Cauchy Problem Research Articles

An inverse Cauchy problem of a stochastic hyperbolic equation

Abstract In this paper, we investigate an inverse Cauchy problem for a stochastic hyperbolic equation. A Lipschitz type observability estimate is established using a pointwise Carleman identity. By minimizing the constructed Tikhonov-type functional, we obtain a regularized approximation to the problem. The properties of the approximation are studied by means of the Carleman estimate and Riesz representation theorem. Leveraging kernel-based learning theory, we simulate numerical algorithms based on the proposed regularization method. These reconstruction algorithms are implemented and validated through several numerical experiments, demonstrating their feasibility and accuracy.

On the Cauchy problem of three‐dimensional compressible magneto‐micropolar fluids with vacuum

AbstractIn this paper, we study the Cauchy problem of three‐dimensional compressible magneto‐micropolar fluids. Applying the energy method and the structural characteristics of the model, the global existence and uniqueness of classical solutions are established provided that is suitable small, where and represent the initial magnetic field and upper bound of initial density. It is worth mentioning that both the vacuum states of initial density and the possible random largeness of initial energy are allowed. Thus, the results obtained particularly extend the one due to Wei et al., where the global well‐posedness of smooth solutions with small perturbations of initial data was proved.

Asymptotic Solution of a Singularly Perturbed Cauchy Problem in the Presence of a Rational Simple Turning Point

Asymptotic Solution of a Singularly Perturbed Cauchy Problem in the Presence of a Rational Simple Turning Point

Strong solution of 3D generalized Navier–Stokes equations with damping

This paper studies the Cauchy problem of the 3D generalized Navier-Stokes equations with damping term α|u|σ−2u (σ≥2). There are two choices for the stress tensor: S(∇u)=|∇u|q−2∇u+∇u and S(∇u)=|∇u|q−2∇u. For both cases, we consider the global existence of strong solutions by using pure energy method.

Well-Posedness for Ohkitani Model and Long-Time Existence for Surface Quasi-geostrophic Equations

We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani, ∂tθ-∇⊥log(10+(-Δ)12)θ·∇θ=0,and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper (Chae et al. in Illposedness via degenerate dispersion for generalized surface quasi-geostrophic equations with singular velocities, arXiv:2308.02120) that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae et al. (Comm Pure Appl Math 65(8):1037–1066, 2012). This well-posedness result can be applied to describe the long-time dynamics of the δ-SQG equations, defined by ∂tθ+∇⊥(10+(-Δ)12)-δθ·∇θ=0,for all sufficiently small δ>00$$\\end{document}]]> depending on the size of the initial data. For the same range of δ, we establish global well-posedness of smooth solutions to the dissipative SQG equations.

Local solutions for a Brinkman equation coupled with heat-convective and concentration-diffusive equations and a volumetric mass source

In this article, we consider a model coupled with the Brinkman heat-convective and concentration-diffusive equations for a mixed gas flow in a porous media. The specificity of this model lies in the presence of a volumetric mass source depending on temperature and concentration in mass balance equation. We will prove the existence and uniqueness of the smooth local solutions for the 3-D Cauchy problem. As a byproduct, we show the convergence of the approximate solutions based on an iteration scheme. For more information see https://ejde.math.txstate.edu/Volumes/2025/23/abstr.html

Global strong solutions to nonlocal Benjamin-Bona-Mahony equations with exponential nonlinearities

Global strong solutions to nonlocal Benjamin-Bona-Mahony equations with exponential nonlinearities

STABILITY OF VARIABLE COEFFICIENTS RAYLEIGH BEAMS WITH INDEFINITE DAMPING UNDER A FORCE CONTROL

In this paper, we continue our study of the damped Rayleigh beam system. The purpose of this work is to add two additional force controls in moment and rotation and see if this remains exponentially stable. Thus we obtain a four-control system. Using the Riesz basis approach, we prove the stability of nonhomogeneous Rayleigh beam via an indefinite damping and the boundary control feedbacks in position, velocity, moment and rotation. We transform the system to an evolution problem and prove that the linear operator of the Cauchy problem generates a semigroup of contractions. Subsequently, we show how the damping term can affect the decay rate asymptotically.

C∞ Well-Posedness of Higher Order Hyperbolic Pseudo-Differential Equations with Multiplicities

In this paper, we study higher order hyperbolic pseudo-differential equations with variable multiplicities. We work in arbitrary space dimension and we assume that the principal part is time-dependent only. We identify sufficient conditions on the roots and the lower order terms (Levi conditions) under which the corresponding Cauchy problem is C∞ well-posed. This is achieved via transformation into a first order system, reduction into upper-triangular form and application of suitable Fourier integral operator methods previously developed for hyperbolic non-diagonalisable systems. We also discuss how our result compares with the literature on second and third order hyperbolic equations.

The modulation instability gain spectrum assessment and generalized propagating wave structures of nonlinear cubic-quartic Schrödinger equation

This study investigates a cubic-quartic nonlinear Schrödinger model with third- and fourth-order dispersions without the disturbance in parabolic law media and group velocity dispersion (GVD). The analytical method is used to produce traveling wave solutions because the inverse scattering transform is unable to solve the Cauchy problem for this equation. The utilized analytical approach is novel and providing generalized families of solutions as compared to other, thus, prior to this study, there was not existing any work in which suck kinds of solutions were exist. The propagating solitons are driven using the unified auxiliary equation approach. As a result, the aforementioned technique is learned in the following areas: singular soliton, shock solution, mixed-singular, mixed trigonometric, mixed shock single solution, plane solutions, exponential, trigonometry, mixed-hyperbolic solution, periodic and mixed periodic solutions, and trigonometry. The stability of the topic under consideration is shown by the modulational instability (MI) study. The appropriate parametric values are shown alongside the graphical representation.

