Cauchy Problem Research Articles

Global well-posedness for the 3D inhomogeneous incompressible magnetohydrodynamics system with axisymmetric data

We focus on the Cauchy problem of 3D inhomogeneous incompressible magnetohydrodynamics (MHD for short) system with axisymmetric initial data. It is shown that this MHD system has a unique global solution provided that ‖1r(1ρ0−1)‖L∞ is sufficiently small. Finally, we establish the L2 decay rate for any given globally smooth solution of the MHD system by using the Schonbek's Fourier splitting method.

The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 1. The Cauchy problem on the real line

The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 1. The Cauchy problem on the real line

Stability of planar diffusion wave for three‐dimensional bipolar Euler–Poisson system with time‐dependent over‐damping

This paper is concerned with the asymptotic behavior of the solution to the Cauchy problem for the three‐dimensional bipolar Euler–Poisson system with time‐dependent over‐damping. We prove that the Cauchy problem admits a unique global‐in‐time smooth solution that tends to the planar diffusion wave as time goes to infinity provided that the initial perturbation is sufficiently small. Moreover, we also obtain the convergence rate of the solution toward the planar diffusion wave. The proof is completed by the time‐weighted energy method.

Global smooth axisymmetric solutions of the damped Boussinesq equations with zero thermal diffusion

In this paper we study the Cauchy problem for the three-dimensional damped Boussinesq system with zero thermal diffusion. We show that the solution of this system with 1\leq\beta\leq\frac{7}{3} is globally well posed if the initial data is axisymmetric without swirl.

Riemann–Hilbert problem and soliton solutions with their asymptotic analysis for the focusing nonlocal Hirota equation with step-like initial data

The Riemann–Hilbert (RH) problem is developed to study the Cauchy problem of focusing space symmetric nonlocal Hirota (fsNH) equation with step-like initial data. Firstly, we analyze the characteristics of the Jost solutions and spectral functions, including analyticity, symmetry, and asymptotics as k→∞ and k→0. Then, we discuss three cases of zero point distributions of the spectral functions, and derive the expressions of spectral functions by constructing the special analytic functions and establishing the scalar RH problem. Furthermore, the constraint relations among the zeros of the spectral functions are analyzed. Next, the corresponding RH problem equipped with residual and singularity conditions is successfully derived. These singularity conditions are then transformed into a general residue condition, which reduces the problem of solving the RH problem with reflection-less potential to solving a system of algebraic equations. Ultimately, the corresponding soliton solutions of the fsNH equation and their asymptotics are derived from the solution of the RH problem.

A lower bound estimate of solutions to the Cauchy problems for a hyperbolic Monge–Ampère equation

This paper investigates the Cauchy problems for a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system via Riemannian invariants. Based on the first a priori estimate of the solutions, the second a priori estimate of the derivation of the solutions, and the third a priori estimate of the continuous mould of the first‐order partial derivatives of the solutions to quasilinear hyperbolic system, a lower bound estimate of classical solutions to the Cauchy problems for a hyperbolic Monge–Ampère equation is derived.

Ill-Posedness for the Cauchy Problem of the Modified Camassa-Holm Equation in $$B_{\infty ,1}^0$$

Ill-Posedness for the Cauchy Problem of the Modified Camassa-Holm Equation in $$B_{\infty ,1}^0$$

Well-posedness of the Cauchy problem for the fourth-order nonlinear Schrödinger equation

The sharp local well-posedness for the one dimensional fourth-order nonlinear Schrödinger equation is established in the Sobolev space Hs(R) for s≥12, which improves the results in Huo and Jia (2007). In addition, we prove that this equation cannot be solved by an iteration scheme based on the Duhamel formula in Hs(R) for s<12. Our method relies upon the Bourgain space and a crucial bilinear estimate, which avoids the tedious classification of the location to the highest dispersion modulation.

