12 votes

Nash–Moser–De Giorgi differences

Here's a slightly more detailed explanation than what Willie Wong fit in the comments. I will not go over the precise statements in detail, but at least some discussion is needed to appreciate the ...
user378654's user avatar
11 votes

"Wild" solutions of the heat equation: how to graph them?

I know this is a late answer, but I think that the other answers posted so far completely fail to illustrate the wildness of the function in question, since they plot it on a domain where not much ...
Hans Lundmark's user avatar
9 votes
Accepted

$L^p$-norm under the heat flow

If I didn't do any miscalculations I believe I have proven the case $1\leq p\leq 2$. I will write $u$ instead of $u_t$. Let $(p)_k=p(p-1)\ldots(p-k+1)$ and $$ w=\begin{pmatrix} pu^{p-1}\Delta^2 u\\ (p)...
Markus Sprecher's user avatar
9 votes

Gradient $L^p$ estimates for heat equation

Let me comment on what I know in an open set $\Omega$, trying to control $C$. First of all, the heat semigroup can be expressed through a kernel $p$ which is pointwise dominated by the heat kernel in $...
Giorgio Metafune's user avatar
8 votes
Accepted

Caffarelli-Kohn-Nirenberg-type inequality with nonradial weight

I'll try to keep your notation except for three things: I prefer to have all parameters non-negative not to get confused myself, I'd rather have $z=(x,y)\in\mathbb R^2$ coordinates on the plane to ...
fedja's user avatar
  • 59.5k
7 votes
Accepted

Injectivity of a Fredholm operator

Surprisingly (to me), the statement is false. My counterexample is a little messy, but the idea is fairly simple. Take $T = 1$ and set $a_n = \frac{1}{n}$ and $b_n = 1 - \frac{1}{n}$ for $n \in \...
Nik Weaver's user avatar
7 votes
Accepted

Weak parabolic maximum principle on Riemannian manifolds

Firstly, you have the wrong inequality (there is a small typo in the paper). Young's inequality is typically written for nonnegative numbers, but for any $a,b\in\Bbb R$ we have \begin{align*} -ab&\...
Ryan Unger's user avatar
7 votes

Explicit solution of a free boundary problem for heat equation

From the maximum principle, the solution $v$ of $$v_t=v_{xx}+1,$$ together with the data $v=0$ at the boundary and initial time, is positive. Therefore your $u$ is nothing but $v$. It turns out that ...
Denis Serre's user avatar
  • 51.5k
7 votes
Accepted

Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight

Here is a functional analytic approach (Kato's book on perturbation theory is a good reference): Let $$ a\colon D(a)\times D(a)\to \mathbb{R},\,(u,v)=\int \nabla u\cdot \nabla v+\int \zeta uv. $$ with ...
MaoWao's user avatar
  • 1,027
7 votes
Accepted

Contractivity of Neumann Laplacian

It might be helpful to point out the following conceptual reasons why ultracontractivity estimates are mainly interesting for times close to $0$. Let us consider the following general setting: We have ...
Jochen Glueck's user avatar
7 votes
Accepted

Strong positivity of Neumann Laplacian

As other users have indicated in the comments, for sufficiently smooth domains one can get it by combining, for instance, elliptic regularity with Hopf's boundary point lemma (and then go from the ...
Jochen Glueck's user avatar
7 votes
Accepted

Existence of solutions to the heat equation on nonsmooth domains

$\newcommand\R{\mathbb R}\newcommand{\Om}{\Omega}$It is convenient to reverse the time by the substitution $t\leftrightarrow T-t$ and also rescale it by a factor of $2$. So, the boundary value problem ...
Iosif Pinelis's user avatar
6 votes
Accepted

$L^\infty$ estimate on heat equation with a lower order term

The estimate as stated is clearly false since $u\to u_0$ as $t\to0$ hence $\|u\|_{L^\infty}\ge \|u_0\|_{L^\infty}$. Anyway you can write the kernel explicitly and extract the information you need ...
Piero D'Ancona's user avatar
6 votes
Accepted

Properties of heat equation

The answer is no. A counterexample has the form $$u(x,t)=\sum_0^\infty \frac{g^{(n)}(t)}{((2n+1)!)^2}(r-R)^{2n+1},$$ where $r>R$ is the distance from the origin in $R^2$. First one shows that this ...
Alexandre Eremenko's user avatar
6 votes
Accepted

Angle of analyticity of semigroup

Too long for a comment. Actually I do not think that it is written anywhere but these kind of counterexamples are usually provided by the Ornstein-Uhlenbeck semigroup, generated by $\Delta+Bx \cdot \...
Giorgio Metafune's user avatar
6 votes
Accepted

