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This paper is concerned with the Cauchy problem of the quadratic nonlinear Schrödinger equation in R × R 2 with the nonlinearity η | u | 2 where η ∈ C ∖ { 0 } and low regularity initial data. If s < − 1 / 4 , the ill-posedness result in the Sobolev space H s (R 2) is known. We will prove the well-posedness in H s (R 2) for − 1 / 2 < s < − 1 / 4 by assuming some angular regularity on initial data. The key tools are the modified Fourier restriction norm and the convolution estimate on thickened hypersurfaces. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying in R n × (0 , T) (n ≥ 1) the Cauchy problem for a semilinear heat equation with source term f (x) non-rarefied at infinity, subjected to zero initial data. To such a problem, the novelty of this work lies in: (a) similar to no source case, non-rarefaction of f at infinity brings about its classical solution blowing up in a finite time T ⁎ > 0 ; (b) we derive the optimal upper and lower bound on the life span T ⁎. This may throw some light on studies of life span. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for the one-dimensional Euler-Poisson (or hydrodynamic) model for semiconductors, where the energy equation is replaced by a pressure-density relation. First, the existence of global entropy solutions is proved by using the vanishing artificial viscosity method, where, a special flux approximate is introduced to ensure the uniform boundedness of the electric field E and the a-priori L ∞ estimate, 0 < 2 δ ≤ ρ ε , δ ≤ M (t) , | u ε , δ | ≤ M (t) , where M (t) could tend to infinity as the time t tends to infinity, on the viscosity-flux approximate solutions (ρ ε , δ , u ε , δ) ; Second, the compensated compactness theory is applied to prove the pointwise convergence of (ρ ε , δ , u ε , δ) as ε , δ go to zero, and that the limit (ρ (x , t) , u (x , t)) is a global entropy solution; Third, a technique, to apply the maximum principle to the combination of the Riemann invariants and ∫ − ∞ x ρ ε , δ (x , t) − 2 δ d x , deduces the uniform L ∞ estimate, 0 < 2 δ ≤ ρ ε , δ ≤ M , | u ε , δ | ≤ M , independent of the time t and ε , δ ; Finally, as a by-product, the known compactness framework [14,29] is applied to show the relaxation limit, as the relation time τ and ε , δ go to zero, for general pressure P (ρ). [ABSTRACT FROM AUTHOR]

We consider the propagation of surface water waves described by the Boussinesq system. Following (Molinet et al. in Nonlinearity 34:744–775, 2021), we introduce a regularized Boussinesq system obtained by adding a non-local pseudo-differential operator define by g λ [ ζ ] ˆ = | k | λ ζ ˆ k with λ ∈ ] 0 , 2 ] . In this paper, we display a twofold approach: first, we study theoretically the existence of an asymptotic expansion for the solution to the Cauchy problem associated to this regularized Boussinesq system with respect to the regularizing parameter ϵ . Then, we compute numerically the function coefficients of the expansion (in ϵ ) and verify numerically the validity of this expansion up to order 2. We also check the numerical L 2 stability of the numerical algorithm. [ABSTRACT FROM AUTHOR]

This study considers the issue for recognizing unknown source within a time-space fractional parabolic equation. This particular issue is characterized by severe ill-posedness, where the solution does not depend continuously on the data. To tackle this issue, the wavelet dual least squares method is extended to handle ill-posedness of the issue under priori rule. Additionally, a posterior wavelet regularization parameter selection rule is introduced to address the Cauchy problem. Furthermore, error estimates between the exact solution and its approximation are established using both a priori and a posteriori methodologies. Numerical examples are given to validate the effectiveness and stability of the proposed method. [ABSTRACT FROM AUTHOR]

