Related papers: On the Nonlinear $\Psi$-Hilfer Fractional Differen…
We study the Cauchy problem for the fractional Schr\"{o}dinger equation $$ i\partial_tu = (m^2-\Delta)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < \alpha < 2$, and $F$ stands for the nonlinearity of…
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schr\"odinger equation with initial data $u_{0}\in X$, where $X\in\{M_{2,q}^{s}(\mathbb R), H^{\sigma}(\mathbb T),…
This paper is in concern with Cauchy problems involving the fractional derivatives with respect to another function. Results of existence, uniqueness, and Taylor series among others are established in appropriate functional spaces. We prove…
We prove maximal Schauder regularity for solutions to elliptic systems and Cauchy problems, in the space $C_b(\mathbb{R}^d;\mathbb{R}^m)$ of bounded and continuous functions, associated to a class of nonautonomous weakly coupled…
In this paper, we study a time-fractional subdiffusion equation with a nonlinear nonlocal initial condition involving the unknown solution at the final time. The considered problem is formulated using the Caputo fractional derivative of…
We consider the Cauchy problem for the inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation \[iu_{t} +\Delta u=|x|^{-b} f(u),\;u(0)\in H^{s} (\mathbb R^{n} ),\] where $n\in \mathbb N$, $0<s<\min \{ n,\; 1+n/2\} $, $0<b<\min \{…
The main goal of this article is to study a Calder\'on type inverse problem for certain viscous nonlocal wave equations. We show that the partial Dirichlet to Neumann map uniquely determines on the one hand linear perturbations and on the…
Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite…
In this paper, we introduce a delayed Mittag-Leffler type function. With the help of the delayed Mittag-Leffler type functions, we give an explicit formula of solutions to linear nonhomogeneous fractional time-delay Langevin equations…
In this paper, we study the solvability of a Cauchy- Dirichlet problem for nonlinear parabolic equation with non standard growths and nonlocal terms. We show the existence of weak solutions of the considered problem under more general…
In this paper, we investigate the existence and uniqueness of solution of nonlinear $\psi$-Caputo fractional differential equation with the help of Banach fixed point theorem. Moreover, by using $\psi$-Gronwall inequality, we studied some…
The main objective of this article is to discuss the local existence of the solution to an initial value problem involving a non-linear differential equation in the sense of Riemann-Liouville fractional derivative of order $\sigma\in(1,2),$…
We consider the Cauchy problem of the higher-order KdV-type equation: \[ \partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_x u = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. The nonlinearity is…
In this paper, we show some results about the existence and the uniqueness of the positive solution for a $p$-Laplacian fractional differential equations with fractional derivative boundary condition. Our results are based on…
This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation $u_t+(-\triangle)^\alpha u= F(u)$ for initial data in the Lebesgue space $L^r(\mr^n)$ with $\ds r\ge r_d\triangleq{nb}/({2\alpha-d})$ or the…
Given a solution of a semilinear dispersive partial differential equation with a real analytic nonlinearity, we relate its Cauchy data at two different times by nonlinear representation formulas in terms of convergent series. These series…
We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic…
A subdiffusion problem in which the diffusion term is related to a stable stochastic process is introduced. Linear models of these systems have been studied in a general way, but non-linear models require a more specific analysis. The model…
In this paper we study the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some spaces of $\omega$-ultradifferentiable functions in the sense of Braun, Meise and Taylor, for…
This paper aims to investigate the Cauchy problem for the semilinear damped wave equation for the fractional sub-Laplacian $(-\mathcal{L}_{\mathbb{H}})^{\alpha}$, $\alpha>0$ on the Heisenberg group $\mathbb{H}^{n}$ with power type…