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In this paper, we consider the Cauchy problem of the hyperbolic Keller-Segel equations in H s (T d) on torus with d ≥ 1. Firstly, developing the dissipative mechanism through translation, we establish the global well-posedness in H s (T d) (s > 1 + d 2) with initial data near some equilibrium state. Secondly, by capturing the feature of the preservation of zero directional derivative, we give a class of initial date that lead to finite time blow-up. It's worth noting that our method of proving blow-up phenomenon does not require any conservation law. Finally, the characterization of this blow-up motivates us to show the ill-posedness of this system in H 3 2 (T d) in the sense of "norm inflation", which implies that our ill-posedness result for this system is sharp on one dimensional torus. [ABSTRACT FROM AUTHOR]

We describe regularizing effects in the linearization of a kinetic equation for nonlinear waves satisfying the Schrödinger equation in terms of weak turbulence and condensate. The problem is first considered in spaces of bounded functions with weights, where existence of solutions and some first regularity properties are proved. After a suitable change of variables the equation is written in terms of a pseudo differential operator. Homogeneity of the equation and classical arguments of freezing of coefficients may then be used to prove regularizing effect in local Sobolev type spaces. [ABSTRACT FROM AUTHOR]

We show that the Cauchy problem for the Camassa–Holm equation has a unique, global, weak, and dissipative solution for any initial data u 0 ∈ H 1 (R) , such that u 0 , x is bounded from above almost everywhere. In particular, we establish a one-to-one correspondence between the properties specific to the dissipative solutions and a solution operator associating to each initial data exactly one solution. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups G with the homogeneous dimension N. We consider the nonlinear function behaves like | u | α or | u | α − 1 u (α > 1) and the initial data u 0 belongs to the Sobolev spaces L s p (G) for 1 < p < ∞ and 0 < s < N / p. Since stratified Lie groups G include the Euclidean space R n as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on R n to G. It should be noted that our proof is very different from it given by Ribaud on R n. We adopt the generalized fractional chain rule on G to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on R n. By using the generalized fractional chain rule on G , we can avoid the discussion of Fourier analysis on G and make the proof more simple. [ABSTRACT FROM AUTHOR]

We define a class of not necessarily linear C 0 -semigroups (P t) t ≥ 0 on C b (E) (more generally, on C κ (E) : = 1 κ C b (E) , for some bounded function κ , which is the pointwise limit of a decreasing sequence of continuous functions) equipped with the mixed topology τ 1 M for a large class of topological (not necessarily Polish) state spaces E. If these semigroups are linear, classical theory of operator semigroups on locally convex spaces as well as the theory of bicontinuous semigroups apply to them. In particular, they are infinitesimally generated by their generator (L , D (L)) and thus reconstructable through an Euler formula from their strong derivative at zero in (C b (E) , τ 1 M). In the linear case, we prove that such (P t) t ≥ 0 can be characterized as integral operators given by measure kernels satisfying certain tightness properties. As a consequence, transition semigroups of Markov processes are C 0 -semigroups on (C b (E) , τ 1 M) , if they leave C b (E) invariant and they are jointly weakly continuous in space and time. Hence, they can be reconstructed from their strong derivative at zero and thus have a fully infinitesimal description. This solves an open problem for Markov processes. We show that our results apply to a large number of Markov processes, e.g., those given as the laws of solutions to SDEs and SPDEs, including the stochastic 2D Navier-Stokes equations and the stochastic fast and slow diffusion porous media equations. Furthermore, we introduce the notion of a Markov core operator (L 0 , D (L 0)) for the above generators (L , D (L)) and prove that uniqueness of the Fokker-Planck-Kolmogorov equations corresponding to (L 0 , D (L 0)) for all Dirac initial conditions implies that (L 0 , D (L 0)) is a Markov core operator for (L , D (L)). As a consequence, we identify the Kolmogorov operators of a large number of SDEs on finite and infinite dimensional state spaces as Markov core operators for the infinitesimal generators of the C 0 -semigroups on (C κ (E) , τ κ M) given by their transition semigroups. Furthermore, if each P t is merely convex, we prove that (P t) t ≥ 0 gives rise to viscosity solutions to the Cauchy problem given by its associated (nonlinear) infinitesimal generator. We also show that value functions of optimal control problems, both, in finite and infinite dimensions are particular instances of convex C 0 -semigroups on (C κ (E) , τ κ M). [ABSTRACT FROM AUTHOR]