A fast numerical algorithm for solving the Cauchy problem for elliptic equations with variable coefficients in standard calculation domains (rectangles, circles, or rings) is proposed. The algorithm is designed to calculate the heat flux at the inaccessible boundary. It is based on the separation of variables method. This approach employs a finite difference approximation and allows obtaining a solution to a discrete problem in arithmetic operations of the order of N ⁢ ln ⁡ N , where 푁 is the number of grid points. As a rule, iterative procedures are needed to solve the Cauchy problem for elliptic equations. The currently available direct algorithms for solving the Cauchy problem have been developed only for (Laplace, Helmholtz) operators with constant coefficients and for use of analytical solutions for problems with such operators. A novel feature of the results of the present paper is that the direct algorithm can be used for an elliptic operator with variable coefficients (of a special form). It is important that in this case no analytical solution to the problem can be obtained. The algorithm significantly increases the range of problems that can be solved. It can be used to create devices for determining in real time heat fluxes on the parts of inhomogeneous constructions that cannot be measured. For example, to determine the heat flux on the inner radius of a pipe made of different materials. [ABSTRACT FROM AUTHOR]

This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class (m , n) ∈ C k (R) ∩ W k , 1 (R) with k ∈ N ∪ { 0 } . This system extends the celebrated Fokas–Olver–Rosenau–Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: ∂ t m (t , x) = ∂ x [ m (t , x) (u (t , x) - ∂ x u (t , x)) (u (- t , - x) + ∂ x (u (- t , - x))) ] , where m (t , x) = 1 - ∂ x 2 u (t , x) . Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class C k ∩ W k , 1 . Moreover, we derive criteria for blow-up of the local solution in this class. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence and uniqueness of solution for a large class of perturbed sweeping processes formulated by fractional differential inclusions in infinite dimensional setting. The normal cone to the (mildly non-convex) prox-regular moving set C(t) is supposed to have a Hölder continuous variation, is perturbed by a continuous mapping, which is both time and state dependent. Using an explicit catching-up algorithm, we show that the fractional perturbed sweeping process has one and only one Hölder continuous solution. Then this abstract result is applied to provide a theorem on the weak solvability of a fractional viscoelastic frictionless contact problem. The process is quasistatic and the constitutive relation is modeled with the fractional Kelvin–Voigt law. This application represents an additional novelty of our paper. [ABSTRACT FROM AUTHOR]

Jeffreys equation and its fractional generalizations provide extensions of the classical diffusive laws of Fourier and Fick for heat and particle transport. In this work, a class of multi-term time-fractional generalizations of the classical Jeffreys equation is studied. Restrictions on the parameters are derived, which ensure that the fundamental solution to the one-dimensional Cauchy problem is a spatial probability density function evolving in time. The studied equations are recast as Volterra integral equations with kernels represented in terms of multinomial Mittag-Leffler functions. Applying operator-theoretic approach, we establish subordination results with respect to appropriate evolution equations of integer order, depending on the considered range of parameters. Analyticity of the corresponding solution operator is also discussed. The main tools in the proofs are Laplace transform and the Bernstein functions' technique, especially, some properties of the sets of real powers of complete Bernstein functions. [ABSTRACT FROM AUTHOR]

We study in this paper the local well‐posedness of classical solutions to the Cauchy problem for a kinetic self‐organized model of Cucker–Smale type for collective motions, which includes two cases of velocity alignment mechanisms: normalized and nonnormalized reorientation cases. The main concern is the a priori estimates for both cases. Our treatments rely on two key ingredients: One point is to deal with the possible singularity and high nonlinearity arising from the normalized/nonnormalized mean velocity, and the other is to control the growth with respect to the microscopic velocity variable in the whole space by employing weighted estimates. [ABSTRACT FROM AUTHOR]

This paper is concerned with the convergence to sharp traveling waves of solutions with semi-compactly supported initial data for Burgers-Fisher-KPP equations with degenerate diffusion. We characterize the motion of the free boundary in the long-time asymptotic of the solution to Cauchy problem and the convergence to sharp traveling wave with almost exponential decay rates. Here a key difficulty lies in the intrinsic presence of nonlinear advection effect. After providing the analysis of the nonlinear advection effect on the asymptotic propagation speed of the free boundary, we construct sub- and super-solutions with semi-compact supports to estimate the motion of the free boundary. The new method overcomes the difficulties of the non-integrability of the generalized derivatives of sharp traveling waves at the free boundary. [ABSTRACT FROM AUTHOR]