A novel explicit fast numerical scheme for the Cauchy problem for integro-differential equations with a difference kernel and its application

The present study focuses on designing a second-order novel explicit fast numerical scheme for the Cauchy problem incorporating memory associated with an evolutionary equation, where the integral term's kernel is a discrete difference operator. The Cauchy problem under consideration is related to a real finite-dimensional Hilbert space and includes a self-adjoint operator that is both positive and definite. We introduce a transformative technique for converting the Cauchy problem incorporating memory, into a local evolutionary system of equations by approximating the difference kernel using the sum of exponentials (SoE) approach. A second-order explicit scheme is then constructed to solve the local system. We thoroughly investigate the stability of this explicit scheme, and present the necessary conditions for the stability of the scheme. Moreover, we extended our investigation to encompass time-fractional diffusion-wave equations (TFDWEs) involving a fractional Caputo derivative with an order ranging between (1,2). Initially, we transform the main TFDWE model into a new model that incorporates the fractional Riemann-Liouville integral. Subsequently, we expand the applicability of our idea to develop an explicit fast numerical algorithm for approximating the model. The stability properties of this fast scheme for solving TFDWEs are assessed. Numerical simulations including a two-dimensional Cauchy problem as well as one-dimensional and two-dimensional TFDWE models are provided to validate the accuracy and experimental order of convergence of the schemes.

A closed-form solution for thermally induced affine deformation in unbounded domains with a temporally accelerated anomalous thermal conductivity

Abstract One of the successful mathematical tools in the 21st century is the distributed-order fractional derivative, particularly, the bi-fractional diffusion equation of natural form (Chechkin, Gorenflo, and Sokolov, Phys. Rev. E 2002) and the bi-fractional diffusion equation of the modified form (Sokolov, Chechkin, and Klafter, Acta Phys. Pol. B 2004). The present study adopts the bi-fractional diffusion heat equation of the modified form that uses two Riemann-Liouville time-fractional derivatives with different fractional orders and the Riesz space-fractional derivative, and bears the property of variable thermal conductivity with temporal acceleration; i.e., the material thermal conduction transits from a low value at the small values of time ($t\to 0$) to a larger value at the long-time domain ($t\to \infty$). The corresponding fractional thermoelasticity theory, that uses the bi-fractional diffusion heat equation of the modified form, is exclusively considered in this work. For a quasi-static unbounded Hookean domain, exact solutions for the temperature, hydrostatic stress, and the displacement fields are derived in both short-time and long-time domains, in terms of the Fox \textit{H}-function. The exact solutions for the linearized problem are derived using the inversion of problem variables from the Laplace-Fourier space. Adding a zero initial condition on the normal stress of the unbounded space transforms the conventional Cauchy problem to a mixed initial-boundary value problem in order to derive a generalized form of the displacement. Additionally, thermal energy is not conserved in the presence of space fractality. The effect of such an accelerating transition in the thermal conduction on the displacement component is focused on, and it is found that the displacement inherits a similar transition behavior.

Cauchy problem for a loaded hyperbolic equation with the Bessel operator

Abstract This work is devoted to the study of the Cauchy problem for a loaded differential equation with the Bessel operator. When studying problems for loaded equations, the properties of Erdélyi-Kober operators are used as transformation operators concerning a relation. We obtain an explicit form of the solution to the Cauchy problem for a loaded one-dimensional differential equation. At the end of the work, we will show several examples on graphs.

On the Cauchy problem for the weakly dissipative modified Camassa–Holm–Novikov equation

On the Cauchy problem for the weakly dissipative modified Camassa–Holm–Novikov equation

On a fractional Cauchy problem with singular initial data

Abstract This article is dedicated to establishing the existence and uniqueness of solutions for the following problem: D α x ( t ) = F ( t , x ( t ) ) x ( 0 ) = x 0 , \left\{\begin{array}{l}{D}^{\alpha }x\left(t)=F\left(t,x\left(t))\hspace{1.0em}\\ x\left(0)={x}_{0},\hspace{1.0em}\end{array}\right. where x 0 {x}_{0} is the singular generalized function and F satisfies L ∞ {L}^{\infty } logarithmic type, D α {D}^{\alpha } is the Caputo derivative of order m − 1 &lt; α &lt; m m-1\lt \alpha \lt m with m ∈ N * m\in {{\mathbb{N}}}^{* } , which we will confirm to be present in Colombeau algebra. The Gronwall lemma is used in Colombeau’s algebra to establish the main results. To illustrate our theoretical analysis, we ended our work with an example.

The Cauchy problem for the fractional nonlinear Schrödinger equation in Sobolev space

The Cauchy problem for the fractional nonlinear Schrödinger equation in Sobolev space

Mild Solutions of the (ρ1,ρ2,k1,k2,φ)-Proportional Hilfer–Cauchy Problem

Inspired by prior research on fractional calculus, we introduce new fractional integral and derivative operators: the (ρ1,ρ2,k1,k2,φ)-proportional integral and the (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional derivative. Numerous previous studied fractional integrals and derivatives can be considered as particular instances of the novel operators introduced above. Some properties of the (ρ1,ρ2,k1,k2,φ)-proportional integral are discussed, including mapping properties, the generalized Laplace transform of the (ρ1,ρ2,k1,k2,φ)-proportional integral and (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional derivative. The results obtained suggest that the most comprehensive formulation of this fractional calculus has been achieved. Under the guidance of the findings from earlier sections, we investigate the existence of mild solutions for the (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional Cauchy problem. An illustrative example is provided to demonstrate the main results.

