In this article, we study the global L^∞ entropy solutions for the Cauchy problem of system of isentropic gas dynamics in a divergent nozzle with a friction. Especially when the adiabatic exponent γ=3, we apply for the maximum principle to obtain the L^∞ estimates w(ρ^δ,ε, u^δ,ε) ≤ B(t) and z(ρ^δ,ε, u^δ,ε) ≤ B(t) for the viscosity solutions (ρ^δ,ε, u^δ,ε), where B(t) is a nonnegative bounded function for any finite time t. This work, in the special case γ ≥ 3, extends the previous works, which provided the global entropy solutions for the same Cauchy problem with the restriction w(ρ^δ,ε, u^δ,ε) ≤ 0 or z(ρ^δ,ε, u^δ,ε) ≤ 0.

In this paper, we obtain an estimate for the lower bound for the dimensions of harmonic functions with polynomial growth and a Liouville type theorem on manifolds with nonnegative Ricci curvature whose tangent cone at infinity is a unique metric cone with a conic measure.

We show that closed shrinking gradient Ricci solitons with positive Ricci curvature and sufficiently pinched Weyl tensor are Einstein. When Weyl tensor vanishes, this has been proved before but our proof here is much simpler.

In this article, the non-self dual extended Harper's model with a Liouville frequency is considered. It is shown that the corresponding integrated density of states is 1/2-Hölder continuous. As an application, the homogeneity of the spectrum is proven.

Let L=-div(A∇) be a second order divergence form elliptic operator, where A is an accretive, n×n matrix with bounded measurable complex coefficients on Rⁿ. Let L^α/2 (0 < α < 1) denotes the fractional differential operator associated with L and (-∆)^α/2b ∈ L^n/α(Rⁿ). In this article, we prove that the commutator[b, L^α/2] is bounded from the homogenous Sobolev space L_α² (Rⁿ) to L²(Rⁿ).

In this paper, we first establish several sharp inequalities of homogeneous expansion for biholomorphic quasi-convex mappings of type B and order α on the unit ball E in a complex Banach space X by applying the method and technique of complex analysis. Then, as the application of these sharp inequalities, we derive the sharp estimate of third homogeneous expansions for the above mappings defined on the unit polydisk Uⁿ in Cⁿ.

Let c be a nonzero constant and f(z) be a transcendental meromorphic function of finite order. Under some conditions, we study the relationships between the exponent of convergence of fixed points of f(z), its shift f(z +c) and forward differences ∆_cⁿ f(z), n ∈ N⁺.

In this paper, we study the initial-boundary value problem for the semilinear parabolic equations u_t -△_Xu=|u|^p-1u, where X=(X₁, X₂, …, X_m) is a system of real smooth vector fields which satisfy the H?rmander's condition, and is a finitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Finally, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy, and also we discuss the asymptotic behavior of the global solutions.

This work deals with approximation solutions to a type of integro-differential equations in several complex variables. It concerns the Cauchy formula on higher dimensional domains. In our study, we make use of multiple power series expansions and an iterative computation method to solve a kind of integro-differential equation. We introduce a symmetrized topology product area which is called a bicylinder. We expand functions and derivatives of them to power series. Moreover we obtain unknown functions by comparing coefficients of the series on both sides of equations. We express the approximation solutions by a regular product of matrixes.

In this paper, we investigate the translative containment measure for a convex domain K_i to contain, or to be contained in the homothetic copy of another convex domain tK_j(t ≥ 0). Via the formulas of translative Blaschke and Poincaré in integral formula, we obtain a Bonnesen-style symmetric mixed isohomothetic inequality. The Bonnesen-style symmetric mixed isohomothetic inequality obtained is known as Bonnesen-style inequality if one of the domains is a disc. As a direct consequence, we attain an inequality which strengthen the result proved by Bonnesen, Blaschké and Flanders. Furthermore, by the containment measure and Blaschke's rolling theorem, we obtain the reverse Bonnesen-style symmetric mixed isohomothetic inequalities. These inequalities are the analogues of the known Bottema's result in 1933.

Let G be a discrete group with a weight w on it. For p>1, we define a class of generalized Figà-Talamanca-Herz algebras A_p(G, w, α, θ) and obtain their (w, α, θ)-Dual spaces. Moreover, we show that the generalized Figà-Talamanca-Herz algebras have an approximation property when G is a proper discrete group and satisfies the p-RD property.

We establish a precise Schwarz lemma for real-valued and bounded harmonic functions in the real unit ball of dimension n. This extends Chen's Schwarz-Pick lemma for real-valued and bounded planar harmonic mapping.

We consider a branching random walk in an independent and identically distributed random environment ξ=(ξ_n) indexed by the time. Let W be the limit of the martingale W_n=∫e^-txZ_n(dx)/E_ξ ∫e^-txZ_n(dx), with Z_n denoting the counting measure of particles of generation n, and E_ξ the conditional expectation given the environment ξ. We find necessary and sufficient conditions for the existence of quenched moments and weighted moments of W, when W is non-degenerate.

In this article we consider positive large solution of cooperative systems of the form -∆u₁=λ₁u₁ + a₁u₁u₂^q₁ -b₁(x)u₁^p₁+1, -∆u₂=λ₂u₂ + a₂u₁^q₂u₂ -b₂(x)u₂^p₂+1 in a bounded smooth domain Ω ⊂ R^N(λ_i ∈ R, a_i, b_i>0, 0 < q_i < p_j, i, j ∈ {1, 2}, i ≠ j), Based on the construction of certain sup and sub-solution, we show existence, uniqueness and blow-up rate of the large solution.

The existence and uniqueness theorem of classical solutions of a coupled system of nonlinear parabolic PDEs arising in modelling of chemical reactions between two polymeric reactants over inhomogeneous surfaces with nonclassical boundary conditions is proved and the long-time behaviour of the solution is studied.

We study the heat flow of equation of H-surface with non-zero Dirichlet boundary in the present article. Introducing the "stable set" M₂ and "unstable set" M₁, we show that there exists a unique global solution provided the initial data belong to M₂ and the global solution converges to zero in H¹ exponentially as time goes to infinity. Moreover, we also prove that the local regular solution must blow up at finite time provided the initial data belong to M₁.

We establish the unilateral global bifurcation result for the following nonlinear operator equation
u=L(λ)u + H(λ, u), (λ, u) ∈ R^m×X
where m is a positive integer, X is a Banach space, L(·) is a positively homogeneous completely continuous operator and H:R^m×X → X is completely continuous with H=o (||u||) near u=0 uniformly on bounded λ sets.

An equivalent condition is derived for g-concave function defined by (static) g-expectation. Several extensions including quadratic generators and (g,h)-concavity are also considered.

In this work, we study the inverse problem stability of the continuous-in-time model which is designed to be used for the finances of public institutions. We discuss this study with determining the Loan measure from algebraic spending measure in Radon measure space M([t_I, Θ_max]), and in Hilbert space L²([t_I, Θ_max]) when they are density measures. For this inverse problem we prove the uniqueness theorem, obtain a procedure for constructing the solution and provide necessary and sufficient conditions for the solvability of the inverse problem in L²([t_I, Θ_max]).

We establish the monotonicity and convexity properties for several special functions involving the generalized elliptic integrals, and present some new analytic inequalities.

In this article, we study the following Klein-Gordon-Maxwell system involving critical exponent
(KGM)

where λ and ω are two positive constants. We found the existence of positive ground state solutions (that is, for solutions which minimizes the action functional among all the solutions) of (KGM) which improves some previous existence result in Carri?o et al.(2012)[8].