Well-posedness of abstract distributed-order fractional diffusion equations
Junxiong Jia Jigen Peng Kexue Li
Communications on Pure & Applied Analysis 2014, 13(2): 605-621 doi: 10.3934/cpaa.2014.13.605
In this paper, based on distributed-order fractional diffusion equation we propose the distributed-order fractional abstract Cauchy problem (DFACP) and study the well-posedness of DFACP. Using functional calculus technique, we prove that the general distributed-order fractional operator generates a bounded analytic $\alpha$-times resolvent operator family or a $C_0$-semigroup under some suitable conditions. In addition, we reveal the relation between two $\alpha$-times resolvent families generated by the sectorial operator $A$ and the special distributed-order fractional operator, $p_1A^{\beta_1}+ p_2A^{\beta_2}+\ldots +p_nA^{\beta_n}$, respectively.
keywords: Distributed-order fractional equation fractional power. $\alpha$-times resolvent family sectorial operator
On the decay and stability of global solutions to the 3D inhomogeneous MHD system
Junxiong Jia Jigen Peng Kexue Li
Communications on Pure & Applied Analysis 2017, 16(3): 745-780 doi: 10.3934/cpaa.2017036

In this paper, we investigative the large time decay and stability to any given global smooth solutions of the 3D incompressible inhomogeneous MHD systems. We prove that given a solution $(a, u, B)$ of (2), the velocity field and the magnetic field decay to zero with an explicit rate, for $u$ which coincide with incompressible inhomogeneous Navier-Stokes equations [1]. In particular, we give the decay rate of higher order derivatives of $u$ and $B$ which are useful to prove our main stability result. For a large solution of (2) denoted by $(a, u, B)$, we show that a small perturbation of the initial data still generates a unique global smooth solution and the smooth solution keeps close to the reference solution $(a, u, B)$. At last, we should mention that the main results in this paper are concerned with large solutions.

keywords: Inhomogeneous MHD system stability of large solution decay rate Besov space
Explosive solutions of parabolic stochastic partial differential equations with lévy noise
Kexue Li Jigen Peng Junxiong Jia
Discrete & Continuous Dynamical Systems - A 2017, 37(10): 5105-5125 doi: 10.3934/dcds.2017221

In this paper, we study the explosive solutions to a class of parbolic stochastic semilinear differential equations driven by a Lévy type noise. The sufficient conditions are presented to guarantee the existence of a unique positive solution of the stochastic partial differential equation under investigation. Moreover, we show that positive solutions will blow up in finite time in mean Lp-norm sense, provided that the initial data, the nonlinear term and the multiplicative noise satisfies some conditions. Several examples are presented to illustrate the theory. Finally, we establish a global existence theorem based on a Lyapunov functional and prove that a stochastic Allen-Cahn equation driven by Lévy noise has a global solution.

keywords: Stochastic reaction-diffusion equation positive solution blow-up of solutions Lévy noise

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