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Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

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42 views

Property of Fixed Point Function

Given an operator $\mathcal{T}$ that maps from a function $f: \mathbb{R}^d\rightarrow \mathbb{R}$ to another function $f': \mathbb{R}^d\rightarrow \mathbb{R}$, we are interested in the fixed point $f^*...
5
votes
1answer
104 views

Finding an asymptotic solution for a first order ODE

Given strictly concave function $f(t)$ that satisfies $f'(t)>0$, $f'(t)=o(1)$ (i.e. $\lim\limits_{t\to\infty}f'(t)=0$) , and $f'(t)=\omega\left(t^{-1}\right)$ (i.e. $\lim\limits_{t\to\infty}tf'(t)=\...
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0answers
20 views

Asymptotic Constancy of solutions of delay/integro differential equations

I have found quite a few papers on asymptotic constancy of solutions of delay differential equations and integral differential equations (see e.g. this reference or this reference). I am however most ...
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0answers
53 views

Solve simple stochastic variational inequality

Let $Z$ be a random variable with finite mean $\mathbb E[Z]$ and let $\phi:(-\infty,\infty) \rightarrow (-\infty,\infty]$ be a convex l.s.c function which is differentiable at $z=1$ with gradient $\...
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3answers
159 views

A Specific Linear Homogeneous System of Differential Equations with Variable Coefficients

Is there an analytical solution satisfying these 3 equations with non-constant z? $$\frac{dx}{dt}=-z\cdot\cos(\omega t)$$ $$\frac{dy}{dt}=z\cdot\sin(\omega t)$$ $$\frac{dz}{dt}=x\cdot\cos(\omega t) - ...
2
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1answer
137 views

Counter example about blow-up solution of DEs

Let $f(\cdot)$ be a continuously differentiable function over $\mathbb{R}$, and $u\in L^2_{loc}(0,\infty)$, $a\in \mathbb{R}$, and $x(t)$ solves the integral of $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...
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0answers
53 views

Slow and fast forming singularities of the mean curvature flow

Let $M \times [0, T) \to \mathbb{R}^{n+1}$ be a mean curvature flow and let $T$ be a singular time. Let $A$ denote the second fundamental form. We have a type I singularity if $$ \max_{p \in M} |A(p,...
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0answers
252 views

Spectral Gap of Elliptic Operator

Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled? The boundary condition is that the ...
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0answers
49 views

Matrix exponential with particular structure

Context I'm trying to numerically solve the following differential equation: $\frac{\mathrm{d} u}{\mathrm{d} t} = -Au + f$, where $u$ and $f$ are vectors, and $A$ is an $N \times N$ matrix, with $N &...
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2answers
188 views

Non-linear Basis for PDE's

Asked this on stack exchange and got no response, so I'll try here. An idea popped into my head awhile ago while doing a project on non-linear effects of systems of coupled oscillators, but I'm not ...
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1answer
51 views

Does differentiating an integro-differential equation results in equivalent stability of the solution?

I have a dynamical system in the form of an integro-differential equation which I want to analyze in terms of stability. To demonstrate my problem consider the following integro-differential equation: ...
4
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0answers
96 views

An embedding question: Morrey spaces

Question. If $u\in L^1$ and $Du$ is in the dual of the Holder space $C^\alpha$, then is it possible to say $u$ belongs to some Morrey space $L^{1, \delta}$?
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0answers
65 views

Understanding the proof that $\Delta u = f(u)$ has a unique critical point on a convex domain

I am struggling to understand step 2 of the proof of Theorem 1 in [1]. It seems that the proof that the critical point is unique relies only on the fact that the nodal curves $$N_\theta = \{x\in\Omega\...
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0answers
46 views

Degeneracy of critical points of solutions to second order elliptic PDEs

Problem: Suppose that $Lu = f(x)$ on a strictly convex domain $\Omega\subset\mathbb{R}^2$ and $u=0$ on $\partial\Omega$ where $L$ is elliptic and second order. I am trying to determine when the ...
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1answer
337 views

Updated background on Hilbert 16th problem?

What is the current situation of the second part of the Hilbert 16th problem? What are the most updated news on this problem?
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1answer
54 views

How can I solve the first ordinary differential equation? [closed]

$\ $ $y'(x)=A(X)B(x)y(x)-B(x)y(x)^2$ where $ A(x)=a(1+(d*x))$, $ B(x)=b/(b+(c*exp(-b*x)))$ What is a solution of $y(x)$?
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0answers
17 views

Are solutions to the second order elliptic PDE $Lu=f(u,\nabla u, x)$ locally $L$-harmonic homogeneous polynomials?