Determination of two unknown functions of different variables in a time‐fractional differential equation

We study the inverse problem for a differential equation of ‐order with the Caputo fractional derivative over time and Schwartz‐type distribution in its right‐hand side. The generalized solution of the Cauchy problem for such an equation, space‐dependent part of a source, and a time‐dependent reaction coefficient in the equation are unknown. We find sufficient conditions for unique local in time solvability of the inverse problem under time‐ and space‐integral overdetermination conditions.

On long-time asymptotic behavior and Painlevé asymptotic to the matrix Hirota equation

Abstract The nonlinear descent method is extended to study the long-time asymptotic behavior of the matrix Hirota equation with $4\times 4$ Lax pair in Schwartz space. The implementation of spectral analysis successfully transforms the Cauchy problem of the matrix Hirota equation into the corresponding high-order Riemann–Hilbert (RH) with $4\times 4$ jump matrix, and further analyses the established oscillation RH problem to study the asymptotic behavior of the solution in the space-time plane. Interestingly, the space-time plane $\{(x,t)|-\infty &amp;lt;x&amp;lt;+\infty , t&amp;gt;0\}$ can be divided into three different asymptotic regions based on the phase function and $\xi =x/t$. The first one is the oscillatory region $\xi &amp;lt;\frac{\alpha ^{2}}{3\beta }$, whose leading-term can be approximated applying the Weber equation with an error of $\mathcal{O}(t^{-1}\log t)$. The second region is the Painlevé region $\xi \approx \frac{\alpha ^{2}}{3\beta }$, whose leading-term can be approximated by the coupled Painlevé II equation, which is related to a $4\times 4$ matrix RH problem with an error of $\mathcal{O}(t^{-\frac{2}{3}})$. The last region is the fast decay region $\xi&amp;gt;\frac{\alpha ^{2}}{3\beta }$, which solution is rapidly decreasing as $t\rightarrow \infty $. Our results provide a detailed proof for the asymptotic analysis for the solution of the matrix Hirota equation on the complete space-time plane.

New decay rates for a weakly damped magneto-thermo-elastic model in [formula omitted]

We consider the Cauchy problem for a linear magneto-thermo-elastic model in R3 without a damping term in the equation of motion. This system has the decay property of regularity-loss type. Working with the system in Fourier space, the dissipative structure becomes very weak in the high frequency region. To circumvent this difficulty, we imposed more regularity on the initial data and adapted ideas introduced in the article of J. E. M. Rivera and R. Racke (2001) [23], using appropriate multipliers and working with the system in terms of its components. We obtain new decay rates for the total energy of the system. The approach developed in this work can be applied in the study of the asymptotic behavior of other linear, weakly dissipative, systems in Rn.

Visco‐elastic damped wave models with time‐dependent coefficient

AbstractIn this paper, we study the following Cauchy problem for linear visco‐elastic damped wave models with a general time‐dependent coefficient : We are interested to study the influence of the damping term on qualitative properties of solutions to () as decay estimates for energies of higher order and the parabolic effect. The main tools are related to WKB‐analysis. We apply elliptic as well as hyperbolic WKB‐analysis in different parts of the extended phase space.

Weakly coupled systems of semilinear σ$$ \sigma $$‐evolution equations with friction and visco‐elastic type damping

In the present paper, we are interested to study global (in time) existence of small data Sobolev solutions to the Cauchy problem for a weakly coupled system of two semilinear ‐evolution equations with friction and visco‐elastic type damping, where for . We study model (1.1) (see below) in several cases with respect to the regularity for the data: First, we assume data . By using estimates of Sobolev solutions to related linear models with vanishing right‐hand side, we explain the admissible range of exponents in (1.1) which allow to prove the global (in time) existence of small data Sobolev solutions. Then we suppose that the data belong to , where now , . So the data have additional regularity to the first assumed integrability. We restrict ourselves to data from energy space (on the basis of ) and from energy space with additional suitable higher regularity. We compare in our statements the admissible set of exponents and in the power nonlinearities with the so‐called modified Fujita exponent. Finally, some blow‐up results complete the paper.