In this paper, we investigate singularity formation for the high-order generalized Fokas–Olver–Rosenau–Qiao equation which include the celebrated Fokas–Olver–Rosenau–Qiao equation (or also called the modified Camassa–Holm equation), and we are devoted to understanding how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we show the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, we investigate the formation of singularities and provide sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solution. [ABSTRACT FROM AUTHOR]

This paper considers the viscous/inviscid Moore–Gibson–Thompson (MGT) equations with memory of type I in the whole space ℝn$$ {\mathrm{\mathbb{R}}}&#x0005E;n $$. For one thing, associating with a new condition on initial data, we derive the optimal L2$$ {L}&#x0005E;2 $$ estimates and the optimal leading term of the acoustic velocity potential for large time, where we analyze different contributions from viscous, thermally relaxing, as well as hereditary fluids on large time asymptotic behavior for the acoustic waves models. For another, by using the multiscale analysis and energy methods in the Fourier space, we demonstrate the L∞$$ {L}&#x0005E;{\infty } $$ inviscid limits (i.e., as the diffusivity of sound tends to zero), which match our Wentzel‐Kramers‐Brillouin (WKB) expansion of the solution. Finally, we give a further application of our results on large time behavior for the nonlinear Jordan‐MGT equation in viscous hereditary fluids. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the Cauchy problem for the three dimensional nonlinear Schrödinger equation with a constant magnetic field. We first establish sufficient conditions that ensure the existence of global in time and finite time blow-up solutions. In particular, we derive sharp thresholds for global existence versus blow-up for the equation with mass-critical and mass-supercritical nonlinearities. We next prove the existence and orbital stability of normalized standing waves which extend the previous known results to the mass-critical and mass-supercritical cases. To show the existence of normalized solitary waves, we present a new approach that avoids the celebrated concentration-compactness principle. Finally, we study the existence and strong instability of ground state standing waves which greatly improve the previous literature. [ABSTRACT FROM AUTHOR]

The main focus of this research is to address the Cauchy problem of the multi-dimensional Helmholtz equation with mixed boundary conditions. This problem is known to be ill-posed according to Hadamard’s definition. To tackle this issue, we propose the mollification regularization method based on exponential decay. Using prior rule and posterior rule to create a regular approximation solution and convergence of the solution is also provided. Furthermore, the proposed method is shown to be robust against data disruption noise, making it a reliable approach for solving the Cauchy problem of the multi-dimensional Helmholtz equation with mixed boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper we consider a modified quantum Boltzmann equation with the quantum effect measured by a continuous parameter \delta that can decrease from \delta =1 for the Fermi-Dirac particles to \delta =0 for the classical particles. In case of soft potentials, for the corresponding Cauchy problem in the whole space or in the torus, we establish the global existence and uniqueness of non-negative mild solutions in the function space L^{\infty }_{T}L^{\infty }_{v,x}\cap L^{\infty }_{T}L^{\infty }_{x}L^1_v with small defect mass, energy and entropy but allowed to have large amplitude up to the possibly maximum upper bound F(t,x,v)\leq \frac {1}{\delta }. The key point is that the obtained estimates are uniform in the quantum parameter 0< \delta \leq 1. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for a time fractional semilinear heat equation where 0 < α < 1 , γ > 1 , N ∈ Z ⩾ 1 and μ (x) belongs to inhomogeneous/homogeneous Besov–Morrey spaces. The fractional derivative C ∂ t α is interpreted in the Caputo sense. We present sufficient conditions for the existence of local/global-in-time solutions to problem (P). Our results cover all existing results in the literature and can be applied to a large class of initial data. [ABSTRACT FROM AUTHOR]

