Large time behavior of solutions for critical and subcritical complex Ginzburg-Landau equations in H<Superscript>1</Superscript>

Considering the Cauchy problem for the critical complex Ginzburg-Landau equation in H¹(Rⁿ), we shall show the asymptotic behavior for its solutions in C(0, ∖;H¹(Rⁿ)) ∩ L²(0, ∖;H^1,2n/(n-2)(R²)), n≥3. Analogous results also hold in the case that the nonlinearity has the subcritical power in H¹(Rⁿ), n≥1. Dedicated to Professor Zhou Yulin for his 80th birthday.     

On Existence and Multiplicity of Solutions for Elliptic Equations Involving the p-Laplacian

In the present paper, the following Dirichlet problem and Neumann problem involving the p-Laplacian

Perturbation theory for convolution semigroups

We deal with convolution semigroups (not necessarily symmetric) in Lp(RN) and provide a general perturbation theory of their generators by indefinite singular potentials. Such semigroups arise in the theory of Lévy processes and cover many examples such as Gaussian semigroups, α-stable semigroups, relativistic Schrödinger semigroups, etc. We give new generation theorems and Feynman-Kac formulas. In particular, by using weak compactness methods in L¹, we enlarge the extended Kato class potentials used in the theory of Markov processes. In L² setting, Dirichlet form-perturbation theory is finely related to L¹-theory and the extended Kato class measures is also enlarged. Finally, various perturbation problems for subordinate semigroups are considered.