These two papers by Caffarelli and Friedman [1], [2] show that a solution $u$ to the elliptic PDE $$ \Delta u = f(u,\,\nabla u,\, x)\,,\quad \text{ on } \Omega\subset\mathbb{R}^n $$ is locally a ...
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0answers
18 views

Rankine Hugoniot Condition for non piecewise smooth solution

I studied the following theorem: (Rankine-Hugoniot condition) Let $u:\mathbb{R} \times [0,+\infty) \to \mathbb{R}$ be a piecewise $C^1$ function. Then $u$ is a weak solution of the conservation law ...
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0answers
26 views

Algebraic connection of the quadratic Lyapunov function existence with the Hurwitz criterion for third order linear system

In what publication the following theorem is proved? Let $a$,$b$, $c$ be real numbers and we choose a Lyapunov function for the system: $$ \left\{ \begin{array}{cr} \dot{x} = y \\ \dot{y} = z \...
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1answer
238 views

Asymptotic solution for a first order ODE

Simplified question*: Given $f(t)$ that satisfies $f'(t)>0$, $f'(t)=\omega\left(t^{-1}\right)$, $\log\left(f'(t)\right)=o\left(f(t)\right)$ we denote $F=\exp\left(f\left(t\right)\right)$. Let $H(...
3
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1answer
131 views

Exponential map/ Lie derivative in variation for constant formula for ODE

In short: The question is how to go from the first equation on page 8, of this paper to the second equation. Some background I'm working in optimization and I am currently reading a paper see page ...
2
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2answers
119 views

Criteria for Schrödinger operator on real line to have simple spectrum

Consider a Schrödinger operator $H:=-\Delta+V$ on $\mathbb R$, where $V$ is such that $H$ has a purely discrete spectrum $-\infty<\lambda_1\leq\lambda_2\leq\cdots$ converging to $+\infty$. Do there ...
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0answers
34 views

First eigenvalue for domains in hyperbolic space

I am interested in examples of bounded open subsets of the hyperbolic space, for which the first eigenvalue of the Dirichlet Laplace operator (acting on functions) is known. In Euclidean space several ...
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0answers
58 views

Limits of a simple damped system

Definition: Let $F_n(s) = \frac{1}{s^{n+1}(1+s)^n}$ be the Laplace transform of $f_n(t)$. Required Result: To show $\lim_{n\rightarrow\infty}f_n(n+n/e) < o(n)$. Ideas: Let $G_n(s)=\frac{1}{s^{n+...
12
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4answers
685 views

History of ODE and PDE reference request

Is there any reference (book or articles) which made the history (up to the modern times) and the conceptual development of Ordinary Differential Equations and Partial Differential Equations? It will ...
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0answers
104 views

Analytical solve for this 4th order Linear PDE

I want to Solve analyticaly or numericaly a biharmonic equation with homogeneous boundary conditions homogeneous in rectangular domain [0,a]*[0,b]? I considered its solution using Fourier analysis (...
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0answers
100 views

Locally Lipschitz sufficiently implies a Gronwall inequality?

In the paper [1], it seems to me the authors implicitly use a local Lipschitz property to deduce a Gronwall's inequality. I am not able to justify/show that this is indeed the case and perhaps someone ...
2
votes
4answers
201 views

Complex differential equations

I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs. Mostly, I'm just ...
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1answer
70 views

Solve nonlinear equation

Suppose that $f:E\to F$(between Banach spaces), is of the form $$f(x)=f(0)+D(x)+N(x).$$ Here $D$ is a linear term, whose kernel is of finite dimension, and admits a right inverse $G$, i.e. $D(G)(\...
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0answers
44 views

Stability of an equilibrium for a third order control system with an integral regulator limited by the stop-type element

In what publication is the following theorem prove carried out? The method of the Lyapunov functions is preferable. Thanks. Consider the system consisting of the controlled object and regulator. The ...
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1answer
28 views

Does stability of equilibrium point preserved by permutation matrix (symmetry)?

Given the following differential equations: \begin{equation} \begin{aligned} \dot{x}_1 &= f_1(x_1,\ldots,x_n) \\ \vdots \\ \dot{x}_n &= f_n(x_1,\ldots,x_n) \end{aligned} \end{equation} In ...
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1answer
43 views

Galerkin Finite element for solving third order time dependent partial differential equation inti weak form

How to solve the third order time dependent partial differential equation (i.e. u_t + 6u_x + u_xxx = 0) into weak form using galerkin finite different method?
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2answers
88 views

One inequality connected with the linear second order ODE

Is the following statement true? Let $ a>0, b>0, h>0 $, $x(t)$ be the solution of the differential equation $ \ddot{x}+a \dot{x}+bx=h$ with initial conditions $x(0)=u<0 , \dot{x}(0)...
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0answers
101 views

Algebraic independence of limit cycles of Lienard equation

It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle. According to this fact, we search for a related ...
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0answers
67 views

What is the unitary $1$-parameter group generated by a vector field on a manifold?