Classical solution of the Cauchy problem for a semilinear wave equation with a Dirac potential

For a one-dimensional semilinear wave equation with a free term that is a solution value at one given point (a Dirac potential), we consider the Cauchy problem in the upper half-plane. We construct the solution using the method of characteristics in implicit analytical form as a solution of some integral equations. The solvability of these equations, as well the smoothness of their solutions, is studied. For the problem in question, we prove the uniqueness of the solution, and establish the conditions under which its classical solution exists.

Existence and uniqueness results for the Cauchy problem of generalized Burgers type equations on stratified Lie groups

Existence and uniqueness results for the Cauchy problem of generalized Burgers type equations on stratified Lie groups

The Navier–Stokes Cauchy Problem in a Class of Weighted Function Spaces

The Navier–Stokes Cauchy Problem in a Class of Weighted Function Spaces

BLOW-UP SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NONLINEAR MEMORY

In this paper, we consider the nonlinear cauchy problem [Formula: see text] where [Formula: see text] and [Formula: see text] [Formula: see text]. We will study the equation for [Formula: see text] We demonstrate the local existence and uniqueness of the solution to this problem by the Banach fixed-point principle. Then, it is shown that local solutions experience blow-up. Finally, we present the blow-up rate when [Formula: see text]

Hyperbolic extensions of constrained PDEs

Systems of partial differential equations (PDEs) comprising a combination of constraints and evolution equations are ubiquitous in physics. For both theoretical and practical reasons, such as numerical integration, it is desirable to have a systematic understanding of the well-posedness of the Cauchy problem for these systems. In this article, we first review the use of hyperbolic reductions, where the evolution equations are singled out for consideration. We then examine in greater detail the extensions, namely, systems in which constraints are evolved as auxiliary variables alongside the original variables, resulting in evolution systems with no constraints. Assuming a particular structure of the original system, we provide sufficient conditions for the strong hyperbolicity of an extension. Finally, this theory is applied to the examples of electromagnetism and a toy model of magnetohydrodynamics.

Concentrated Couple in a Plane and in a Half-Plane in the Framework of Fractional Nonlocal Elasticity

In nonlocal elasticity, the constitutive equation for the stress tensor is written in an integral form, with the weight function, referred to as the nonlocality kernel, often being the Green’s function for the partial differential equation. In this paper, we obtain solutions to elasticity problems for a concentrated couple in a plane and on the boundary of a half-plane within framework of a new theory of nonlocal elasticity, where the nonlocal kernel is the Green’s function of the Cauchy problem for the fractional diffusion equation. The obtained solutions are free from nonphysical singularities that appear in the classical local elasticity solutions.

A Modified Cauchy Problem With Nonlocal Boundary Condition for a Degenerate Hyperbolic Equation

A Modified Cauchy Problem With Nonlocal Boundary Condition for a Degenerate Hyperbolic Equation

The Cauchy problem for doubly degenerate parabolic equations with weights

We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, roughly speaking of the type of the exponential of a power of the distance from the origin. We assume here the solutions of the Cauchy problem to be globally integrable in space (in the appropriate weighted sense) and non-negative. Under suitable assumptions, we prove for the solutions sup estimates, i.e., the decay rate at infinity, the property of finite speed of propagation and support estimates. All our estimates are given explicitly in terms of the weight appearing in the equation.

The Cauchy problem for a class of nonlinear telegraph equation

In this paper, we consider the initial value problem for a class of nonlinear telegraph equation appearing in lossless transmission circuits in n-dimensional space. Different from the classical damping term | u t | μ − 1 u t , the presence of nonlinear disspative term | u | μ − 1 u t causes more difficulties. On one hand, based on a fixed point argument, we establish the global existence and decay of small amplitude solutions in Sobolev space. In addition, we investigate the influence of dissipative term | u | μ − 1 u t on the decay of the solutions. On the other hand, the existence and uniqueness of local mild solutions are obtained for large initial value. In order to obtain the global existence of the solutions, we need to balance the lower regularity of solutions and higher order derivative of nonlinear term. To this end, the fundamental solution of a certain type of the Poisson equation is introduced. Then the global solutions can be established by continuous argument under suitable conditions.

Nonlocal separable elliptic and parabolic equations and applications

The regularity properties of nonlocal anisotropic elliptic equations with parameters are investigated in abstract weighted Lp spaces. The equations include the variable coefficients and abstract operator function A = A (x) in a Banach space E in leading part. We find the sufficient growth assumptions on A and appropriate symbol polynomial functions that guarantee the uniformly separability of the linear problem. It is proved that the corresponding anisotropic elliptic operator is sectorial and is also the negative generator of an analytic semigroup. Byusing these results, the existence and uniqueness of maximal regular solution of the nonlinear nonlocal anisotropic elliptic equation is obtained in weighted Lp spaces. In application, the maximal regularity properties of the Cauchy problem for degenerate abstract anisotropic parabolic equation in mixed Lp norms, the boundary value problem for anisotropic elliptic convolution equation, the Wentzel-Robin type boundary value problem for degenerate integro-differential equation and infinite systems of degenerate elliptic integro-differential equations are obtained.