This paper focuses on the Cauchy problem for a one-dimensional quasilinear hyperbolic–parabolic coupled system with initial data given on a line of parabolicity. The coupled system is derived from the Poiseuille flow of full Ericksen–Leslie model in the theory of nematic liquid crystals, which incorporates the crystal and liquid properties of the materials. The main difficulty comes from the degeneracy of the hyperbolic equation, which makes that the system is not continuously differentiable and then the classical methods for the strictly hyperbolic–parabolic coupled systems are invalid. With a choice of a suitable space for the unknown variable of the parabolic equation, we first solve the degenerate hyperbolic problem in a partial hodograph plane and express the smooth solution in terms of the original variables. Based on the smooth solution of the hyperbolic equation, we then construct an iterative sequence for the unknown variable of the parabolic equation by the fundamental solution of the heat equation. Finally, we verify the uniform convergence of the iterative sequence in the selected function space and establish the local existence and uniqueness of classical solutions to the degenerate coupled problem. [ABSTRACT FROM AUTHOR]

The primary objective of this work is to examine the Kuralay equation, which is a complex integrable coupled system, in order to investigate the integrable motion of induced curves. The soliton solutions derived from the Kuralay equation are thought to be the supremacy study of numerous significant phenomena and extensive applications across a wide range of domains, including optical fibres, nonlinear optics and ferromagnetic materials. The inverse scattering transform is unable to resolve the Cauchy problem for this equation, so the analytical method is used to produce exact travelling wave solutions. The modified auxiliary equation and Sardar sub-equation approaches are used to find solitary wave solutions. As a result, singular, mixed singular, periodic, mixed trigonometric, complex combo, trigonometric, mixed hyperbolic, plane and combined bright–dark soliton solution can be obtained. The derived solutions are graphically displayed in 2-D and 3-D glances to demonstrate how the fitting values of the system parameters can be used to predict the behavioural responses to pulse propagation. This study also provides a rich platform for further investigation. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem of a two-phase flow system consisting of the compressible isothermal Euler equations and the incompressible Navier-Stokes equations coupled through the drag force, which can be formally derived from the Vlasov-Fokker-Planck/incompressible Navier-Stokes equations. When the initial data is a small perturbation around an equilibrium state, we prove the global well-posedness of the classical solutions to this system and show the solutions tends to the equilibrium state as time goes to infinity. In order to resolve the main difficulty arising from the pressure term of the incompressible Navier-Stokes equations, we properly use the Hodge decomposition, spectral analysis, and energy method to obtain the L 2 time decay rates of the solution when the initial perturbation belongs to L 1 space. Furthermore, we show that the above time decay rates are optimal. [ABSTRACT FROM AUTHOR]

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L 2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L 2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L -traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the wave equation on infinite graphs. On one hand, in contrast to the wave equation on manifolds, we construct an example for the non-uniqueness for the Cauchy problem of the wave equation on graphs. On the other hand, we obtain a sharp uniqueness class for the solutions of the wave equation. The result follows from the time analyticity of the solutions to the wave equation in the uniqueness class. In the last part, we extend the result to a wide class of linear evolution equations. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for the Einstein-Boltzmann system with soft potentials in a cosmological setting. We assume the Bianchi I symmetry to describe a spatially homogeneous, but anisotropic universe and consider a cosmological constant Λ > 0 to describe an accelerated expansion of the universe. For the Boltzmann equation we introduce a new weight function and apply the method of Illner and Shinbrot to obtain the future global existence of spatially homogeneous, small solutions. For the Einstein equations we assume that the initial value of the Hubble variable is close to (Λ / 3) 1 / 2. We obtain the future global existence and asymptotic behavior of spatially homogeneous solutions to the Einstein-Boltzmann system with soft potentials. [ABSTRACT FROM AUTHOR]