If $M$ is a (compact, Riemannian) manifold, and one complexifies its tangent bundle, is it possible to give a meaning to the "unitary group"-like expression $t \mapsto \exp (\mathrm i t X)$ when $X$ ...
6
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1answer
118 views

Ordinary differential operators satisfying braid relation?

Let $W$ be the algebra of linear ordinary differential operators with analytic coefficients $C^{\omega}(\mathbb{R})[\partial_x]$ (with multiplication given by composition). Do there exist two elements ...
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0answers
25 views

Mean of a periodic velocity field and trajectory displacement bound

Suppose $u(t,x)$ is a smooth velocity field on $[0,\infty)\times \mathbb{R}$ and periodic in space, i.e., $u(t,0)=u(t,1)$ $\forall t$. Assume that $\int_0^1 u(t,x) \,dx = c$, independent of time. Let $...
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1answer
2k views

Differential equation changing sign almost everywhere

Conjecture: Let $f:\mathbb{R}→\mathbb{R}$ be an everywhere differentiable function and assume that $f(x)+f′(x)∈ \{-1,1\}$ almost everywhere and $f'(0)=0$. Then is $f$ necessarily a constant function? ...
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1answer
112 views

Spectral decomposition of a specific operator

To understand a crucial example in representation theory, I need the explicit spectral decomposition of the differential operator $$ Df(x)=(1+x^2)f''(x)+2xf'(x) $$ on $L^2({\mathbb R})$. I'm not an ...
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0answers
256 views

Is the $\hat{A}$-genus invariant under crepant birational maps between smooth algebraic varieties?

The degree of the Todd class of the tangent bundle of a variety $Y$ gives the holomorphic genus of $Y$, which is a birational invariant. We also know that the $\hat{A}$-roof of a smooth variety $Y$ ...
6
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1answer
539 views

(In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations $$ \dot{x}_1(t) = -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\ \dot{x}_2(t) = -\gamma x_2(t) - \cos(\...
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0answers
152 views

A question about whether an operator can be lipschitz or not

Let assume $ X= \Omega \times (0,T) $ where $\Omega \subset \mathbb{R} $ is a bounded open set and $T>0$. Now define the operator $ \mathcal{A} : C^{‎\sigma‎, \sigma‎/2‎}(‎X‎) \to C^{‎\sigma‎, \...
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0answers
81 views

2nd oder evolution equations and regularity results of their solution

I am interested in regularity results for solutions to 2nd order evolution equations in the shape of $$ u''(t) + A(t)u(t) = f(t) \text{ in } V^*\text{ a.e. in }S, \\ u(0) = u_0 \text{ in } H, u'(0)...
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1answer
80 views

Behaviour of solutions to $(A-r)f=0$ in the limit $r \to \infty$

Define the second order linear differential operator associated with $X$ (Here $X$ is the unique strong solution to appropriate Ito SDE) by $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \...
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1answer
166 views

Classification of a system of two second order PDEs with two dependent and two independent variables

If we have a second order quasilinear PDE of the form $A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$...
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1answer
121 views

Positive Solutions of second-order ODE

Consider second-order ordinary differential equations of the form $u''(t)=a(t)u(t)-2$ I'm interested in general criteria on the function $a(t)$, which guarantee respectively rule out the existence ...
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0answers
51 views

Well-posedness for a general linear grad-div PDE

Let $U \subset \mathbb{R}^n$ be an open bounded domain with smooth boundary ($\partial U$ is a closed manifold) and let $A(x)\in C^1(U, M_{n \times n}(\mathbb{R}))$, $b(x) \in C^0(U, \mathbb{R}^n)$ ...
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1answer
55 views

Well-posedness for equations of the form $u_t = grad[V(u)]$ and $u_{tt}=grad[V(u)]$?

Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE: $$u_t = grad[V(u)]$$ For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-...
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0answers
111 views

Conditions to the existence of periodic orbits of non vanishing vector fields on $\mathbb{T}^2$

I'm doing a research about Filippov systems on $\mathbb{S}^3$ with discontinuities on $\displaystyle\frac{1}{\sqrt{2}}\cdot\mathbb{T^2} =\left\{\displaystyle\frac{1}{\sqrt{2}}x ; \ x \in \mathbb{S}^1\...
3
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0answers
66 views

Parametrix of external product of elliptic operators

Let $S$ and $T$ be Dirac-type operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. ...