Related papers: Nonlinear Fokker-Planck equations with time-depend…
This paper develops solutions of fractional Fokker-Planck equations describing subdiffusion of probability densities of stochastic dynamical systems driven by non-Gaussian L\'evy processes, with space-time-dependent drift, diffusion and…
We prove global existence, uniqueness and regularity of the mild, Lp and classical solution of a non-linear Fokker-Planck equation arising in an adaptive importance sampling method for molecular dynamics calculations. The non- linear term…
This work is concerned with the existence and uniqueness of generalized (mild or distributional) solutions to (possibly degenerate) Fokker-Planck equations $\rho_t-\Delta\beta(\rho)+{\rm div}(Db(\rho)\rho)=0$ in…
The focus of this paper is a non-local singular non-linear Fokker-Planck partial differential equation (PDE). The peculiarity of this PDE feature is in its divergence coefficient, which presents a product between a Besov distribution and a…
We approximate the solution $u$ of the Cauchy problem $$ \frac{\partial}{\partial t} u(t,x)=Lu(t,x)+f(t,x), \quad (t,x)\in(0,T]\times\bR^d, $$ $$ u(0,x)=u_0(x),\quad x\in\bR^d $$ by splitting the equation into the system $$…
In this paper, we study the following semilinear Schr\"odinger equation with periodic coefficient: $$-\triangle u +V(x)u=f(x,u), u\in H^{1}(\mathbb{R}^{N}).$$ The functional corresponding to this equation possesses strongly indefinite…
For the nonlocal quasilinear fractional $p$-Laplace operator $(-\Delta)^s_p$ with $s\in (0,1)$ and $p\in(1,\infty)$, we investigate the nonexistence and existence of nontrivial nonnegative solutions $u$ in the local fractional Sobolev space…
We derive analytic solutions for the full time dependence of space-fractional Fokker-Planck equations corresponding to stochastic Langevin equations with additive tempered-stable L\'{e}vy noise terms. The drift terms are generalised to be…
We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes $$ \partial_{t}^{\alpha}u=\left( \phi(\Delta) u +f(u) \right) + \partial_t^\beta \sum_{k=1}^\infty \int_0^t…
In this work, we consider a FDE (fractional diffusion equation) $${}^C D_t^\alpha u(x,t)-a(t)\mathcal{L} u(x,t)=F(x,t)$$ with a time-dependent diffusion coefficient $a(t)$. For the direct problem, given an $a(t),$ we establish the…
In this paper the feasibility of funnel control techniques for the Fokker-Planck equation corresponding to a multi-dimensional Ornstein-Uhlenbeck process on an unbounded spatial domain is explored. First, using weighted Lebesgue and Sobolev…
This paper is concerned with the asymptotic behavior of bounded solutions of the Cauchy problem \begin{equation*} \left\{ \begin{array}{ll} u_t=u_{xx} +f(t,u), & x\in\mathbb{R},\,t>0,\\ u(x,0)= u_0, & x\in\mathbb{R}, \end{array}\right.…
We start with a global Maxwellian $M_{k}$, which is a stationary solution, with the constant total density ($\rho(t)\equiv \wt \rho$), of the Fokker-Planck equation. The notion of distance between the function $M_{k}$ and an arbitrary…
We consider the linear Schr\"odinger equation under periodic boundary condition, driven by a random force and damped by a quasilinear damping: $$ \frac{d}{dt}u+i\big(-\Delta+V(x)\big) u=\nu \Big(\Delta u-\gr |u|^{2p}u-i\gi |u|^{2q}u \Big)…
It is known that for some time periodic potentials $q(t, x) \geq 0$ having compact support with respect to $x$ some solutions of the Cauchy problem for the wave equation $\partial_t^2 u - \Delta_x u + q(t,x)u = 0$ have exponentially…
We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equation $$ u_{xt}+\alpha\beta^2u-2i\alpha\beta u_x-\alpha u_{xx}-i\alpha\beta^2|u|^2u_x=0 \label{cs} $$ with generic initial data in a Sobolev space which…
The Vlasov-Nordstr\"{o}m-Fokker-Planck system describes the evolution of self-gravitating matter experiencing collisions with a fixed background of particles in the framework of a relativistic scalar theory of gravitation. We study the…
The Cauchy- and periodic boundary value problem for the nonlinear Schroedinger equations in $n$ space dimensions [u_t - i\Delta u = (\nabla \bar{u})^{\beta}, |\beta|=m \ge 2, u(0)=u_0 \in H^{s+1}_x] is shown to be locally well posed for $s…
We investigate the Cauchy problem for a semilinear spatio--temporal fractional diffusion equation with a time-dependent forcing term: \[ \partial_t^\alpha u + (-\Delta)^{\mathsf{s}} u = |u|^p + t^{\sigma}\,\mathbf{w}(x), \quad (t,x) \in…
We deal with a planar differential system of the form \begin{equation*} \begin{cases} \, u' = h(t,v), \\ \, v' = - \lambda a(t) g(u), \end{cases} \end{equation*} where $h$ is $T$-periodic in the first variable and strictly increasing in the…