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 ...
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 ...
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
$$
w=\begin{pmatrix}
pu^{p-1}\Delta^2 u\\ (p)...
9
votes
Gradient $L^p$ estimates for heat equation
Let me comment on what I know in an open set $\Omega$, trying to control $C$. First of all, the heat semigroup can be expressed through a kernel $p$ which is pointwise dominated by the heat kernel in $...
8
votes
Accepted
Caffarelli-Kohn-Nirenberg-type inequality with nonradial weight
I'll try to keep your notation except for three things: I prefer to have all parameters non-negative not to get confused myself, I'd rather have $z=(x,y)\in\mathbb R^2$ coordinates on the plane to ...
7
votes
Accepted
Injectivity of a Fredholm operator
Surprisingly (to me), the statement is false. My counterexample is a little messy, but the idea is fairly simple.
Take $T = 1$ and set $a_n = \frac{1}{n}$ and $b_n = 1 - \frac{1}{n}$ for $n \in \...
7
votes
Accepted
Weak parabolic maximum principle on Riemannian manifolds
Firstly, you have the wrong inequality (there is a small typo in the paper). Young's inequality is typically written for nonnegative numbers, but for any $a,b\in\Bbb R$ we have
\begin{align*}
-ab&\...
7
votes
Explicit solution of a free boundary problem for heat equation
From the maximum principle, the solution $v$ of
$$v_t=v_{xx}+1,$$
together with the data $v=0$ at the boundary and initial time, is positive. Therefore your $u$ is nothing but $v$. It turns out that ...
7
votes
Accepted
Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight
Here is a functional analytic approach (Kato's book on perturbation theory is a good reference):
Let
$$
a\colon D(a)\times D(a)\to \mathbb{R},\,(u,v)=\int \nabla u\cdot \nabla v+\int \zeta uv.
$$
with ...
7
votes
Accepted
Contractivity of Neumann Laplacian
It might be helpful to point out the following conceptual reasons why ultracontractivity estimates are mainly interesting for times close to $0$.
Let us consider the following general setting: We have ...
7
votes
Accepted
Strong positivity of Neumann Laplacian
As other users have indicated in the comments, for sufficiently smooth domains one can get it by combining, for instance, elliptic regularity with Hopf's boundary point lemma (and then go from the ...
7
votes
Accepted
Existence of solutions to the heat equation on nonsmooth domains
$\newcommand\R{\mathbb R}\newcommand{\Om}{\Omega}$It is convenient to reverse the time by the substitution $t\leftrightarrow T-t$ and also rescale it by a factor of $2$. So, the boundary value problem ...
6
votes
Accepted
$L^\infty$ estimate on heat equation with a lower order term
The estimate as stated is clearly false since $u\to u_0$ as $t\to0$ hence $\|u\|_{L^\infty}\ge \|u_0\|_{L^\infty}$.
Anyway you can write the kernel explicitly and extract the information you need ...
6
votes
Accepted
Properties of heat equation
The answer is no. A counterexample has the form
$$u(x,t)=\sum_0^\infty \frac{g^{(n)}(t)}{((2n+1)!)^2}(r-R)^{2n+1},$$
where $r>R$ is the distance from the origin in $R^2$.
First one shows that this ...
6
votes
Accepted
Angle of analyticity of semigroup
Too long for a comment. Actually I do not think that it is written anywhere but these kind of counterexamples are usually provided by the Ornstein-Uhlenbeck semigroup, generated by $\Delta+Bx \cdot \...
6
votes
Accepted
Compactness for initial-to-final map for heat equation
Parabolic regularity show that $u$ is regular for all positive times; in particular $u(t,\cdot) \in W^{1,2}(M)$ for all $t > 0$. Interior parabolic estimates additionally show that there is a ...
6
votes
Nontrivial invariant transformations for heat equations
Yes, how about the Appell transform, see here.
$$v(t,x)\mapsto\Gamma(t,x)v(-\frac1t,\frac xt)$$
where $\Gamma$ is the heat kernel.
Truly nontrivial if you ask me. Of course, this inverts time and not ...
5
votes
Accepted
Growth at infinity of a solution to a parabolic PDE
What about $u(t,x) = x e^{t x^2}$, which is a solution of $$\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}$$ with initial value $u_0(x) = x$...
5
votes
Motivation behind the parabolic metric
Here is one reason, the core of which can be found at page 17 of Moser's book mentioned in the comments. Many theorems about the heat equation are valid on "cylindrical" domains of the form $(0,T) \...
5
votes
Accepted
Reference for De Giorgi-Nash-Moser theory
The proof of Harnack's inequality using De Giorgi method has a great flexibility and it can be exteneded even to doubling metric measure spaces that support Poincaré inequalities.
The elliptic case ...
5
votes
Accepted
Why is this test function admissible? [Paper explanation]
The reason I'm asking is because characteristic\indicator functions have no smooth derivatives and plus I don't understand in which function space of test functions the authors define the weak ...
5
votes
Reference request: Long-term behaviour of the heat equation for bounded initial data
First of all, with initial data in $C_b$, classical solution exist, so there is no need for quotation marks. It is easy to see that the convolution of the initial data $f(x)$ with the Gauss–...
5
votes
Accepted
Neumann/Robin Laplacian semigroup well-known estimate
The estimate the OP is looking for is called an ultracontractivity estimate. A characterisation of semigroups that satisfy such an estimate can be found in the Theorem on page 65, Subsection 7.3.2, of ...
5
votes
Accepted
Reaction-diffusion systems treated as dynamical systems
For your basic question (topic name) the classical reference is
Henry D. Geometric Theory of Semilinear Parabolic Equations. SpringerVerlag, 1981.
However, you will not find there general recipes for ...
5
votes
Accepted
Gluing of two solutions to the same parabolic equation
Absolutely not! Taking the difference $v=u_1-u_2$, you see that $v(x,t)$ solves
$$
\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,\,...
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 ...
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 ...
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 ...
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 ...
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 ...
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