Here, we consider here Hyperbolic Nonlinear Schrödinger Equations (HNLS) that occur as asymptotic models in the modulational regime when the Hessian of the dispersion relation is not positive (or negative) definite. We review classical examples, well-known results, and main open questions. [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]

This present paper is dedicated to the study of the Cauchy problem of the two-dimensional Euler-Boussinesq-Bénard equations which couple the incompressible Euler equations for the velocity and a transport equation with critical dissipation for the temperature. We show that there is a global unique solution to this model with Yudovich's type data. This settles the global regularity problem which was remarked by Wu and Xue (2012) [44]. [ABSTRACT FROM AUTHOR]

We study BV solutions for a 2 × 2 system of hyperbolic balance laws. We show that when initial data have small total variation on (− ∞ , ∞) and small amplitude, and decay sufficiently fast to a constant equilibrium state as | x | → ∞ , a Cauchy problem (with generic data) has a unique admissible BV solution defined globally in time. Here the solution is admissible in the sense that its shock waves satisfy the Lax entropy condition. We also study asymptotic behavior of solutions. In particular, we obtain a time decay rate for the total variation of the solution, and a convergence rate of the solution to its time asymptotic solution. Our system is a modification of a Keller-Segel type chemotaxis model. Its flux function possesses new features when comparing to the well-known model of Euler equations with damping. This may help to shed light on how to extend the study to a general system of hyperbolic balance laws in the future. [ABSTRACT FROM AUTHOR]

We consider spin manifolds with an Einstein metric, either Riemannian or indefinite, for which there exists a Killing spinor. We describe the intrinsic geometry of nondegenerate hypersurfaces in terms of a PDE satisfied by a pair of induced spinors, akin to the generalized Killing spinor equation.Conversely, we prove an embedding result for real analytic pseudo-Riemannian manifolds carrying a pair of spinors satisfying this condition. [ABSTRACT FROM AUTHOR]

In this paper, we study a blow-up radial solution to the fully parabolic chemotaxis system with the higher dimensional two species Cauchy problem. In addition, we show that the set of positive radial functions in L¹(R) ∩ L∞ (RN) × L1(R) ∩ L∞ ( RN) × W1,1 ( RN) ∩ W1,∞ ( RN) has a dense subset composed of positive radial initial data causing blow-up in finite time with respect to topology Lp (RN) × Lp (RN) × H¹ (RN) ∩ W1,1 (RN) for p ∈ [ 1, 2N/N+2 ) . [ABSTRACT FROM AUTHOR]

We devote to the analysis of blow-up solutions for the nonlinear Schrödinger equation with inhomogeneous perturbation. Firstly, we get the sharp threshold of global existence and blow-up for the Cauchy problem. Then, we show the mass concentration properties and the limiting behavior of blow-up solutions. At last, we prove the nonexistence of the minimal mass blow-up solution by using the limiting behavior of blow-up solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider three nonlinear equations of the theory of magnets with gradient nonlinearities ∇ u q , ∂ t ∇ u q , and ∂ t 2 ∇ u q are considered. For the corresponding Cauchy problems, we obtain results on local-in-time unique solvability in the weak sense and on blow-up for a finite time. These three equations have the same critical exponent q = 3/2 since weak solutions behave differently for 1 < q ≤ 3/2 and for q > 3/2. By the method of nonlinear capacity proposed by S. I. Pokhozhaev, we obtain a priori estimates, which imply the absence of local and global weak solutions. [ABSTRACT FROM AUTHOR]