In this article, we examine the general space‐time fractional diffusion equation for left‐invariant hypoelliptic homogeneous operators on graded Lie groups. Our study covers important examples such as the time fractional diffusion equation, the space‐time fractional diffusion equation when diffusion is under the influence of sub‐Laplacian on the Heisenberg group, or general stratified Lie groups. We establish the global well‐posedness of the Cauchy problem for the general space‐time fractional diffusion equation of the Rockland operator on a graded Lie group in the associated Sobolev spaces and also develop some regularity estimates for it. [ABSTRACT FROM AUTHOR]

We consider a degenerate/singular wave equation in lone dimension, with drift and in presence of a leading operator that is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem. [ABSTRACT FROM AUTHOR]

This paper considers a Cauchy problem for the multi-dimensional modified Helmholtz equation with inhomogeneous Dirichlet and Neumann data. The Cauchy problem is severely ill-posed, and a general mollification method is introduced to solve the problem. Both the a priori and a posteriori choice strategies of the regularization parameter are proposed, and error estimations of the corresponding regularization solutions are also presented. Finally, two numerical examples are introduced to show the effectiveness of the general mollification regularization method. [ABSTRACT FROM AUTHOR]

We revisit the theory of first-order quasilinear systems with diagonalizable principal part and only real eigenvalues, what is commonly referred to as strongly hyperbolic systems. We provide a self-contained and simple proof of local well-posedness, in the Hadamard sense, of the Cauchy problem. Our regularity assumptions are very minimal. As an application, we apply our results to systems of ideal and viscous relativistic fluids, where the theory of strongly hyperbolic equations has been systematically used to study several systems of physical interest. [ABSTRACT FROM AUTHOR]

This paper investigates a class of planar Schrödinger–Poisson systems with critical exponential growth. The conditions for the local well‐posedness of the Cauchy problem in the energy space are defined. By introducing innovative ideas and relaxing some of the classical growth assumptions on nonlinearity, this study shows that such a system has at least two standing waves with a prescribed mass. One wave is a ground‐state standing wave with positive energy, and another one is a high‐energy standing wave with positive energy. In addition, with the help of the local well‐posedness, it is shown that the set of ground‐state standing waves is orbitally stable. [ABSTRACT FROM AUTHOR]

Let f be a bounded and uniformly continuous function from \mathbb {R} to a Banach space X and \mathbf {sp}(f) be its Carleman spectrum. A classical Tauberian theorem states that f is constant if and only if \mathbf {sp}(f) \subset \{ 0 \}, and f is \omega-periodic if and only if \mathbf {sp}(f) \subset \frac {2\pi }{\omega } \mathbb {Z} for some \omega >0. However, one cannot expect analogous results on \mathbb {R}^+ since there is a counterexample showing that the case of \mathbb {R}^+ contrasts dramatically with the case of \mathbb {R}. In this paper, we succeed in extending the above classical Tauberian theorem to \mathbb {R}^+ and obtain an extension of the well-known Ingham theorem. We also apply our Tauberian theorems to abstract Cauchy problems and improve a result in [Russian Math. 58 (2014), pp. 1–10]. Moreover, as an application, we present an extension of a Katznelson-Tzafriri theorem in [J. Funct. Anal. 103 (1992), pp. 74–84] with weaker assumptions. In addition, it is interesting to note that several of our results and examples show that \mathcal {S}-asymptotically \omega-periodic functions on \mathbb {R}^{+} is just the "natural" analogue of periodic functions on \mathbb {R}. [ABSTRACT FROM AUTHOR]

