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Let $\mathcal M$ be a smooth manifold. A linear PDE on $\mathcal M$ is a sum of terms $$ X_1\dots X_N u,\quad\text{where the $X_j$ are smooth vector fields.} $$ We may use the convention that if $N=0$ …

A most important result is missing in the previous answers, namely the characterization by Lars Hörmander of hypoellipticity in his seminal paper, On the theory of general partial differential operato …

Let $Ai$ be the classical Airy function and let $(a_j)_{j\ge 1}$ be the strictly decreasing sequence of its zeroes: we have $a_{j+1}<a_j<\dots <a_2<a_1<0$, $\lim_{j\rightarrow +\infty}a_j=-\infty$. I …

(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE: $$ \dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N. $$ Instea …

I want to consider the solutions of the following fourth-order ODE: $$ f^{(4)}(t)+a tf^{(1)}(t)+b f(t)=0, \tag{$\ast$}$$ where $a,b$ are complex parameters. It turns out that with a Fourier transforma …

I understand that your ODE is one-dimensional: in that case you can avoid Lipschitz continuity and replace it by transversality: The autonomous equation $$ \dot x =f(x),\quad x(0)=x_0, $$ has a uniqu …

The operator $\frac{\partial}{\partial t}+t\Delta_x$ is not locally solvable near $t=0$. It could be seen as a quasi-homogeneous version of Mizohata operator $D_t+it\vert D_x\vert$ since its symbol is …

Consider the (stationary) Schrödinger equation $$ -\Delta u+ Vu=0, $$ or the differential inequality $\vert\Delta u \vert\le \vert V u\vert$, where $V$ is some "potential" function. The following uniq …

No. Take for instance the following ODE where $0$ is a regular singular point, $$x y'=\lambda y.$$ The functions $cx_+^{\lambda}$ are solutions and are not analytic. If $\lambda$ is not a non-negative …

Forget about finding a closed analytical expression for the fundamental matrix. Think about the simple case of a second order scalar equation such as the Airy equation $\ddot x=tx.$ There is neverthe …

Denis Serre's answer is just perfect. Let me add a couple of examples of distributions that can be squared: With $H$ the Heaviside function, define $\operatorname{Log}(x+i0)=\ln(\vert x\vert)+i\pi H( …

Let $P$ be a pseudodifferential operator with symbol $p(x,\xi)$ belonging to $S^m_{1,0}$ and let $a(x)$ be a $C^\infty$ function. Then the operator $Pa$ defined by $Pa u=P(au)$ is a pseudodifferential …

Let us work in $\mathbb R^n$. The distribution solutions of the equation $$ (x\cdot \partial _x) u=0 $$ are the distributions which are homogeneous of degree 0. Here $x\cdot \partial_x=\sum_{1\le j\le …

Analytic regularity of a $C^\infty$ function can be characterized by using an infinite number of derivatives. A function $f\in C^\infty(\Omega)$ where $\Omega$ is an open subset of $\mathbb R^n$ is r …

Let $E$ be a fondamental solution of $\partial_{x_1}\square$. Then you have for $u$ compactly supported $$ u=u\ast \delta=u\ast (\partial_{x_1}\square E)= (\partial_{x_1}\square u)\ast E, $$ so that $ …