In this note, we show that the area integral of positive harmonic functions on a constant negative curvature space form is almost everywhere finite with respect to a harmonic measure on S(∞).

This paper deals with the Cauchy problem for the system of semilinear wave equations with small initial data. We give the upper bounds for the lifespan of the classical solution to the systems.

In this paper, a series of sharp criterions of global existence and collapse for coupled nonlinear Schrödinger equations are derived out in terms of the characteristics of the ground state and the local theories. And the conclusion that how small the initial data are, the global solutions exist is proved.

For quasilinear hyperbolic systems with characteristics of constant mul- tiplicity, suppose that characteristics of constant multiplicity(> 1) are linearly degen- erate, by means of generalized normalized coordinates we get the global existence and the blow-up phenomenon of the C¹ solution to the Cauchy problem under an additional hypothesis.

The uniqueness and existence of BV solutions to Dirichlet problem of doubly degenerate parabolic equations of the following form \frac{∂u}{∂t} = div(A(|∇B(u)|)∇B(u)) in Q_T = Ω × (0, T) are studied.

This paper is concerned with the large time behavior for solutions of the nonlinear parabolic equations in whole spaces R^n. The spectral decomposition methods of Laplace operator are applied and it is proved that if the initial data u_0 ∈ L² ∩ L^r for 1 ≤ r ≤ 2, then the solutions decay in L² norm at t^{-\frac{n}{2}(\frac{1}{r}-\frac{1}{2})}. The decay rates are optimal in the sense that they coincide with the decay rates of the solutions to the heat equations with the same initial data.

We consider the compactness theorem for the positive solutions of the equation Δu + h_1u^α + h_2u^β = 0 in Ω ⊂ R^n and obtain the measure estimate of the blow up set for positive smooth solutions {u_i} of the above equation with {||u_i||_{H¹(Ω)} + ||u_i||_{L^{α+1}(Ω)} + ||u_i||_L^{β+1}(Ω)} bounded.

In this paper, we prove the existence and uniqueness of global solutions in H^s(R^3) (s ∈ R, s ≥ 0) for the initial value problem of the bipolar Schrödinger-Poisson systems.