In this paper, we study an initial-boundary value problem of three-dimensional (3D) compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in a bounded simply connected domain, whose boundary has a finite number of two-dimensional connected components. We derive the global existence and uniqueness of classical solutions provided that the initial total energy is suitably small. Our result generalizes the Cauchy problems of compressible Navier-Stokes equations with Coulomb force [S. Liu, X. Xu and J. Zhang, Global well-posedness of strong solutions with large oscillations and vacuum to the compressible Navier-Stokes-Poisson equations subject to large oscillations and non-flat doping profile, J. Differ. Equ. 269 (2020) 8468-8508] and compressible MHD equations [H. Li, X. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal. 45 (2013) 1356-1387] to the case of bounded domains although tackling many surface integrals caused by the slip boundary condition are complex. The main ingredient of this paper is to overcome the strong nonlinearity caused by Coulomb force, magnetic field, and rotation effect of micro-particles by applying piecewise-estimate method and delicate analysis based on the effective viscous fluxes involving velocity and micro-rotational velocity. [ABSTRACT FROM AUTHOR]

Keywords: Compressible Navier-Stokes; cauchy problem; weak solutions; spherical symmetry.GraphBy Zhilei LiangReported by Author [Extracted from the article]

Keywords: Regularity criterion; bootstrap argument; Green's function; global existence; decay rate of solution.GraphBy Jie Qi and Weike WangReported by Author; Author [Extracted from the article]

In this paper, we consider Cauchy problem for the 2D incompressible Keller-Segel-Navier-Stokes equations with the fractional diffusion { ∂ t n + u ⋅ ∇ n − Δ n = − ∇ ⋅ (n ∇ c) − n 3 , ∂ t c + u ⋅ ∇ c − Δ c = − c + n , ∂ t u + u ⋅ ∇ u + ∧ 2 α u + ∇ P = − n ∇ ϕ , where ∧ : = (− Δ) 1 2 and α ∈ [ 1 2 , 1 ] . We get the global well-posedness for the above system with the rough initial data by a new priori estimate of the solutions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the inverse problem for a differential equation of 2b$$ 2b $$‐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. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Cauchy problem for linear visco‐elastic damped wave models with a general time‐dependent coefficient g=g(t)$g=g(t)$: ★ utt−Δu−g(t)Δut=0,(t,x)∈(0,∞)×Rn,u(0,x)=u0(x),ut(0,x)=u1(x),x∈Rn.$$\begin{equation} {\begin{cases} u_{tt}- \Delta u {- g(t)\Delta u_t}=0, &(t,x) \in (0,\infty) \times \mathbb {R}^n,\\ u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x), & x \in \mathbb {R}^n. \end{cases}} \end{equation}$$We are interested to study the influence of the damping term −g(t)Δut$-g(t)\Delta u_t$ 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. [ABSTRACT FROM AUTHOR]

We consider a two-point boundary-value problem for a system of differential equations with generalized piecewise-constant argument. To solve the problem, we propose to use a constructive method based on the Dzhumabaev parametrization method and a new approach to the concept of general solution. The interval is partitioned with regard for the singularities of the argument. The values of the solution at the interior points of the partition are regarded as additional parameters, and the differential equation is transformed into a system of Cauchy problems with parameters given on subintervals of the partition. By using the solutions of these problems, we obtain a new general solution of the differential equation with piecewise-constant argument and establish its properties. This new general solution, the boundary conditions, and the conditions of continuity of the solution at the interior points of the partition are used to construct a linear system of algebraic equations for the introduced parameters. The coefficients and the right-hand side of the system are found as a result of the solution of the Cauchy problems for linear ordinary differential equations on the subintervals of the partition. It is shown that the solvability of the boundary-value problem is equivalent to the solvability of the constructed system. We propose algorithms of the parametrization method for solving the analyzed boundary-value problem and establish necessary and sufficient conditions for the well-posedness of this problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose three new types of the integrable nonlocal combined nonlinear Schrödinger–Gerdjikov–Ivanov (NLS-GI) models. By the Riemann–Hilbert approach, we discuss the Cauchy problem of the reverse-space-time nonlocal combined NLS-GI model with step-like initial data: u(z, 0) = o(1) for z → −∞ and u(z, 0) = A + o(1) for z → +∞, where A is an arbitrary positive constant. First of all, we give an integrable nonlocal combined NLS-GI model and its Lax pair. Then, we consider the analytical and asymptotic behaviors, symmetries, and scattering matrix of the Jost solutions. Finally, we discuss the Cauchy problem for the nonlocal combined NLS-GI model with step-like initial data. [ABSTRACT FROM AUTHOR]

