All Questions
87 questions
9
votes
2
answers
418
views
Reference request: Parabolic Equations
I am a PhD student working mainly on Elliptic Equations. With the other PhDs of my department, we organised a reading group, meaning that we agreed on a book we were all interested in, we meet weekly ...
- 452
9
votes
3
answers
2k
views
Real analyticity of solution of heat equation
Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
- 1,407
7
votes
1
answer
439
views
About the convergence rate for an approximation to the heat kernel
Let $G(t,x)$ be the heat kernel
$$
G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}.
$$
Here is one approximation to $G(t,x)$:
$$
G_\epsilon(t,x)=e^{-t/\epsilon} \...
- 1,649
6
votes
2
answers
1k
views
Properties of heat equation
** I simplified the question: **
On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.
I ...
user121558
6
votes
0
answers
187
views
Gaussian lower heat kernel bounds on non-convex bounded domain
I am looking for a proof the following theorem.
Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ ...
- 61
6
votes
0
answers
110
views
Heat Flows and spatial singularities
While working on an abstract problem, I came up with the following question:
Let $\Omega_1 := \mathrm B(-1, 1)$ and $\Omega_2 := \mathrm B(1, 1)$, where $\mathrm B(x, r) \subseteq \mathbb R^2$ denotes ...
5
votes
3
answers
1k
views
Decay estimate for the heat equation: $\sup_{t>0}\int_{\mathbb{R}} t^\alpha |u_x|^2\ dx$
Let $u$ be a solution of the heat equation $$u_t - u_{xx} = 0, \quad t>0, x \in \mathbb{R}$$
with initial data $u(0,\cdot) = u_0$.
Fix $\alpha >0$. How can I estimate (without using explicitly ...
- 839
5
votes
1
answer
1k
views
$L^\infty$ estimate on heat equation with a lower order term
Let $u$ be the weak solution on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ (for $n \leq 3$) of
$$u_t - \Delta u = f$$
$$u(0) = u_0$$
$$\partial_\nu u = 0 \quad\text{on $\partial\Omega$}$$
...
- 53
5
votes
1
answer
153
views
Why is density and separability needed for uniqueness of weak (time) derivatives?
Let $X,Y$ be Banach spaces with $X \subset Y$. Recall that $u \in L^1(0,T;X)$ has weak derivative $g \in L^1(0,T;Y)$ if
$$\int_0^T u(t)\phi'(t) = -\int_0^T g(t)\phi(t) \qquad\forall \phi \in C_c^\...
5
votes
1
answer
279
views
Connecting PDE notions for functions $[0,T] \to (\Omega \to \mathbb{R})$ to related notions for functions $[0,T] \times \Omega \to \mathbb{R}$
Fix $\Omega \subseteq \mathbb{R}^N$ a bounded domain (of whatever smoothness you end up needing, let's say $C^1$ domain for definiteness) and fix some $0 <T < \infty$. In considering evolution ...
- 324
5
votes
1
answer
361
views
Exponential decay of solution in $L^p$ with $p>2$
Consider the following evolution equation
$$u_t=\Delta u$$
in a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$, with smooth initial conditions $u_0\geq 0$ and homogeneous Dirichlet ...
- 161
5
votes
1
answer
462
views
Backward uniqueness for a heat equation with a drift
Consider heat equation with a drift (=reaction-diffusion equation)
$$
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1]
$$
with periodic or ...
- 931
4
votes
2
answers
447
views
Heat equation and evolution of number of critical points
Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...
4
votes
1
answer
458
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
4
votes
1
answer
418
views
Periodicity and Burger's equation
Consider the 1-dimensional Burger's equation on a finite interval $I=(0,1)$,
$$u_t+uu_x=u_{xx}$$
with initial condition
$$u(x,0)=f(x)$$
and boundary conditions
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$
...
- 43.2k
4
votes
1
answer
2k
views
Crandall & Rabinowitz Theorem, bifurcation curves
Crandall & Rabinowitz Theorem states what follows. We have got a Banach Space $(X,||\cdot||)$ and an equation of the following type:
$$
F(\lambda,u) = \lambda u - G(u) = 0,
$$
where $G \in C^1(X,X)...
4
votes
1
answer
1k
views
Where to learn about parabolic Hölder spaces and when to use them
Is there a good resource from where I can learn about parabolic Hölder spaces? I see quite a few different definitions of this space in different papers. I am clueless about why, for example, one may ...