With ∂ ¯ -generalization of the Deift-Zhou steepest descent method, we investigate the long-time asymptotics of the solution to the Cauchy problem for the Hunter-Saxton (HS) equation u t x x − 2 ω u x + 2 u x u x x + u u x x x = 0 , x ∈ R , t > 0 , u (x , 0) = u 0 (x) , where u 0 ∈ H 3 , 4 (R) and ω > 0 is a constant. Via a series of deformations to a Riemann-Hilbert problem associated with the Cauchy problem, we obtain the long-time asymptotic approximations of the solution u (x , t) in two kinds of space-time regions under a new scale (y , t). The solution of the HS equation decays as a speed of O (t − 1 / 2) in the region y / t > 0 ; while in the region y / t < 0 , the solution of the HS equation is depicted by the solution of a parabolic cylinder model with an residual error order O (t − 1 + 1 2 p ) with 2 < p < ∞. • We extend Monvel's result to the solution under a weighted Sobolev initial data u 0 ∈ H 3 , 4 (R). • We establish a scattering map between initial data and reflection coefficient. • We obtain a complete long-time asymptotics for HS equation on whole (x , t) -plane. [ABSTRACT FROM AUTHOR]

We discuss the integration of the Cauchy problem for the negative-order nonlinear Schrödinger equation in the class of rapidly decreasing functions via the inverse scattering problem method. In particular, we obtain the time dependence of scattering data of the Zakharov–Shabat system with the potential that is a solution of the considered problem. We give an explicit representation of the one-soliton solution of the negative-order nonlinear Schrödinger equation based on the obtained results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order systems of differential-operator equations. The existence and uniqueness theorems in appropriate topological vector spaces are also provided. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties. [ABSTRACT FROM AUTHOR]

We study singularity formation for the heat flow of harmonic maps from R d . For each d ≥ 4 , we construct a compact, d-dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly shrinker) that represents a stable blowup mechanism for the corresponding Cauchy problem. [ABSTRACT FROM AUTHOR]

This paper is devoted to the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa–Holm equation (also called Fokas–Olver–Rosenau–Qiao equation) and the Novikov equation as two special cases. On the one hand, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions in the Gevrey–Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map for the system. On the other hand, we prove the persistence properties in weighted spaces of the solution, provided that the initial potential satisfies a certain sign condition. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global existence and finite time blow-up of solution for the Cauchy problem of one-dimensional fifth-order Boussinesq equation with logarithmic nonlinearity. Fist we prove the existence and uniqueness of local mild solutions in the energy space by means of the contraction mapping principle. Further under some restriction on the initial data, we establish the results on existence and uniqueness of global solutions and finite time blow-up of solutions by using the potential well method. Moreover, the sufficient and necessary conditions of global existence and finite time blow-up of solutions are given. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Oldroyd-B model with stress tensor diffusion in R d with d ≥ 2. We first establish the global well-posedness of solutions to this model for small initial data (u 0 , τ 0) ∈ ( B ˙ p , 1 d p − 1 ∩ B ˙ p , 1 d p + 1) d × ( B ˙ p , 1 d p ) d × d with 1 ≤ p < ∞. Furthermore, under some additional L 2 type conditions on (u 0 , τ 0) , but without any more smallness restrictions, we get the L 2 decay rates of all derivatives of (u , τ). It is shown that the velocity u decays as fast as the solution to the corresponding homogeneous linear heat equation, and the symmetry τ decays as fast as ∇ u. In particular, when d = 3 the velocity u admits the decay rate faster than (1 + t) − 3 4 in L 2. [ABSTRACT FROM AUTHOR]

This paper concerns the polynomial decay of the dissipative wave equation subject to Kinetic boundary condition and non-neglected density in the square. After reformulating this problem into an abstract Cauchy problem, we show the existence and uniqueness of the solution. Then, by analyzing a family of eigenvalues of the corresponding operator, we prove that the rate of energy decay decreases in a polynomial way. [ABSTRACT FROM AUTHOR]

