Related papers: Temporal asymptotics for fractional parabolic Ande…

We study the existence of {weak} solutions for fractional elliptic equations of the type, \begin{equation*} (-\Delta)^{\frac{1}{2}} u+ V(x) u= h(u), u> 0 \;\textrm{in} \;\mathbb R, \end{equation*} %where $1<q<2,\;p>2,\;1<\beta\leq2\;,…

In this paper, we investigate the following fractional Choquard type equation: \[ (- \Delta)_p^s\, u = \lambda\frac{|u|^{r-2}u}{|x|^\alpha}\,+\gamma \big(\int_\Omega \frac{|u|^q}{|x-y|^\mu}dy\big) |u|^{q-2}u \ \ \text{in } \Omega,\ \ u = 0…

A new class of fractional-order stochastic evolution equations of the form $(\partial_t + A)^\gamma X(t) = \dot{W}^Q(t)$, $t\in[0,T]$, $\gamma \in (0,\infty)$, is introduced, where $-A$ generates a $C_0$-semigroup on a separable Hilbert…

The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and…

We consider the fractional stochastic heat type equation \begin{align*} \frac{\partial}{\partial t} u_t(x)=-(-\Delta)^{\alpha/2}u_t(x)+\xi\sigma(u_t(x))\dot{F}(t,x),\ \ \ x\in D, \ \ t>0, \end{align*} with nonnegative bounded initial…

In this paper, we use local fraction derivative to show the H\"older continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation:…

We consider the parabolic equation $$u_t-\Delta u=F(x,u),\quad (t,x)\in\R_+\times\R^n\tag{P}$$ and the corresponding semiflow $\pi$ in the phase space $H^1$. We give conditions on the nonlinearity $F(x,u)$, ensuring that all bounded sets of…

In this paper, we investigate Harnack estimates for weak solutions to the following nonlocal equation: $$ \partial_t u = \Delta^{\alpha/2} u + b \cdot \nabla u + f, $$ where $\Delta^{\alpha/2}$ denotes the fractional Laplacian, $b$ is a…

We study the following equation \begin{equation*} \frac{\partial u(t,\,x)}{\partial t}= \Delta u(t,\,x)+b(u(t,\,x))+\sigma \dot{W}(t,\,x),\quad t>0, \end{equation*} where $\sigma$ is a positive constant and $\dot{W}$ is a space-time white…

We study the existence and the properties of the reduced measures for the parabolic equations $\partial_tu-\Delta u+g(u)=0$ in $\Omega\times (0,\infty)$ subject to the conditions ($P$): $u=0$ on $\partial\Omega\times (0,\infty)$,…

We study the asymptotic behavior of nonnegative solutions of the semilinear parabolic problem {u_t=\Delta u + u^{p}, x\in\mathbb{R}^{N}, t>0 u(0)=u_{0}, x\in\mathbb{R}^{N}, t=0. It is known that the nonnegative solution $u(t)$ of this…

We show that if $u$ solves the fractional parabolic equation $(\partial_t - \Delta )^s u = Vu$ in $B_5 \times (-25, 0]$ ($0<s<1$) such that $u(\cdot, 0) \not\equiv 0$, then the maximal vanishing order of $u$ in space-time at $(0,0)$ is…

Consider the stochastic partial differential equation $\partial_t u = Lu+\sigma(u)\xi$, where $\xi$ denotes space-time white noise and $L:=-(-\Delta)^{\alpha/2}$ denotes the fractional Laplace operator of index…

We consider a one-dimensional nonlocal nonlinear equation of the form: $\partial_t u = (\Lambda^{-\alpha} u)\partial_x u - \nu \Lambda^{\beta}u$ where $\Lambda =(-\partial_{xx})^{\frac 12}$ is the fractional Laplacian and $\nu\ge 0$ is the…

We consider the Cauchy problem of fractional pseudo-parabolic equation on the whole space $R^n,n\geq 1$. Here, the fractional order $\alpha$ is related to the diffusion-type source term behaving as the usual diffusion term on the high…

In this article, we study the stochastic wave equation in arbitrary spatial dimension $d$, with a multiplicative term of the form $\sigma(u)=u$, also known in the literature as the Hyperbolic Anderson Model. This equation is perturbed by a…

Let $\Omega$ be a bounded domain in ${\mathbb R}^N$ and $T>0$. We study the problem \begin{equation} (P)\left\{ \begin{array}{lll} u_t - \Delta u \pm g(u) &= \mu \quad &\text{in } Q_T:=\Omega \times (0,T) \\ \phantom{------,} u&=0 &\text{on…

In this note we consider the parabolic Anderson model in one dimension with time-independent fractional noise $\dot{W}$ in space. We consider the case $H<\frac{1}{2}$ and get existence and uniqueness of solution. In order to find the…

We investigate forward and backward problems associated with abstract time-fractional Schr\"odinger equations $\mathrm{i}^\nu \partial_t^\alpha u(t) + A u(t)=0$, $\alpha \in (0,1)\cup (1,2)$ and $\nu\in\{1,\alpha\}$, where $A$ is a…

We study the regularity up to the boundary of solutions to fractional heat equation in bounded $C^{1,1}$ domains. More precisely, we consider solutions to $\partial_t u + (-\Delta)^s u=0 \textrm{ in }\Omega,\ t > 0$, with zero Dirichlet…