Inspired by prior research on fractional calculus, we introduce new fractional integral and derivative operators: the (ρ 1 , ρ 2 , k 1 , k 2 , φ) -proportional integral and the (ρ 1 , ρ 2 , k 1 , k 2 , 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 (ρ 1 , ρ 2 , k 1 , k 2 , φ) -proportional integral are discussed, including mapping properties, the generalized Laplace transform of the (ρ 1 , ρ 2 , k 1 , k 2 , φ) -proportional integral and (ρ 1 , ρ 2 , k 1 , k 2 , 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 (ρ 1 , ρ 2 , k 1 , k 2 , φ) -proportional Hilfer fractional Cauchy problem. An illustrative example is provided to demonstrate the main results. [ABSTRACT FROM AUTHOR]

We investigate the Cauchy problem for the fractional nonlinear Schrödinger (fNLS) equation. First, we establish the local well-posedness in the fractional Sobolev spaces Hγ with γ> 0 for the fNLS equation under less regularity assumptions for the nonlinear term than previous work. Based on the local well-posedness result, we then prove that the fNLS equation is globally wellposed in Hγ> with γ> > 0 if 4s/d < n > nc(g) and the initial data is sufficiently small. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate the Cauchy problem of an integrable focusing fifth‐order modified Korteweg–de Vries (KdV) equation, which contains the fifth‐order dispersion and relevant higher order nonlinear terms. The long‐time asymptotics of solution is established in the case of initial conditions that lie in some low regularity weighted Sobolev spaces and allow for the presence of discrete spectrum. Our method is based on a ∂¯$\bar{\partial }$ generalization of the nonlinear steepest descent method of Deift and Zhou. We show that the solution decomposes in the long time into three main regions: (i) an expanding oscillatory region where solitons and breathers travel with positive velocities, the leading order term has the form of a multisoliton/breather and soliton/breather–radiation interactions; (ii) a Painlevé region, which does not have traveling solitons and breathers, the asymptotics can be characterized with the solution of a fourth‐order Painlevé II equation; (iii) a region of breathers traveling with negative velocities. Employing a global approximation via PDE techniques, the asymptotic behavior of solution is extended to lower regularity spaces with weights. [ABSTRACT FROM AUTHOR]

The analytic relation between solutions of the original Cauchy problem and a corresponding perturbed problem is established. In the representation formula of solution, the effects of the discontinuous initial condition and perturbation of the initial data are revealed. [ABSTRACT FROM AUTHOR]

In this paper, we pursue our series of papers aiming to show the applicability of the concept of very weak solutions. We consider a wave model with irregular position-dependent mass and dissipation terms, in particular, allowing for δ -like coefficients and prove that the problem has a very weak solution. Furthermore, we prove the uniqueness in an appropriate sense and the coherence of the very weak solution concept with classical theory. A special case of the model considered here is the so-called telegraph equation. [ABSTRACT FROM AUTHOR]

We study the local classical solvability of the Cauchy problem to the equations of one-dimensional nonlinear thermoelasticity. The governing model is a coupled system of a nonlinear hyperbolic equation for the displacement and a parabolic equation for the temperature of the elastic material. We allow the hyperbolic equation to degenerate at spacial infinity, which results that the coefficients of the coupled system are not uniformly bounded and then the previous methods for the strictly hyperbolic-parabolic coupled systems are invalid. We introduce a suitable weighted norm to establish the local existence and uniqueness of classical solutions by the contraction mapping principle. The existence time T of the solution is independent of the spatial variable. [ABSTRACT FROM AUTHOR]

