamura and Ozawa (1999). The main feature of this model is that the exponential growth nonlinearity is essentially different from the polynomial growth one, and harder in controlling. We first prove the local existence and uniqueness of mild solutions in an Orlicz space by deriving a nonlinear estimate and Lp−Lq estimates the source term and solution operators, respectively. Furthermore, by additionally using specific techniques for estimating integrals, we show that small data solutions exist globally over time in such Orlicz space. The last theoretical result is about the second global-in-time existence of solutions in Besov spaces. Such a result is based on a different nonlinear estimate which is derived from a logarithmic interpolation inequality. The main ingredients of proof are the efficient application of function spaces such as Lebesgue spaces, Orlicz space,Besov spaces and computational techniques involving generalized integrals. In addition, some numerical examples are provided to illustrate theoretical results.

Theoretical Analysis of Boundary Value Problems for Generalized Boussinesq Model of Mass Transfer with Variable Coefficients

A boundary value problem is formulated for a stationary model of mass transfer, which generalizes the Boussinesq approximation in the case when the coefficients in the model equations can depend on the concentration of a substance or on spatial variables. The global existence of a weak solution of this boundary value problem is proved. Some fundamental properties of its solutions are established. In particular, the validity of the maximum principle for the substance’s concentration has been proved. Sufficient conditions on the input data of the boundary value problem under consideration, which ensure the local existence of the strong solution from the space H2, and conditions that ensure the conditional uniqueness of the weak solution with additional property of smoothness for the substance’s concentration are established.

Local existence, global existence and decay results of a logarithmic wave equation with delay term

In this article, we deal with a logarithmic wave equation with strong damping and delay. Firstly, we prove the local existence by utilizing the semigroup theory. Later, we obtain the global existence of solutions by using the well‐depth method. Moreover, under appropriate assumptions on the weight of the delay and that of strong damping, we get the exponential decay.