In this paper, we shall investigate the large-time behavior of solutions to the Cauchy problem for the three-dimensional bipolar isentropic Euler–Poisson equations with time-dependent damping effects $ -\frac {\mu }{(1+t)^{\lambda }}n_iu_i(i=1, 2) $ − μ (1 + t) λ n i u i (i = 1 , 2) for $ -1 − 1 < λ < 1 , μ > 0. Here, we consider a more general case that the two pressure functions are different and the doping profile is non-zero. Under the assumption that the initial data are close to the constant equilibrium states, we show that the smooth solutions to the Cauchy problem exist uniquely and globally. The algebraic time-decay rates of the solutions toward the constant equilibrium states are also obtained. The main ingredient of the proof is the time-weighted energy method with artfully chosen time-weighted functions. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the Keller-Segel system of consumption type coupled with the incompressible Euler equations in R 2. This coupled system describes a biological phenomenon in which aerobic bacteria living in slightly viscous fluids (such as water) move towards a higher oxygen concentration to survive. We firstly prove the local existence of smooth solutions for arbitrary smooth initial data. Then we show that these smooth solutions can be extended globally if the initial density of oxygen is sufficiently small. The main ingredient in the proof is the W 1 , q -energy estimate (q > 2) motivated by the partially inviscid two-dimensional Boussinesq system. Our result improves the well-known global well-posedness of the two-dimensional Keller-Segel system of consumption type coupled with the incompressible Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper, we prove a new blowup criterion for the strong solution to the Cauchy problem of three-dimensional micropolar fluid equation with vacuum. Specifically, we establish a blowup criterion in terms of L t ∞ L x q of the density, where 1 < q < ∞ , and it is independent on the velocity of rotation of the microscopic particles. [ABSTRACT FROM AUTHOR]

We show global existence of a measure-valued solution to the Cauchy problem for the rectilinear Euler-Poisson equations, which govern flow of pressureless dust experiencing linear damping in the gravitational field generated by its own mass, and a given background charge field modeling exterior gravitational force. The velocity of the dust is a Borel measurable function, and the distribution of mass is given by time-dependent probability measures on the real line. Upon adapting the method established in Hynd (2020) [25] , we obtain the solution by construction of a probability measure on the path space, through approximation of finite sticky particles. [ABSTRACT FROM AUTHOR]

In this paper, we consider a spatially size–stage-structured population dynamics model with resting phase. The primary objective of this model is to study size structure, stage structure, resting phase and spatial location simultaneously in a single population system. At first, we reformulate the problem as an abstract nondensely defined Cauchy problem. Then, taking advantage of the integrated semigroup and bifurcation theories, we investigate the stability and Hopf bifurcation at the positive equilibrium of the model. Finally, numerical simulations are presented as evidence to support our analytical results. A discussion of related problems is also presented briefly. [ABSTRACT FROM AUTHOR]

We study a nonlinear Cauchy problem modeling the motion of an extensible beam | y t | r y tt + γ Δ 2 y tt + Δ 2 y - a + b | | ∇ y | | 2 + c (∇ y , ∇ y t) Δ y + Δ 2 y t + d (x) h (y t) + f (y) = 0 , in a bounded domain of R N , with clamped boundary conditions in either cases: when r = γ = 0 or else when r and γ are positive. We prove, in both cases, the existence of solutions and the exponential decay of energy. [ABSTRACT FROM AUTHOR]

We consider transport processes on metric graphs with time-dependent velocities and show that, under continuity assumption of the velocity coefficients, the corresponding non-autonomous abstract Cauchy problem is well-posed by means of evolution families and evolution semigroups. [ABSTRACT FROM AUTHOR]

