Questions tagged [ca.classical-analysis-and-odes]
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
20 questions from the last 30 days
0
votes
0
answers
15
views
Linear and non-linear intersection to solve ODE
Consider a linear operator $$L(u(t)) = \dfrac{d}{dt}u(t)+p(t)u(t)$$ for known function $p(t)$. It is well known the homogeneous equation $$L(u) = 0 ~~\text{or}~~\dfrac{d}{dt}u(t)+p(t)u(t)= 0$$ has ...
1
vote
0
answers
30
views
Explicit rate of decay of the positive standing wave of the subcritical nonlinear Schrödinger equation
Consider the following semilinear problem:
$$
\begin{cases}
- \Delta u + u = u |u|^{p - 2}
&\text{in} ~ \mathbb{R}^N;
\\
u (x) \to 0 &\text{as} ~ |x| \to \infty,
\end{cases}
$$
where $N \geq 2$...
- 144
-1
votes
0
answers
114
views
Stability of flow map
$\DeclareMathOperator\Diff{Diff}$Setting:
Let $(M,g)$ be a compact and connected $C^{\infty}$-Riemannian manifold. Let $d_g$ denote the induced shorted path metric and equip $C^{\infty}(M)$ with the ...
- 5,405
0
votes
1
answer
104
views
If $u \in H^2(\mathbb{R}^3)$, does $r^{-1} u \in H^{\alpha}(\mathbb{R}^3)$ for some $\alpha > 0$?
Let $u$ belong to the Sobolev space $H^1(\mathbb{R}^3)$. We have the classical Hardy inequality
\begin{equation*}
\int_{\mathbb{R}^3} \frac{|u|^2}{|x|^2} dx \le 4\int_{\mathbb{R}^3} |\nabla u(x)|^2 dx,...
- 481
0
votes
0
answers
34
views
Existence and uniqueness of heteroclinic solution of Allen–Cahn on $\mathbb R$ with driving-damping term
The Allen–Cahn equations on $\mathbb R$ are $u'' = u^3 - u$. It is well-known that all the solutions of this equation which satisfy the asymptotic boundary conditions $\lim_{x \to \pm \infty} u\left(x\...
- 395
2
votes
0
answers
70
views
+50
When is a first-order delay differential equation equivalent to a higher-order ordinary differential equation?
The proportional delay differential equation
$$
xf'(x)+2xf'(x/2)+C+4f(x/2)-5f(x)=0
$$
with initial condition $f(0)=C$ expresses that Simpson's rule exactly integrates $f$ over any interval $[0,x]$ and ...
- 3,065
0
votes
0
answers
71
views
Fourier decay implies what kind of regularity
We consider a function $f:\mathbb R^2 \to \mathbb C$ that is compactly supported and bounded. In addition, we know that
$$\lim_{\vert x\vert \to \infty} \vert x \vert^2 \vert \hat{f}(x)\vert =0,$$
...
0
votes
1
answer
74
views
Singularities at the circle of convergence: generalization of Cauchy-Hadamard theorem
Consider a series $\sum a_n z^n$ with finite radius of convergence $R$. Cauchy-Hadamard theorem gives $1/R = lim\ sup |a_n|^{1/n}$.
Q: Suppose for some reason (e.g. numerical) we know that there is ...
- 593
0
votes
1
answer
115
views
Fourier transform of exponential over torus
I found the following formula for the Fourier transform on a flat 2-torus, but I don't quite know how to derive it. We have a variable $q=(q_x,q_y) \in [0,2\pi)^2$ and by considering it in polar ...
7
votes
1
answer
160
views
When is a non-linear first-order ODE equivalent to a linear second-order ODE?
The Riccati equation $y'(x)+y(x)^2=f(x)$ is non-linear, but can be transformed into the linear equation $-u''(x)+f(x)u(x)=0$ via $y(x)=\frac{u'(x)}{u(x)}$.
Is there a general statement known about ...
- 3,065
0
votes
1
answer
77
views
When is Laplace transform of a function power-law and relation to the behavior of the function near zero?
I want to see when the Laplace transform of a non-negative function $f$ defined on $[0, +\infty)$ is a power function in the loose sense, i.e.,
$$g(s) = \mathcal L\{f\}(x) = \int_0^\infty f(x) e^{-sx} ...
- 103
8
votes
1
answer
436
views
Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 3,065
1
vote
0
answers
100
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Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$
$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$.
I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
- 178
7
votes
1
answer
553
views
Example of continuous function which is not differentiable everywhere in a strong sense
Is there a continuous function
$$u\colon (0,1)\to \mathbb{R}$$
such that at every point $x\in (0,1)$ one has
$$\lim\sup_{y\to x+0}\frac{u(y)-u(x)}{y-x}=+\infty?$$
In particular $u$ is not ...
- 21.8k
0
votes
0
answers
96
views
Derivative bounds for self convolution of the spherical measure in $R^d$
While reading this article on near $L^1$ estimates for the spherical lacunary maximal function, I came across the estimate
$$ |\partial^{\gamma} (\widetilde{\sigma} \ast \sigma)(x)| \lesssim |x|^{-(1 +...
- 1
0
votes
1
answer
71
views
Upper bound on higher order derivatives of $\frac{1}{v(t)}$
Suppose that $ v(t) >l>0$ and
$$
\vert v^{(k)}(t) \vert \leq c \frac{k!}{r^k}.
$$
Can we give an upper bound for
$$
(\frac{1}{v(t)})^{(k)}
$$
?
Attempt:
We first compute the first fourth order ...
- 269
5
votes
0
answers
204
views
A proof for an $L^p$-$L^p$ inequality
This is a transfer of the question
https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality
Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
- 852
3
votes
0
answers
95
views
Deeper reason for why classical orthogonal polynomials have simple generating functions?
Is there a known reason why all classical families of orthogonal polynomials have simple generating functions? I was wondering whether one could get an explanation using the connection with Sturm-...
- 312
0
votes
2
answers
148
views
Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines
I would like to understand the asymptotic behaviour of the following integrals with fixed $x_0>0$:
$$J_m=\int^{+\infty}_{x_0}|H_m(x)|^2 e^{-x^2}dx,$$
where $H_m(x)$ is the $m-$th Hermite polynomial....
- 21
4
votes
4
answers
473
views
A certain inequality involving square roots of polynomials
I want to prove the inequality
$$\begin{aligned}
&\sqrt{(x - 1)^2 + y^2}\Big[y^2(9x - 6) - 9x^2 + 9x^3\Big]+ y^2(16x^2 - 16x + 7)\\
&- \sqrt{x^2 + y^2}\Big[9x + y^2(9x - 3) + \sqrt{(x - 1)^2 + ...
- 1,109