Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary (null) controllability and the local boundary (weak) observability for a coupled system of 1-D quasilinear wave equations with various types of boundary conditions.

The authors provide a relaxation result in BV × L ^q, 1 ≤ q < + ∞ as the first step towards the analysis of thermochemical equilibria.

The author considers a new class $SH_{\lambda \mu }^m (\alpha )$ of normalized analytic functions defined by a differential operator. Several basic properties and characteristics of the functions belonging to the class $SH_{\lambda \mu }^m (\alpha )$ are investigated. These include integral representations, coefficient bounds, the Fekete-Szegö problem, class-preserving operators and T _δ-neighborhoods.

In this paper, a one-dimensional bipolar Euler-Poisson system (a hydrodynamic model) from semiconductors or plasmas with boundary effects is considered. This system takes the form of Euler-Poisson with an electric field and frictional damping added to the momentum equations. The large-time behavior of uniformly bounded weak solutions to the initial-boundary value problem for the one-dimensional bipolar Euler-Poisson system is firstly presented. Next, two particle densities and the corresponding current momenta are verified to satisfy the porous medium equation and the classical Darcy’s law time asymptotically. Finally, as a by-product, the quasineutral limit of the weak solutions to the initial-boundary value problem is investigated in the sense that the bounded L ^∞ entropy solution to the one-dimensional bipolar Euler-Poisson system converges to that of the corresponding one-dimensional compressible Euler equations with damping exponentially fast as t → +∞. As far as we know, this is the first result about the asymptotic behavior and the quasineutral limit for the one-dimensional bipolar Euler-Poisson system with boundary effects and a vacuum.

The Hardy space H ^p is not locally convex if 0 < p < 1, even though its conjugate space (H ^p)* separates the points of H ^p. But then it is locally p-convex, and its conjugate cone (H ^p) _p* is large enough to separate the points of H ^p. In this case, the conjugate cone can be used to replace its conjugate space to set up the duality theory in the p-convex analysis. This paper deals with the representation problem of the conjugate cone (H ^p) _p ^* of H ^p for 0 < p ≤ 1, and obtains the subrepresentation theorem (H ^p) _p ^* ≃ L ^∞(T, C _p ^*).

In this paper, for the full Euler system of the isothermal gas, we show that a globally stable supersonic conic shock wave solution does not exist when a uniform supersonic incoming flow hits an infinitely long and curved sharp conic body.

The authors consider ±(Φ, J)-holomorphic maps from Sasakian manifolds into Kähler manifolds, which can be seen as counterparts of holomorphic maps in Kähler geometry. It is proved that those maps must be harmonic and basic. Then a Schwarz lemma for those maps is obtained. On the other hand, an invariant in its basic homotopic class is obtained. Moreover, the invariant is just held in the class of basic maps.

The author studies the boundary value problem of the classical semilinear parabolic equations $u_t - \Delta u = \left| u \right|^{p - 1} u in \Omega \times (0,{\rm T}),$, and u = 0 on the boundary ∂Ω × [0, T) and u = φ at t = 0, where Ω ⊂ ℝ ⁿ is a compact C ¹ domain, 1 < p ≤ pS is a fixed constant, and φ ∈ C ₁ ⁰ (Ω) is a given smooth function. Introducing a new idea, it is shown that there are two sets $\tilde W$ and $\tilde Z$, such that for $\varphi \in \tilde W$, there is a global positive solution $u(t) \in \tilde W$ with H ¹ omega limit 0 and for $\varphi \in \tilde Z$, the solution blows up at finite time.

A new system of generalized nonlinear variational-like inclusions involving A-maximal m-relaxed η-accretive (so-called, (A, η)-accretive in [36]) mappings in q-uniformly smooth Banach spaces is introduced, and then, by using the resolvent operator technique associated with A-maximal m-relaxed η-accretive mappings due to Lan et al., the existence and uniqueness of a solution to the aforementioned system is established. Applying two nearly uniformly Lipschitzian mappings S ₁ and S ₂ and using the resolvent operator technique associated with A-maximal m-relaxed η-accretive mappings, we shall construct a new perturbed N-step iterative algorithm with mixed errors for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping Q = (S ₁, S ₂) which is the unique solution of the aforesaid system. We also prove the convergence and stability of the iterative sequence generated by the suggested perturbed iterative algorithm under some suitable conditions. The results presented in this paper extend and improve some known results in the literature.

Let Ω be a bounded domain in ℝ ⁿ with a smooth boundary, and let h _p,q be the space of all harmonic functions with a finite mixed norm. The authors first obtain an equivalent norm on h _p,q, with which the definition of Carleson type measures for h _p,q is obtained. And also, the authors obtain the boundedness of the Bergman projection on h _p,q which turns out the dual space of h _p,q. As an application, the authors characterize the boundedness (and compactness) of Toeplitz operators T _µ on h _p,q for those positive finite Borel measures µ.