We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well. [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to the study of the compressible viscous Baer-Nunziato (BN) system in multi-dimensional spaces with $ d\geq2 $. Compared to previous findings, the (BN) system for compressible two-phase flows is led in two pressure state laws that vary with different phases. This property adds a practical physics background to this model. We will investigate the global existence of strong solutions to the Cauchy problem within the $ L^p $ critical regularity framework. Furthermore, we develop a Lyapunov-type energy argument that provides time-decay estimates of solutions without requiring additional smallness assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller-Segel system with a fractional dissipation and advection by a weakly mixing flow (see Definition 2.5). Here the attractive kernel has a strong singularity, namely, the derivative appearing in the nonlinear term by singular integral. Without advection, the solution of system blows up in finite time. Under a suitable mixing condition on the advection, we show the global existence of classical solution with large initial data in the case that the order of the derivative of dissipative term is higher than that of nonlinear term. Since the attractive kernel has a strong singularity, the weakly mixing has both destabilizing effects and enhanced dissipation effects, which makes the problem more complicated and difficult. In this paper, we establish the $ L^\infty $-criterion and obtain the global $ L^\infty $ estimate of the solution by introducing some new ideas and techniques. [ABSTRACT FROM AUTHOR]

В работе рассматривается линейное обыкновенное дифференциальное уравнение с производной дробного порядка, которое содержит оператор инволюции в подчиненном слагаемом. Рассматриваемое уравнение является модельным и относится к классу дифференциальных уравнений, к необходимости исследовать которые приводит изучение краевых задач для дифференциальных уравнений дробного порядка, содержащих композицию лево- и правосторонних операторов дробного дифференцирования. Последние возникают при моделировании различных физических и геофизических процессов, и, в частности, имеет важное значение при описании диссипативных колебательных систем. Для рассматриваемого уравнения исследуется начальная задача в единичном интервале. Основной результат работы – теорема существования и единственности решения изучаемой задачи. В терминах ограничений на коэффициент и правую часть рассматриваемого уравнения сформулированы достаточные условия, обеспечивающие однозначную разрешимость исследуемой задачи. Построено фундаментальное решение, получены его различные представления, изучены его основные свойства. В терминах фундаментального решения найдено явное представление решения исследуемой задачи.

This article is dedicated to establishing the existence and uniqueness of solutions for the following problem: Dαx(t)=F(t,x(t))x(0)=x0,\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 x0{x}_{0} is the singular generalized function and F satisfies L∞{L}^{\infty } logarithmic type, Dα{D}^{\alpha } is the Caputo derivative of order m−1

This paper investigates the well-posedness and the optimally complete regularity of the weak solutions, the existence of strong bi-space global attractors and their upper semicontinuity for the structurally damped wave equation with two nonlinearities on $ \mathbb{R}^{N} (N\geqslant 3): u_{tt}-\Delta u+(-\Delta)^\alpha u_{t}+h(x, u_{t})+g(x, u) = f $, where $ \alpha\in [1/2, 1) $ is a perturbed dissipative index determining the dissipative strength. We show that when the nonlinearity $ g(x, u) $ is of critical growth: $ p^*: = \frac{N+2}{N-2} $ when $ \alpha\in (1/2, 1) $, and optimally subcritical growth $ p0 $, which ensures that the weak solution is just a strong one; (ii) the related dynamical system $ (S^{\alpha}(t), \mathcal{H}) $ possesses a strong $ (\mathcal{H}, \mathcal{H}_{2\alpha}) $-global attractor $ \mathcal{A}_\alpha $ for each $ \alpha\in [1/2, 1) $, where $ \mathcal{H} $ is the energy space while $ \mathcal{H}_{2\alpha} $ is the strong solution space, and it is also the standard global attractor of optimal regularity of $ (S^{\alpha}(t), \mathcal{H}) $; (iii) the family of strong $ (\mathcal{H}, \mathcal{H}_{2\alpha}) $-global attractors $ \mathcal{A}_\alpha $ is upper semicontinuous at each point $ \alpha_0\in (1/2, 1) $ in $ \mathcal{H}_{2\alpha_0} $-topology. [ABSTRACT FROM AUTHOR]