We construct the Hamiltonian Feynman, Lagrangian Feynman, and Feynman–Kac formulas for the solution of the Cauchy problem with the Schrödinger operator −MgDα − V, where Dα is the Vladimirov operator and Mg is the operator of multiplication by a real-valued function g defined on the d-dimensional space A K d of finite adeles over the algebraic number field K. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation with a large gradient of the initial function and a small dispersion parameter. The renormalization method is used to construct an asymptotic solution in the explicit form of integral convolution. An asymptotic analogue of the renormalization group property is established under scaling transformations determined by the dispersion parameter. In the case of a negative focusing coefficient, a clarifying expression is obtained for the asymptotic solution in terms of known elliptic special functions. [ABSTRACT FROM AUTHOR]

The domain of science and engineering relies heavily on an in-depth comprehension of fluid dynamics, given the prevalence of fluids such as water, air, and interstellar gas in the universe. Euler equations form the basis for the study of fluid motion. This paper is concerned with the Cauchy problem of isentropic compressible Euler equations away from the vacuum. We use the integration method with the general test function f = f (r) , proving that there exist the corresponding blowup results of C 1 irrotational solutions for Euler equations and Euler equations with time-dependent damping in R n ( n ≥ 2 ), provided the density-independent initial functional is sufficiently large. We also provide two simple and explicit test functions f (r) = r and f (r) = 1 + r , to demonstrate the blowup phenomenon in the one-dimensional case. In particular, our results are applicable to the non-radial system. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to discuss the Cauchy problem of integrable nonlocal (reverse-space-time) Kundu–Eckhaus (KE) equation through the Riemann–Hilbert (RH) method. Firstly, based on the zero-curvature equation, we present an integrable nonlocal KE equation and its Lax pair. Then, we discuss the properties of eigenfunctions and scattering matrix, such as analyticity, asymptotic behavior, and symmetry. Finally, for the prescribed step-like initial value: u (z , t) = o (1) , z → - ∞ and u (z , t) = R + o (1) , z → + ∞ , where R > 0 is an arbitrary constant, we consider the initial value problem of the nonlocal KE equation. The paramount techniques is the asymptotic analysis of the associated RH problem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for a three-component generalization of Camassa–Holm equation (G3CH equation henceforth) with analytic initial data. The analyticity of its solutions is proved in both variables, globally in space and locally in time. [ABSTRACT FROM AUTHOR]

We study the accuracy of shock-capturing schemes for the shallow water Cauchy problems with piecewise smooth discontinuous initial data. We consider the second order balance-characteristic (CABARETM) scheme, the third order finite-difference Rusanov–Burstein–Mirin (RBM) scheme and the fifth order in space, the third order in time weighted essentially non-oscillatory (WENO5) scheme. We have shown that the maximum loss of accuracy occurs in the centered rarefaction waves of the exact solutions, where all these schemes have the first order of convergence and fairly close values of the numerical disbalances (errors), regardless of their formal approximation order on the smooth solutions. In the same time, inside the shock influence areas the considered schemes can have different convergence orders and, as a result, significantly different accuracy. In particular, when solving the Cauchy problem with periodic initial data, when the exact solution has no centered rarefaction waves, the RBM scheme has a significantly higher accuracy inside the shock influence areas, compared to the CABARETM and WENO5 schemes. It means that the combined scheme, in which the RBM scheme is a basic scheme and the CABARETM scheme is an internal one, can be effectively used to compute weak solutions of such type Cauchy problems. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the continuity of the solution map for the Cauchy problem of the Whitham equation. First, the continuity dependence of solution is established in B2,rs$\mathrm{B}^{s}_{2,r}$ in the sense of Hadamard. Next, by constructing approximate solutions, we show that the data‐to‐solution map is not uniformly continuous in Besov spaces B2,rs$\mathrm{B}^{s}_{2,r}$ (s>32,1≤r≤∞$s>\frac{3}{2}, 1\le r\le \infty$) on the periodic case and on the line. The crucial technical tool used to prove this result is Lemma 5.1, which generalizes the result of Lemma 3 in [23]. [ABSTRACT FROM AUTHOR]

