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Results tagged with ap.analysis-of-pdes
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user 61629
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
12
votes
4
answers
2k
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 …
- 1,759
7
votes
1
answer
555
views
Sobolev spaces are smooth? Their dual is strictly convex?
Do you know any reference which says something about the:
Smoothness of the Sobolev space $W^{1,p}(\Omega)$ i.e. if the duality mapping $J\colon W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ is a singleton.
…
- 1,759
6
votes
1
answer
216
views
Question about Bochner measurability
When I study parabolic pde's I often came across the following type of Bochner spaces $L^p([a,b];L^{q}(\Omega),\ W^{1,p}([a,b];L^{q}(\Omega))$ and $L^{q}([a,b];W^{1,p}(\Omega))$ where $p,q\geq 1$ and …
- 1,759
4
votes
1
answer
255
views
Reaction-diffusion systems treated as dynamical systems
I wonder if there is a good reference on reaction-diffusion systems on $\mathbb{R}^N$, that treats them as dynamical systems.
I have the books of Alain Haraux – Systèmes dynamiques dissipatifs et appl …
- 1,759
4
votes
0
answers
706
views
Maximum Principles in Parabolic PDE with Neumann Condition
I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in D …
- 1,759
4
votes
0
answers
106
views
The logistic elliptic equation
Studying the Fisher-KPP evolution equation I came across the steady state elliptic problem which can be written in the following form:
$$
\begin{cases} -d\Delta Y(x)=r(x)Y(x)\left (1-\dfrac{Y(x)}{K(x) …
- 1,759
4
votes
1
answer
360
views
Strong positivity of Neumann Laplacian
There are many places in the literature where the positivity of some semigroups is treated. However I did not know anyone which states and proves the strong positivity even for the basic semigroups li …
- 1,759
4
votes
1
answer
237
views
When is $W^{1,p}(\Omega)$ a Banach algebra?
Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$.
My question: knowing that $f,g\in W^{1,p}(\Omega)$ for what $p$ can we conclude that $f\cdot g\in …
- 1,759
4
votes
3
answers
271
views
Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$
Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open and c …
- 1,759
4
votes
1
answer
450
views
Contractivity of Neumann Laplacian
I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on semigroups of linear operators, I found on many places properties of the Neumann Laplacian.
In …
- 1,759
3
votes
1
answer
383
views
Neumann/Robin Laplacian semigroup well-known estimate
Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:
$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \pa …
- 1,759
3
votes
0
answers
101
views
A special type of differential equations
Working in optimal control of PDEs, I came across a type of evolution problem that has instead of an initial condition a link between the initial state and the final state.
Here is a simplified versio …
- 1,759
3
votes
1
answer
278
views
Are Neumann Laplacian eigenfunctions in $C(\overline{\Omega})$?
Consider that $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ (in the distributional sense) such that for some $\lambda>0$ we have that:
$$\begin{cases} \Delta u(x)=\lambda u(x), & x\in\Omega\\ \dfr …
- 1,759
3
votes
1
answer
253
views
$L^{\infty}$ estimate for heat equation with $L^2$ initial data
Is there a way to show in a relatively simple manner that for a Lipschitz bounded, connected, open domain $\Omega\subset\mathbb{R}^N$ for any $f\in L^2(\Omega)$ the solution of the problem:
$$\begin{c …
- 1,759
3
votes
1
answer
163
views
Question about Lebesgue Bochner spaces
Let $T>0$ and $\Omega\subset\mathbb{R}^N$ be a bounded domain. Also $p\in (1,\infty)$ is any number.
I know that $u\in L^{p}((0,T);L^p(\Omega))$ and $\nabla u\in L^{p}((0,T);L^p(\Omega))^N$. How can I …
- 1,759