In this note we study the nonexistence of nontrivial global solutions on S=R^N×(0,∞) for the following inequalities: |u|_t≥Δ(|u|^{m-1}u)+|u|^q, and  |u|_t≥div(|∇u|^{p-2}∇|u|)+|u|^q. When m, p, q satisfy some given conditions, the nonexistence of nontrivial global solution is proved, without taking their traces on the hyperplans t=0 into account.

In this paper, we study a generalized thin film equation which is relevant to capillary driven flows of thin films of power-law fluids. We prove that the generalized thin film equation in dimension d ≥ 2 has a unique C^1 source type radial self-similar nonnegative solution if 0 < n < 2p-1 and has no solution of this type if n ≥ 2p-1.

In this paper we investigate time-dependent Ginzburg-Landau equations come from the superfluid atomic Fermi-gases near the Feshbach resonance from the Fermion-Boson model. We obtain the global existence and uniqueness of solution to the equations in various spatial dimensions.

In the present paper the regularity of solutions to Dirichlet problem of degenerate elliptic Monge-Ampère equations is studied. Let Ω⊂R^2 be smooth and convex. Suppose that u∈C^2(Ω) is a solution to the following problem: det(u_{ij}) = K(x) f (x,u,Du) in Ω with u = 0 on ∂Ω. Then u∈C^∞(\bar{Ω}) provided that f (x,u,p) is smooth and positive in \bar{Ω}×R×R^2, K > 0 in Ω and near ∂Ω, K=d^m\tilde{K}, where d is the distance to ∂Ω, m some integer bigger than 1 and \tilde{K} smooth and positive on \bar{Ω}.

This paper is concerned with the asymptotic behavior of solution to the following model involving two species all with chemotaxis: \frac{∂p}{∂t}=D_p∇(p∇ln\frac{p}{ω}), \frac{∂q}{∂t}=D_q∇(q∇ln\frac{q}{ω}), \frac{∂ω}{∂t}=βp-δω, p∇ln(\frac{p}{ω}·\vec{n}=q∇ln\frac{q}{ω})·\vec{n}=0. We prove that the solution exists globally as β ≥ 0. As β < 0, whether the solution exists globally or not depends on the initial data. By function transformation and comparison, the asymptotical behavior of the solution is studied.

In this paper, we study the higher-order semilinear parabolic system u_t+(-Δ)^mu=a|v|^{p-1}v, t(x)∈R^1_+×R^N, v_t+(-Δ)^mv=b|u|^{q-1}u, t(x)∈R^1_+×R^N, u(0,x)=φ(x), v(0,x)=ψ(x), x∈R^N, where m, p, q > 1, a,b∈R. We prove that the global existence of mild solutions for small initial data with respect to certain norms. Some of these solutions are proved to be asymptotically self-similar.