IPI
Backward problem for a time-space fractional diffusion equation
Junxiong Jia Jigen Peng Jinghuai Gao Yujiao Li
Inverse Problems & Imaging 2018, 12(3): 773-799 doi: 10.3934/ipi.2018033

In this paper, a backward problem for a time-space fractional diffusion process has been considered. For this problem, we propose to construct the initial function by minimizing data residual error in Fourier space domain with variable total variation (TV) regularizing term which can protect the edges as TV regularizing term and reduce staircasing effect. The well-posedness of this optimization problem is obtained under a very general setting. Actually, we rewrite the time-space fractional diffusion equation as an abstract fractional differential equation and deduce our results by using fractional operator semigroup theory, hence, our theoretical results can be applied to other backward problems for the differential equations with more general fractional operator. Then a modified Bregman iterative algorithm has been proposed to approximate the minimizer. The new features of this algorithm is that the regularizing term altered in each step and we need not to solve the complex Euler-Lagrange equation of variable TV regularizing term (just need to solve a simple Euler-Lagrange equation). The convergence of this algorithm and the strategy of choosing parameters are also obtained. Numerical implementations are provided to support our theoretical analysis to show the flexibility of our minimization model.

keywords: Backward time-space fractional diffusion fractional operator semigroup Bregmann iterative method variable TV regularization
CPAA
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
DCDS
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
CPAA
On general fractional abstract Cauchy problem
Zhan-Dong Mei Jigen Peng Yang Zhang
Communications on Pure & Applied Analysis 2013, 12(6): 2753-2772 doi: 10.3934/cpaa.2013.12.2753
This paper is concerned with general fractional Cauchy problems of order $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ in infinite-dimensional Banach spaces. A new notion, named general fractional resolvent of order $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ is developed. Some of its properties are obtained. Moreover, some sufficient conditions are presented to guarantee that the mild solutions and strong solutions of homogeneous and inhomogeneous general fractional Cauchy problem exist. An illustrative example is presented.
keywords: General fractional abstract Cauchy problem general Riemann-Liouville fractional derivative general fractional resolvent.
CPAA
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

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