Compactness for initial-to-final map for heat equation

Parabolic regularity show that $u$ is regular for all positive times; in particular $u(t,\cdot) \in W^{1,2}(M)$ for all $t > 0$. Interior parabolic estimates additionally show that there is a ...
Leo Moos's user avatar
  • 4,968
6 votes

Nontrivial invariant transformations for heat equations

Yes, how about the Appell transform, see here. $$v(t,x)\mapsto\Gamma(t,x)v(-\frac1t,\frac xt)$$ where $\Gamma$ is the heat kernel. Truly nontrivial if you ask me. Of course, this inverts time and not ...
Funktorality's user avatar
5 votes
Accepted

Growth at infinity of a solution to a parabolic PDE

What about $u(t,x) = x e^{t x^2}$, which is a solution of $$\partial_t u = \Delta u + b(t,x) \partial_x u$$ with $$b(t,x) = \frac{x^3 - 6 t x - 4 t^2 x^3}{1 + 2 t x^2}$$ with initial value $u_0(x) = x$...
Mateusz Kwaśnicki's user avatar
5 votes

Motivation behind the parabolic metric

Here is one reason, the core of which can be found at page 17 of Moser's book mentioned in the comments. Many theorems about the heat equation are valid on "cylindrical" domains of the form $(0,T) \...
Alex M.'s user avatar
  • 5,197
5 votes
Accepted

Reference for De Giorgi-Nash-Moser theory

The proof of Harnack's inequality using De Giorgi method has a great flexibility and it can be exteneded even to doubling metric measure spaces that support Poincaré inequalities. The elliptic case ...
Piotr Hajlasz's user avatar
5 votes
Accepted

Why is this test function admissible? [Paper explanation]

The reason I'm asking is because characteristic\indicator functions have no smooth derivatives and plus I don't understand in which function space of test functions the authors define the weak ...
DCM's user avatar
  • 778
5 votes

Reference request: Long-term behaviour of the heat equation for bounded initial data

First of all, with initial data in $C_b$, classical solution exist, so there is no need for quotation marks. It is easy to see that the convolution of the initial data $f(x)$ with the Gauss–...
Mateusz Kwaśnicki's user avatar
5 votes
Accepted

Neumann/Robin Laplacian semigroup well-known estimate

The estimate the OP is looking for is called an ultracontractivity estimate. A characterisation of semigroups that satisfy such an estimate can be found in the Theorem on page 65, Subsection 7.3.2, of ...
Jochen Glueck's user avatar
5 votes
Accepted

Reaction-diffusion systems treated as dynamical systems

For your basic question (topic name) the classical reference is Henry D. Geometric Theory of Semilinear Parabolic Equations. SpringerVerlag, 1981. However, you will not find there general recipes for ...
demolishka's user avatar
5 votes
Accepted

Gluing of two solutions to the same parabolic equation

Absolutely not! Taking the difference $v=u_1-u_2$, you see that $v(x,t)$ solves $$ \begin{cases} (L-\partial_t) v=0 & \mbox{for }(x,t)\in(0,1/2)\times(0,T]; \\ v(0,t)=f(t) & \mbox{for }x=0,\,...
leo monsaingeon's user avatar
5 votes
Accepted

Are there any researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$?

The theorem you stated can be true only for genus zero (that is for the sphere), if $K(x)<0$ at some point $x$); this follows from the Gauss Bonnet theorem that integral of the curvature $-K$ is ...
Alexandre Eremenko's user avatar
5 votes
Accepted

Periodicity and Burger's equation

Of course not. For instance if $A=B\equiv0$ (these are periodic), then the solution decays to $0$ as $t\to+\infty$. Instead, and this is classical in dynamical system theory, if $A$ and $B$ are ...
Denis Serre's user avatar
  • 51.5k
5 votes
Accepted

Iterated Duhamel's formula for solutions of Boltzmann equation

I took a closer look at the manuscript. If one lets $f_{[n]}$ denote the quantity implicitly defined by (1.15), then it appears to me that this is indeed slightly different from $f_{(n)}$ in that ...
Terry Tao's user avatar
  • 108k
5 votes

Heat equation with nonlocal boundary condition

A short observation, which is too long for a comment. Let's assume that $\Omega$ has unit measure, i.e. $|\Omega|=1$. We define $$ w = u - \int u \,\mathrm{d}x . $$ By doing so, $w$ solves the ...
André Schlichting's user avatar
4 votes

$L^p$–$L^q$ estimates for heat equation - regularizing effect

This is standard, but the argument is short enough to fit in an answer. It is not restrictive to assume that $0\in\Omega$. Denote by $Q(t,x,y)$ the heat kernel associated with the Dirichlet Laplacian ...
Piero D'Ancona's user avatar

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