We consider the boundedness of the Fourier multipliers σt(ξ)=sin(tϕ(h(ξ)))ϕ(h(ξ))$\sigma _{t}(\xi) = \frac{\sin (t\phi (h(\xi)))}{\phi (h(\xi))}$ on modulation spaces, where h$h$ defined on Rd${\mathbb {R}}^{d}$ is a C∞(Rd∖0)$C^{\infty }({\mathbb {R}}^{d}\setminus \left\lbrace 0\right\rbrace)$ positive homogeneous function with degree λ>0$\lambda >0$ and ϕ:R+→R+$\phi : {\mathbb {R}}^{+} \rightarrow {\mathbb {R}}^{+}$ is a smooth function satisfying some decay conditions. We prove the boundedness of this kind of Fourier multipliers and obtain its asymptotic estimates as t$t$ goes to infinity. We remote the restriction λ>1/2$\lambda >1/2$ in Deng, Ding, and Sun's result in [Nonlinear Anal. 85 (2013), 78–92], and we consider the more general form of this multiplier. As applications, we obtain the grow‐up rate of the solutions for the Cauchy problems for the generalized wave and Klein–Gordon equations, and we obtain the local well‐posedness of nonlinear wave and Klein–Gordon equations in modulation spaces. [ABSTRACT FROM AUTHOR]

The paper examines stochastic diffusion within an expanding space–time framework motivated by cosmological applications. Contrary to other results in the literature, for the considered general stochastic model, the expansion of space–time leads to a class of stochastic equations with non-constant coefficients that evolve with the expansion factor. The Cauchy problem with random initial conditions is posed and investigated. The exact solution to a stochastic diffusion equation on the expanding sphere is derived. Various probabilistic properties of the solution are studied, including its dependence structure, evolution of the angular power spectrum and local properties of the solution and its approximations by finite truncations. The paper also characterizes the extremal behaviour of the random solution by establishing upper bounds on the probabilities of large deviations. Numerical studies are carried out to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we would like to consider the Cauchy problem for semilinear σ$$ \sigma $$‐evolution equations with time‐dependent damping for any σ≥1$$ \sigma \ge 1 $$. Motivated strongly by the classification of damping terms in some previous papers, the first main goal of the present work is to make some generalizations from σ=1$$ \sigma &#x0003D;1 $$ to σ>1$$ \sigma >1 $$ and simultaneously to investigate decay estimates for solutions to the corresponding linear equations in the so‐called effective damping cases. For the next main goals, we are going not only to prove the global well‐posedness property of small data solutions but also to indicate blow‐up results for solutions to the semilinear problem. In this concern, the novelty which should be recognized is that the application of a modified test function combined with a judicious choice of test functions gives blow‐up phenomena and upper bound estimates for lifespan in both the subcritical case and the critical case, where σ$$ \sigma $$ is assumed to be any fractional number. Finally, lower bound estimates for lifespan in some spatial dimensions are also established to find out their sharp results. [ABSTRACT FROM AUTHOR]

In this paper, we study the well‐posedness of the Cauchy problem to the compressible inertial Qian–Sheng model. Inspired by Jiang–Luo–Ma–Tang's work in [J. Hyperbolic. Differ. Equ 18:221‐256, 2021], we introduce a novel Condition (H) which ensures the energy dissipation, and then, we prove that when both the viscosity coefficients β5+β6=0$$ {\beta}_5&#x0002B;{\beta}_6&#x0003D;0 $$ and Condition (H) are satisfied, the local‐in‐time solutions exist for large initial data. Otherwise, we can only obtain small data local solutions. Finally, the local solution can be extended globally under at least one of the two conditions: (μ˜2−μ2)2<8β4μ1$$ {\left({\tilde{\mu}}_2-{\mu}_2\right)}&#x0005E;2<8{\beta}_4{\mu}_1 $$ or μ˜2=μ2$$ {\tilde{\mu}}_2&#x0003D;{\mu}_2 $$. [ABSTRACT FROM AUTHOR]