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The nonlocal Fokker–Planck equations for a class of stochastic differential equations with non-Gaussian α -stable Lévy motion in Euclidean space are studied. The existence and uniqueness of weak solution are obtained with rough drift. The solution is shown to be smooth on spatial variable if all derivatives of the drift are bounded. Moreover, the solution is jointly smooth on spatial and time variable if we assume further that the drift grows like a power of logarithm function at infinity. [ABSTRACT FROM AUTHOR]

We study the quantitative unique continuation property of some higher order elliptic operators. We prove a lower bound for nontrivial solutions of the equation ( − Δ ) m u + V ( x ) u = 0 , m ∈ N , V ( x ) is bounded. The bound shows that nontrivial solutions can not decay faster than e − | x | 4 / 3 ln ⁡ | x | at infinity. Moreover, we obtain an improved lower bound of nontrivial solutions for a special forth order elliptic operators in dimension 2, the bound is shown to be essentially sharp by constructing a Meshkov-type example. [ABSTRACT FROM AUTHOR]

Abstract: The long time behavior of solutions of 2D quasi-geostrophic equations with damping on the whole space is studied. The existence of the global attractor in is proved. New difficulties come from nonlocal terms and the Banach space . We overcome them by using commutator estimates and positivity inequalities associated with fractional Laplacian. [Copyright &y& Elsevier]

Abstract: In this paper we consider the long time behavior of solutions of a version of gKdV equation on the real line. The existence of the global attractor in is proved when the force f belongs to . Moreover, we investigate the change of the attractor according to the change of f and the coefficient γ of damping term. Finally, several numerical experiments confirm all the theoretical conclusions. [Copyright &y& Elsevier]

In this paper we study the critical exponents of the Cauchy problem in Rn of the quasilinear singular parabolic equations: ut = div(&z.sfnc;∇u&z.sfnc;m − 1∇u) + ts&z.sfnc;x&z.sfnc;σup, with non-negative initial data. Here s ≥ 0, (n − 1)/(n + 1) < m < 1, p > 1 and σ > n(1 − m) − (1 + m + 2s). We prove that pc ≡ m + (1 + m + 2s + σ)/n > 1 is the critical exponent. That is, if 1 < p ≤ pc then every non-trivial solution blows up in finite time, but for p > pc, a small positive global solution exists. [Copyright &y& Elsevier]

For every p belongs to [ 1 , ∞) and n dimensional vector b , it is shown that − Λ α − b ⋅ ∇ generates an analytic semigroup in L p (R n) if 1 ≤ α ≤ 2 , and a Gevrey semigroup of order δ , for any δ > 1 α , in L p (R n) if 0 < α < 1. Moreover, it is proved that the semigroup is not a Gevrey semigroup of order δ with δ ≤ 1 α if 0 < α < 1. This shows that our result is sharp. [ABSTRACT FROM AUTHOR]

Abstract: Under investigation in this paper is the Hirota–Maccari equation, which is a generalized -dimensional model in fluid dynamics, plasma physics and optical fiber communication. With the aid of the Hirota bilinear method and symbolic computation, the corresponding N-soliton solutions are given and illustrated. The characteristic face method and asymptotic analysis are applied to discuss the solitonic propagation and collision, including the bidirectional solitons, elastic interactions and inelastic interactions. Finally, a kind of special phenomenon with the parameters varying is investigated, which might provide people with useful information on the dynamics of the relevant fields. [Copyright &y& Elsevier]