Related papers: Decay at infinity of caloric functions within char…

For $s \in [1/2, 1)$, let $u$ solve $(\partial_t - \Delta)^s u = Vu$ in $\mathbb R^{n} \times [-T, 0]$ for some $T>0$ where $||V||_{ C^2(\mathbb R^n \times [-T, 0])} < \infty$. We show that if for some $0< c< T$ and $\epsilon>0$…

In this paper we prove that if $u$ is a solution to second order hyperbolic equation $\partial^2_tu+a(x)\partial_tu-(div_x\left(A(x)\nabla_x u\right)+b(x)\cdot\nabla_x u+c(x)u)=0$ and $u$ is flat on a segment $\{x_0\}\times (-T,T)$ then $u$…

In this short note we prove that if $u$ solves $(\partial_t - \Delta)^s u = Vu$ in $\mathbb R^n_x \times \mathbb R_t$, and vanishes to infinite order at a point $(x_0, t_0)$, then $u \equiv 0$ in $\mathbb R^n_x \times \mathbb R_t$. This…

Let $u\left( t,y\right) $ be a polyharmonic function of order $N$ defined on the strip $\left( a,b\right) \times\mathbb{R}^{d}$ satisfying the growth condition $$ \sup_{t\in K}\left\vert u\left( t,y\right) \right\vert \leq o\left(…

We construct functions $u: \mathbb{R}^2 \to \mathbb{C}$ that satisfy an elliptic eigenvalue equation of the form $-\Delta u + W \cdot \nabla u + V u = \lambda u$, where $\lambda \in \mathbb{C}$, and $V$ and $W$ satisfy $|V(x)| \lesssim…

We obtain a vanishing result for solutions of the inequality $|\Delta u|\le q_1|u|+q_2|\nabla u|$ that decay to zero along a very general warped cylindrical end of a Riemannian manifold. The appropriate decay condition at infinity on $u$ is…

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{equation} \begin{cases} u_{t} =\Delta u - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v =…

It is shown that a function $u$ satisfying $|\partial_tu+\sum_{i,j}\partial_i(a^{ij}\partial_ju)|\leq N(|u|+|\nabla u|)$, $|u(x,t)|\leq Ne^{N|x|^2}$ in $\mathbb{R}^n_+\times[0,T]$ and $u(x,0)=0$ in $\mathbb{R}^n_+$ under certain conditions…

We consider an elliptic differential inequality: $\vert \Delta u(x) \vert \le C_0(\YYYY^{-\gamma}\vert u(x)\vert + \YYYY^{-\theta}\vert \nabla u(x)\vert)$ in an exterior domain $\R^n \setminus \ooo{U}$, where $U$ is a simply connected…

In this paper we obtain quantitative bounds on the maximal order of vanishing for solutions to $(\partial_t - \Delta)^s u =Vu$ for $s\in [1/2, 1)$ via new Carleman estimates. Our main result Theorem 1.1 and Theorem 1.3 can be thought of as…

Let $0\le u_0(x)\in L^1(\R^2)\cap L^{\infty}(\R^2)$ be such that $u_0(x) =u_0(|x|)$ for all $|x|\ge r_1$ and is monotone decreasing for all $|x|\ge r_1$ for some constant $r_1>0$ and ${ess}\inf_{\2{B}_{r_1}(0)}u_0\ge{ess}…

We construct a real-valued solution to the eigenvalue problem $-\text{div}(A\nabla u)=\lambda u$, $\lambda>0,$ in the cylinder $\mathbb{T}^2\times \mathbb{R}$ with a real, uniformly elliptic, and uniformly $C^1$ matrix $A$ such that…

This paper deals with the equation $-\Delta u+\mu u=f$ on high-dimensional spaces $\mathbb{R}^m$ where $\mu$ is a positive constant. If the right-hand side $f$ is a rapidly converging series of separable functions, the solution $u$ can be…

Let $M = M_0 \times \R^2$ be a pp--wave type spacetime endowed with the metric $<\cdot,\cdot>_z = <\cdot,\cdot>_x + 2 du dv + H(x,u) du^2$, where $(M_0, <\cdot,\cdot>_x) $ is any Riemannian manifold and $H(x,u)$ an arbitrary function. We…

We study the decay (at infinity) of extremals of Morrey's inequality in $\mathbb{R}^n$. These are functions satisfying $$ \displaystyle \sup_{x\neq y}\frac{|u(x)-u(y)|}{|x-y|^{1-\frac{n}{p}}}= C(p,n)\|\nabla u\|_{L^p(\mathbb{R}^n)} , $$…

We prove a quantified Tauberian theorem for functions under a new kind of Tauberian condition. In this condition we assume in particular that the Laplace transform of the considered function extends to a domain to the left of the imaginary…

We characterize the set of functions $u\_0\in L^2(R^n)$ such that the solution of the problem $u\_t=\mathcal{L}u$ in $R^n\times(0,\infty)$ starting from $u\_0$ satisfy upper and lower bounds of the form $c(1+t)^{-\gamma}\le \|u(t)\|\_2\le…

In this paper, we prove Huang et al.'s conjecture stated that if $f$ is a holomorphic function on $\Delta^+:=\{z\in \mathbb C \colon |z|<1,~\mathrm{Im}(z)>0\}$ with $\mathcal{C}^\infty$-smooth extension up to $(-1,1)$ such that $f$ maps…

We study the existence of separable infinite harmonic functions in any cone of R N vanishing on its boundary under the form u(r, $\sigma$) = r --$\beta$ $\omega$($\sigma$). We prove that such solutions exist, the spherical part $\omega$…

Let $n\geq 3$, $0< m<\frac{n-2}{n}$ and $T>0$. We construct positive solutions to the fast diffusion equation $u_t=\Delta u^m$ in $\mathbb{R}^n\times(0,T)$, which vanish at time $T$. By introducing a scaling parameter $\beta$ inspired by…