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We provide the classification of eternal (or ancient) solutions of the two-dimensional Ricci flow, which is equivalent to the fast diffusion equation $ \frac{\partial u}{\partial t} = \Delta \log u $ on $ \R^2 \times \R.$ We show that,…

Let $N\ge 3$. We are concerned with a Cauchy problem of the semilinear heat equation \[ \begin{cases} \partial_tu-\Delta u=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] where $f(0)=0$, $f$ is…

The main goal of this work is to prove that every non-negative {\it strong solution} $u(x,t)$ to the problem $$ u_t+(-\Delta)^{\alpha/2}u=0 \ \quad\mbox{for } (x,t)\in\mathbb{R}^{n}\times(0,T), \quad 0<\alpha<2, $$ can be written as…

In this paper we study the partial differential equation \begin{equation} \begin{split} \partial_tu &= k(t)\Delta_\alpha u - h(t)\varphi(u), u(0) &= u_0. \end{split} \end{equation} Here $\Delta_\alpha$ is the fractional Laplacian,…

Let $u(t, x) = (u_1(t, x), \dots, u_d(t, x))$ be the solution to the systems of nonlinear stochastic heat equations \[ \begin{split} \frac{\partial}{\partial t} u(t, x) &= \frac{\partial^2}{\partial x^2} u(t, x) + \sigma(u(t, x)) \dot{W}(t,…

We study the positivity and regularity of solutions to the fractional porous medium equations $u_t+(-\Delta)^su^m=0$ in $(0,\infty)\times\Omega$, for $m>1$ and $s\in (0,1)$ and with Dirichlet boundary data $u=0$ in…

In this paper we identify, for small $t$ and a fixed $T>0,$ the order $\alpha>0$ in the abstract fractional differential equation $$\partial^\alpha u(t)=Au(t),$$ where the time-fractional derivative $\partial^\alpha$ is understood in the…

In this paper, we discuss the uniqueness for solution to time-fractional diffusion equation $\partial_t^\alpha (u-u_0) + Au=0$ with the homogeneous Dirichlet boundary condition, where an elliptic operator $-A$ is not necessarily symmetric.…

In this paper we consider the problem: $\partial_{t} u- \Delta u=f(u),\; u(0)=u_0\in \exp L^p(\R^N),$ where $p>1$ and $f : \R\to\R$ having an exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in $\exp L^p_0(\R^N)$…

Let u = {u(t, x), t $\in$ [0, T ], x $\in$ R d } be the solution to the linear stochastic heat equation driven by a fractional noise in time with correlated spatial structure. We study various path properties of the process u with respect…

Supposing only that $\displaystyle\lim_{t \to 0} \frac{f(t)}{t} = 0$ and $\displaystyle\lim_{t \to \infty} \frac{f(t)}{t^{p}} = 0$, for some $p \in \left(1,\frac{N+1}{N-1}\right)$, we prove that solutions to the extension problem…

In this paper, we investigate pointwise time analyticity of solutions to fractional heat equations in the settings of $\mathbb{R}^d$ and a complete Riemannian manifold $\mathrm{M}$. On one hand, in $\mathbb{R}^d$, we prove that any solution…

For parabolic equations of the form $$ \frac{\partial u}{\partial t} - \sum_{i,j=1}^n a_{ij} (x, u) \frac{\partial^2 u}{\partial x_i \partial x_j} + f (x, u, D u) = 0 \quad \mbox{in } {\mathbb R}_+^{n+1}, $$ where ${\mathbb R}_+^{n+1} =…

In this paper, we consider the following indefinite fully fractional heat equation involving the master operator \begin{equation} (\partial_t -\Delta)^{s} u(x,t) = x_1u^p(x,t)\ \ \mbox{in}\ \R^n\times\R , \end{equation} where $s\in(0,1)$,…

This paper deals with nonlinear singular partial differential equations of the form $t \partial u/\partial t=F(t,x,u,\partial u/\partial x)$ with independent variables $(t,x) \in \mathbb{R} \times \mathbb{C}$, where $F(t,x,u,v)$ is a…

In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$ (-\Delta)^s u= h(x_N)f(u) \quad \hbox{in }\mathbb{R}^{N} $$ where $(-\Delta)^s$ stands for the fractional Laplacian with $s\in (0,1)$, and the…

We address the quantitative uniqueness properties of the solutions of the parabolic equation $ \partial_t u - \Delta u = w_j (x,t) \partial_j u + v(x,t) u $ where $v$ and $w$ are bounded. We prove that for solutions $u$, the order of…

When $P$ is the fractional Laplacian $(-\Delta )^a$, $0<a<1$, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set $\Omega \subset{\Bbb R}^n$:…

We consider the equation $(\partial_t + \rho(\sqrt{-\Delta}))f(t,x) = \mathbf 1_\omega u(t,x)$, $x\in \mathbb R$ or $\mathbb T$. We prove it is not null-controllable if $\rho$ is analytic on a conic neighborhood of $\mathbb R_+$ and…

We examine the short and long-time behaviors of time-fractional diffusion equations with variable space-dependent order. More precisely, we describe the time-evolution of the solution to these equations as the time parameter goes either to…