For questions on nonlinear analysis, a branch of mathematical analysis that deals with nonlinear mappings.
Questions tagged [nonlinear-analysis]
500 questions
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Noether's theorem in the critical heat equation
I am watching a serie of lectures on "Blow up solution for the energy critical heat equation" from Monica Musso on YouTube and at some point she states a result I do not understand. Let met recall the setting.
We are studying the following…
Falcon
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votes
3 answers
Functional equation $f(px)+p=[f(x)]^2$
Let $p\in\mathbb{N}$ and $p>1$. Consider the functional equation
$$f(px)+p=[f(x)]^2$$
I need to find all functions $f:\mathbb{R}\to\mathbb{R}$ that is continuous at $0$ and satisfies above functional equation for all $x\in\mathbb{R}$.
For $p=2$, it…
Sahan Manodya
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Statements on the behavior of solutions to $y' = \sin(xy)$ for large $y(0)$
Consider the following initial-value problem involving a nonlinear first-order ODE:
$y' = \sin(xy), \quad y(0) = y_0$.
For large enough values of $y_0$, the solutions to this equation appear to converge onto a specific shape under the appropriate…
aghostinthefigures
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Is $y’=x^y\implies xy’’-x\ln(x)y’^2-yy’=0,y’=y^x\implies xy'(x)^4-y(x)y'(x)^2 \left(xy''(x)+2y'(x)\right)+y(x)^2\left(y'''(x)y'(x)-y''(x)^2\right)=0$?
Here is a partial inspiration for this question:
ODE: $$y''y+ax+by+c=0,y=k\pm\sqrt2\int\sqrt{a\int\ln(y)dx-(ax+c)\,\ln(y)-by+K}dx,\int\frac{dy}{\sqrt{K-(ax+c)\,\ln(y)+a\int\ln(y)dx-by}}=k\pm\sqrt2x$$
and
Is there an algebraic solution for the…
Tyma Gaidash
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What is the most general Carathéodory-type global existence theorem?
I am looking for a general theorem that guarantees the existence of a global solution for an ODE system in $\mathbb{R}^n$
$$
\begin{equation}
\left\{ \begin{aligned}
x'(t) &= f(t, x(t)), \qquad t \in [a,b] \\
x(a) &= x_0
\end{aligned}…
A. Pesare
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Nonlinear funtionals of smooth maps between Riemannian manifolds
Let two smooth Riemannian manifolds $M$ and $N$, and let $C^{\infty}(M, N)$ be the family of smooth maps between them.
I would like to study functionals of the form $$E[\cdot]: C^{\infty}(M, N) \to \mathbb{R},$$ where for each $F \in…
them
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2 answers
A Learning Roadmap to the "foundations" of Nonlinear Analysis (and certain specific topics)
I'm searching for throughout references that -- in the long term -- can help me gradually gain a solid background and firm foundations to understand the main methods and theorems to deal with nonlinear problems (in particular, wave equations,…
Dal
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Can this non-linear boundary-value problem be solved analytically?
I am trying to solve
$$
[y^2(x)]''+\frac{x}{2} y'(x) = 0
$$
on $x\in\mathbb{R}$, with the conditions
$$
\lim_{x\rightarrow-\infty} y(x) = a,\qquad
\lim_{x\rightarrow\infty} y(x) = b.
$$
I am particularly interested in non-negative solutions,…
ArrhythmicTiling
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votes
1 answer
Maximum of $ F(f)=\int_0^1 |f(x)|^2\; dx-\left(\int_0^1 f(x)\; dx\right)^2 $ over a subset of continuous functions on $[0,1]$
Let $X$ be a subset of $C([0,1])$ with
$$
X=\big\{f\in C([0,1]): 0\le f(x)\le x,\text{$f$ is a polynomial}\big\}
$$
where $C([0,1])$ denotes the space of continuous real-valued functions on $[0,1]$. Define a nonlinear functional $F$ on $C([0,1])$…
user1010411
7
votes
1 answer
Surveys: problems, conjectures, and questions in some areas of nonlinear analysis
I would like to create a "big-list" of resources (e.g., survey papers, webpages, conference proceedings, monographs, etc.) that collect and offer some context and overviews of:
open problems, conjectures, and questions that have recently arisen in…
Dal
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1 answer
Existence and uniqueness of $-\Delta u + u^3 =f$
I want to show the existence and uniqueness of the equation of the form
$-\Delta u+u^3=f\in L^2(\Omega),\quad u=0~\mbox{on}~ \partial\Omega,~$
$\Omega\subset\mathbb{R}^3$-bounded
I know one proof using Minty-Browder theorem and by direct method of…
whereabouts
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Are limit cycles the same as periodic solutions?
I'm just having a small trouble with the nomenclature. So basically we spent a lot of time in class discussing limit cycles in the phase plane. However it never really became 100% clear to me whether all periodic solutions of a nonlinear ODE…
Jim
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votes
2 answers
Contractive Operators on Compact Spaces
Suppose that $T: M \to M$ is a compact contractive Operator on a nonempty compact subset $M$ of a complete metric space $X$. Show that $T$ has a unique fixed point. Further show that the sequence defined by $x_{n+1}=Tx_n$ converges to the fixed…
Aarna
- 382
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5
votes
3 answers
Solving equations with log
Is there any easy way (no numerical methods, just a straightforward way to do it by hand) to solve an equation like:
\begin{equation}\frac{-34450}{x} - 2.01\ln(x) + 33.74 = 0\end{equation}
It is a step in a bigger problem that I'm trying to solve…
c.leblanc
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5
votes
2 answers
Convexity of infimum function
This question asks about the convexity of the function
$$ g(x)=\inf_{y\in\Re^n} f(x,y) $$
where $f \colon \Re^n \times \Re^m \to \Re$ is convex in $(x,y)$. I would ask a more advanced case that if the set given in the inner minimization problem…
Keith
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