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Let $z_1,\ldots,z_n$ be $n\ge 1$ distinct points of $\mathbb R^2$. Define the potential function $U: \mathbb R^2 \to\mathbb R$ by $$U(x):=\sum_{1\le i\le n} \log(|x-z_i|),$$ where $|\cdot|$ denotes th …

Consider the Schrödinger equation on $\mathbb R_+\times\mathbb R^n$ as follows: $$i\partial_t\varphi(t,x)=-\frac12\Delta_x\varphi(t,x),\quad \varphi(0,x)=\varphi_0(x).$$ Denote by $\varphi$ its soluti …

Let $X$ be the solution to (real-valued) stochastic differential equation : $$dX_t = b(t,X_t)dt + a(t,X_t)dW_t, \quad \forall t\ge 0.$$ Let $\Delta t>0$ be given. Under suitable conditions (on $b,a, X …

Following the previous post Asymptotic behaviour of the solution to some delayed stochastic differential equation I consider a deterministic version (as no answer is received) : $$\frac{d x^\theta}{d …

Consider the one-dimensional logarithmic diffusion equation for $u: \mathbb R_+ \times [0,1]\ni (t,x)\to u(t,x)\in \mathbb R$ : $$(\ast)\quad\quad \begin{cases} 2u_t = \big(\log(u)\big)_{xx} & \text …

For $i=1,\ldots, n$, let $c_i:\mathbb R\to [0,1]$ be continuous, $a_i, b_i>0$ and $\mu_i \ge 0$. Consider the following Volterra equation: $$f(t)=\sum_{i=1}^n c_i(t)\left[\mu_i + \int_{-\infty}^t a_ie …

Following the hints of Christian Remling, my claim is not true. Take $n=2$ for example. A straightforward computation implies that the only equilibrium point $x^*$, i.e. $F(x^*)=0$, is given as $$x^*= …

Let $M$ be the set of continuous and increasing functions $h: [0,1)\to\mathbb R_+$ s.t. $h(0)=0$ and $h(1-)=+\infty$. Consider the equation as follows: $$ 1-t= \int\limits_0^\infty p(x)\Phi\left(\frac …