We prove that both local and non-local formulations of the Degasperis–Procesi equation possess a pseudospherical nature. As a result, solutions determined by Cauchy problems with non-trivial initial data and a minimal specific regularity define a coframe for a pseudospherical metric within a designated strip. While the region is entirely described by the initial condition, the one-forms are determined by the corresponding solutions. Moreover, we establish that any non-trivial initial condition yields a second fundamental form, which is locally determined within the strip. The surface may collapse within a finite-height strip when the initial condition leads to wave breaking. We determine conditions for the coframe to be defined on the upper plane. We investigate possible integrability structures related to the triad of fundamental one-forms. [ABSTRACT FROM AUTHOR]

This paper concerns the Cauchy problem to the compressible magnetohydrodynamic equations in R 2 with the constant state of density at far field being vacuum or nonvacuum. Under the conditions that the adiabatic constant γ > 1 , the shear viscosity coefficient μ is a positive constant, and the bulk one λ (ρ) = ρ β with β > 4 / 3 , we establish the global existence and uniqueness of strong solutions. In particular, the initial data can be arbitrarily large and the density is allowed to vanish initially. These results generalize and improve previous ones by Huang-Li (2022) and Jiu-Wang-Xin (2018) for compressible Navier-Stokes equations. This paper introduces some key weighted estimates on H and presents some delicate analysis to exploit the decay properties of solutions due to the strong coupling and interplay interaction. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mathematical model motivated by the studies of coral broadcast spawning { ∂ t n + u ⋅ ∇ n − Δ n = − χ ∇ ⋅ (n ∇ c) + n − ϵ n q ∂ t c + u ⋅ ∇ c − Δ c = − c + n in R d × R + , where d = 2 , 3 , ϵ > 0 , and q ≥ 2. We establish global-in-time well-posedness and boundedness of the solution to the Cauchy problem of this system by developing local-in-space estimates. The crux point of our proof depends intensely on localization in the space of solutions induced by "local effect" of the L ∞ (R d) -norm. [ABSTRACT FROM AUTHOR]

In this paper, we develop a Haar wavelet-based reconstruction method to recover missing data on an inaccessible part of the boundary from measured data on another accessible part. The technique is developed to solve inverse Cauchy problems governed by the Poisson equation which is severely ill-posed. The new method is mathematically simple to implement and can be easily applied to Cauchy problems governed by other partial differential equations appearing in various fields of natural science, engineering, and economics. To take into account the ill-conditioning of the obtained linear system due to the ill-posedness of the Cauchy problem, a preconditioning strategy combined with a regularization has been developed. Comparing the numerical results produced by a meshless method based on the polynomial expansion with those produced by the proposed technique illustrates the superiority of the latter. Other numerical results show its effectiveness. [ABSTRACT FROM AUTHOR]

This work is devoted to a study of a compressible generic two-fluid model with possibly unequal velocities for arbitrarily large initial data. This model can describe some interesting physical phenomena, such as counter-current flow in which two fluids move in opposite direction. Less is known about the global existence and uniqueness of large solutions. Even for the one-fluid model, it is still open when the adiabatic parameter belongs to $ (1, \frac{3}{2}] $. The degeneracy of viscosities and the non-conservative form of the pressure terms due to unequal velocities are the main issues. In this work, we obtain the global existence and uniqueness of the solution for arbitrarily large initial data via applying the two-fluid effective velocities to a decomposition of implicit pressure functions and deriving a series of new global a priori $ L^p $ estimates of both velocities and effective velocities. The adiabatic parameters are allowed to be bigger than 1. [ABSTRACT FROM AUTHOR]