In this paper we consider the Cauchy problem of a generalization of time-fractional diffusion equation with variable coefficients in R n+1 + , where the time derivative is replaced by a regularized hyper-Bessel operator. The explicit solution of the inhomogeneous linear equation for any n ∈ Z + and its uniqueness in a weighted Sobolev space are established. The key tools are Mittag-Leffler functions, M-Wright functions and Mikhlin multiplier theorem. At last, we obtain the existence of solution of the semilinear equation for n = 1 by using a fixed point theorem.

In this paper, we study the well posed‐ness of Cauchy problem for a class of hyperbolic equation with characteristic degeneration on the initial hyperplane. By a delicate analysis of two integral operators in terms of Bessel functions, we give the uniform weighted estimates of solutions to the linear problem with a parameter m∈(0,1) and establish local and global existences of solution to the semilinear equation. Meanwhile, we derive the existence of solutions to semilinear generalized Euler‐Poisson‐Darboux equation with a negative parameter α∈(−1,0).

In this paper, we consider direct problem and inverse source problem of time‐fractional Black‐Scholes type model involving hyper‐Bessel operator. Analytical solutions to these problems are constructed based on appropriate eigenfunction expansion and Erdélyi‐Kober fractional integrals whose kernel has double singularities; then, existence and uniqueness are established. At last, the results are demonstrated by explicit solutions of some examples using appropriate choice of the given data.

By studying a weakly singular integral whose kernel involves Mittag‐Leffler functions, we obtain some new Gronwall‐type integral inequalities. Applying these inequalities and fixed point theorems, existence and uniqueness of positive solution of initial value problem to nonlinear fractional differential equation with Caputo‐like counterpart hyper‐Bessel operators are established.

We focus on the nonexistence of global weak solutions of nonlinear Keldysh type equation with one derivative term. In terms of the analysis of the first Fourier coefficient, we show the solution of singular initial value problem and singular initial-boundary value problem of the nonlinear equation with positive initial data blow-up in some finite time interval.