In this paper, a fully parabolic chemotaxis-haptotaxis model with non-local logistic source terms is considered in a bounded domain with a smooth boundary. We prove that the system has a globally bounded classical solution, which means that the solution will not blow up over time. Our results extend previous work in Ren, Nonlinear Differ. Equations Appl. 31(2), 17 (2024).

We consider an isolated macroscopic quantum system in a pure state ψt evolving unitarily in a separable Hilbert space H and take for granted that different macro states ν correspond to mutually orthogonal subspaces Hν⊂H. Let Pν be the projection to Hν. It was recently shown that for all Hamiltonians with no highly degenerate eigenvalues and gaps most ψ0∈Hμ are such that for most t ≥ 0, ‖Pνψt‖2 is close to a t-and ψ0-independent value Mμν provided that Mμν is not too small. Here, “most” refers to the uniform distribution on the sphere S(Hμ). In the present work, we generalize this result from the uniform distribution, corresponding to the micro-canonical ensemble, to the much more general class of Gaussian adjusted projected (GAP) measures. For any density matrix ρ on H, GAP(ρ) is the most spread out distribution on S(H) with density matrix ρ. We show that also for GAP(ρ)-most ψ0∈H for most t ≥ 0, ‖Pνψt‖2 is close to a fixed value MρPν (which must not be too small). Moreover, we prove a generalization for certain operators B instead of Pν and for finite times. Since certain GAP measures are quantum analogs of the (grand-)canonical ensemble, our result expresses a version of equivalence of ensembles.

The newly discovered exponential and algebraic double-soliton solutions of the massive Thirring model in laboratory coordinates are placed in the context of the inverse scattering transform. We show that the exponential double-solitons correspond to double isolated eigenvalues in the Lax spectrum, whereas the algebraic double-solitons correspond to double embedded eigenvalues on the imaginary axis, where the continuous spectrum resides. This resolves the long-standing conjecture that multiple embedded eigenvalues may exist in the spectral problem associated with the massive Thirring model. To obtain the exponential double-solitons, we solve the Riemann–Hilbert problem with the reflectionless potential in the case of a quadruplet of double poles in each quadrant of the complex plane. To obtain the algebraic double-solitons, we consider the singular limit where the quadruplet of double poles degenerates into a symmetric pair of double embedded poles on the imaginary axis.

In this paper, we use the steady isentropic Euler system with linear damping to describe Chaplygin gas flow in a two-dimensional expansion nozzle. Firstly, the unique existence of subsonic and supersonic solutions of the damped radial isentropic Euler flow in a two-dimensional expansion and contraction nozzle are established. Then take the radial subsonic solution in a expansion nozzle as the background solution, the structural stability problem of the background solution under small perturbations of the inlet Bernoulli function, inlet velocity ratio, and outlet density is transformed into the unique existence problem of the solution to the boundary value problem of a nonlinear hyperbolic-elliptic coupled system. After performing a Lagrange transformation on the decoupled system, we transform it into a fixed boundary problem, linearize it, and construct a suitable iteration set and a iteration scheme. Finally the existence of a unique fixed point can be proved, which yields the structural stability of the background solution.

For the family of Hénon maps (x,y)↦(a(1−x2)−by,x) of R2, the so-called anti-integrable (AI) limit concerns the limit a → ∞ with fixed Jacobian b. At the AI limit, the dynamics reduces to a subshift of finite type. There is a one-to-one correspondence between sequences allowed by the subshift and the AI orbits. The theory of anti-integrability says that each AI orbit can be continued to becoming a genuine orbit of the Hénon map for a sufficiently large (and fixed Jacobian). In this paper, we assume b is a smooth function of a and show that the theory can be extended to investigating the limit lima→∞b/a=ν for any ν > 0 provided that the one dimensional quadratic map x↦1ν(1−x2) is hyperbolic.

In this paper, we study the three-dimensional primitive equations for atmosphere dynamics with only horizontal diffusivity. This system is considered to have full viscosities in the momentum equations, and have horizontal diffusions in the temperature equation, specific humidity equation and liquid water content equation. First, for any H2 initial data, we use energy estimate method to establish the higher order estimations of state functions in local time. Then applying the contraction mapping principle and the Aubin–Lions lemma, we obtain the well-posedness of local strong solution. Furthermore, we use a logarithmic Sobolev inequality of the limiting type and a classic logarithmic type Gronwall inequality to obtain the higher order estimations of state functions in global time. Applying the continuity method, we prove the existence and uniqueness of global strong solution.

A previous construction of regular rational extensions of the trigonometric Darboux-Pöschl-Teller potential, obtained by one-step Darboux transformations using seed functions associated with the para-Jacobi polynomials of Calogero and Yi, is generalized by considering m-step Darboux transformations. As a result, some novel families of exceptional orthogonal polynomials depending on m discrete parameters, as well as m continuous real ones λ1, λ2, …, λm, are obtained. The restrictions imposed on these parameters by the rational extensions regularity conditions are studied in detail.

In this paper, we investigate the instability and stability of three-dimensional incompressible micropolar fluids in a bounded domain. Under the instability condition, we establish the linear and nonlinear instability around the steady-state solution. For linearized problem, we construct an exponential growth solution in H2 by a modified variational method. Based on the linear instability and the Gronwall-type global estimates of perturbation problem, taking the micro-rotation viscosity μr to be suitably small such that N<32 in (2.7), we show the equilibrium state is nonlinear unstable in the sense of Hadamard. On the other hand, when the steady density is lighter with increasing height, we show the steady state is linearly globally stable. In particular, if ρ̄′ is a constant, the steady state is nonlinearly asymptotically stable.

In this paper, we obtain an explicit upper bound for the life span of solutions to a nonlinear Schrödinger equation with initial data that are non-rarefied at infinity. Our result is applicable to cases involving a broader range of nonlinear terms and initial data that are constant at infinity. The key technique employed is a refined test function method.

We have studied the bound states of exactly solvable complex PT symmetric potentials Rosen Morse II potential and Scarf II potential from the perspective of exact quantization rule, especially the derivation of energy levels formulas. It was found that in the case of complex potentials, the two turning points on the complex plane, but it does not affect us to calculate the energy levels of Rosen Morse II and Scarf II potentials. We obtain the bound states energy levels by integrating the momentum in the ground state and the momentum in the general state between their two turning points.