Blow-up behaviour for the fourth-order quasilinear porous medium equation with source, u_t=(|u|^nu)_{xxxx}+|u|^{p-1}u in R×R_+, where n > 0, p > 1, is studied. Countable and finite families of similarity blow-up patterns of the form u_S(x,t)=(T-t)^{-\frac{1}{p-1}}f(y), where y=\frac{x}{T-t}^β, β=\frac{p-(n+1)}{4(p-1)}, which blow-up as t→T^- < ∞, are described. These solutions explain key features of regional (for p=n+1), single point (for p > n+1), and global (for p∈(1,n+1)) blowup. The concepts and various variational, bifurcation, and numerical approaches for revealing the structure and multiplicities of such blow-up patterns are presented.

In this paper we consider regularities of radial solutions to the degenerate complex Hessian equations. Our results generalize some results for Monge-Ampere equation in [Monn, Math. Ana. 275 (1986), pp. 501-511] and [Delanoe, J. Diff. Eqn. 58 (1985), pp. 318-344].

In this paper we investigate the equations for light-like extremal surfaces in Minkowski space R^{1+(1+n)}. We show that the light-like assumption is compatible with the Cauchy problem and give a necessary and sufficient condition on the global existence of classical solutions of the Cauchy problem. Based on this, we obtain entire light-like extremal surfaces by solving the Cauchy problem explicitly when such necessary and sufficient condition holds. Finally, some discussions and related remarks are given.

This paper is concerned with the existence theory of a semilinear elliptic system. In particular, we will prove that the system has a nontrivial positive solution in some appropriate solution spaces.

In this article, we will investigate the viscous Burgers equation with boundary feedback. The existence of the solution is proved by constructing a convergence sequence inductively. Moreover, the decay property of the solution is shown based on the maximum principle for nonlinear parabolic equations.

We define a kind of KdV (Korteweg-de Vries) geometric flow for maps from a real line or a circle into a Kähler manifold (N, J,h) with complex structure J and metric h as the generalization of the vortex filament dynamics from a real line or a circle. By using the geometric analysis, the existence of the Cauchy problems of the KdV geometric flows will be investigated in this note.