It is proved that given a conformal metric e^{u0}g_0, with e^{u0} ∈ L∞, on a 2-dim closed Riemannian manfold (M, g_0), there exists a unique smooth solution u(t) of the Ricci flow such that u(t) → u_0 in L² as t → 0.

This paper studies the Cauchy problem of the dissipative quasi-geostrophic equation in pseudomeasure space PM^{n+1-2α}(\mathbb{R}^n) or Lorentz space L\frac{n}{2α-1, ∞}(\mathbb{R}^n), which admit the singular solutions. The global well-posedness is established provided initial data θ_0(x) are small enough in these spaces. Moreover, the asymptotic stability of solutions in pseudomeasure space is proved. In particular, if the initial data are homo-geneous functions of degree 1 - 2α, the self-similar solutions are also obtained.

This paper is devoted to studying the initial value problem of the Ginzburg-Landau type equations. We treat the case where the nonlinear interaction function is a general continuous function, not required to satisfy any smoothness conditions. Local and global existence results of solutions of the problem are given. Decay estimates are also shown.

In this article, we prove the Strichartz type estimate for the solutions of linear heat equation with initial data in Hardy space H¹(R^d). As an application, we obtain the full space-time estimate to the solutions of heat equation with initial data in L^p(R^d) for 1 ‹ p ‹ ∞.

This paper considers the stability of the solitary waves for the generalized Zakharov system. By applying the abstract theory of Grillakis M. et al. and detailed spectral analysis, we obtain the stability of the solitary waves.

In this paper, the author gives the uniform estimates for the solutions of the semilinear elliptic equation Δu+λe^u = 0 with zero Dirichlet boundary condition, and then derive the uniqueness of the nonminimal solution assuming some conditions satisfied.