- 45
4
votes
1
answer
346
views
Reference for a Heat Process in a Wedge
I would like to ask about an explicit suggestion/reference for the following type of heat processes:
Roughly, assume we have a "wedge" $W$ of the following form - a domain in $\mathbb{R}^n$ with a ...
- 337
4
votes
1
answer
442
views
Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$
Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required.
Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy
$$0 < a \leq g(x,t) \leq b < \...
- 183
4
votes
1
answer
220
views
parabolic PDE with almost-monotone elliptic operator, existence results?
Are there any existence results for parabolic PDE of the type $$u_t - Au = f$$ in some Gelfand triple setting ($V \subset H \subset V^*$) with $A$ an operator that it is not quite monotone but close: ...
- 85
4
votes
0
answers
129
views
Trace-class heat semigroups
Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator.
Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$
$$T_{\varphi}(u) :=...
user516086
4
votes
0
answers
194
views
$L^\infty$ solutions for parabolic Neumann problem (heat equation)
Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs:
$$u_t - \Delta u = f$$
$$\partial_\nu u = 0$$
$$u|_{t=0} = u_0$$
where $f \in L^p(0,T;L^r(\Omega))$ and $...
- 307
4
votes
0
answers
220
views
improved regularization for $\lambda$-convex gradient flows
It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
- 5,380
4
votes
0
answers
746
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 ...
- 1,759
4
votes
0
answers
198
views
Relationships between fractional Sobolev space, Bessel spacse and Hajłasz–Sobolev space
It is known that for $\alpha\in(0,1)$ and $p>1$,
the fractional Sobolev space $W^{\alpha,p}(R^n)$ is defined by
$$
W^{\alpha,p}(R^n):=\{f\in L^p(R^n):\int_{R^n}\int_{R^n}\frac{|f(x)-f(y)|^p}{|x-y|^...
- 411
4
votes
0
answers
500
views
Properties of the solution of the heat equation
Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention.
Note 2: This is my research problem, but the original problem ...
- 679
3
votes
1
answer
215
views
Checking initial data in parabolic PDE with no control on time derivative
It is possible to define a weak solution of a parabolic PDE
$$u_t - Au = f$$
$$u(0) = u_0$$
as $u \in L^2(0,T;H^1)$ such that
$$-\int_0^T\int_\Omega u(t)\varphi'(t) + \int_0^T\int_\Omega Au(t)\varphi(...
- 155
3
votes
1
answer
552
views
Lax-Milgram and the existence of solution to parabolic equation
I think it is standard and common to use Lax-Milgram theorem to prove the existence of solution to elliptic equation. However, can we use it to establish the existence of parabolic equation? I do not ...
- 54
3
votes
1
answer
356
views
Initial data and heat equation
We assume all solutions to be bounded here!
Let $y_{+},y_{-} \in C_c^{\infty}$ be two positive functions.
If we then consider the heat equation
$$\partial_t u(t,x) = \Delta u(t,x)$$ for two ...
user121558
3
votes
1
answer
627
views
Compact embedding between parabolic Hölder spaces
My question is about the following compact embedding:
\begin{equation}
C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T).
\end{equation}
what condition should be put ...
- 31
3
votes
1
answer
219
views
Positivity of generalised heat kernels
Let $K_\alpha(t,x)$ be the (generalised or fractional) heat kernel which corresponds to the fractional heat equation (I'm not sure that's the right name) in $\mathbb R^n$
$$
u_t=(-\Delta)^\alpha u, \...
- 2,933
3
votes
1
answer
563
views
Uniqueness of weak solutions of a heat equation
Let $M$ be a smooth compact closed manifold.
Let $u \in H^1(0,T;H^{-1}(M)) \cap L^2(0,T;H^1(M))$ be a solution of
$$u_t - \Delta u - u = 0$$
$$u(0)=u(T)$$
satisfying $\int_M u(t) = 0$ for all $t$. Is ...
- 33
3
votes
0
answers
196
views
Parabolic smoothing for semilinear PDE
Consider the semilinear energy-critical parabolic PDE in $\mathbb{R}^3$
\begin{align}
\partial_t u &= \Delta u + |u|^{4/(n-2)}u = \Delta u + u^5\\
u(0,x) &= u_0\in \smash{\dot{H}}^1(\mathbb{R}^...
- 537
3
votes
0
answers
94
views
Harmonic heat flow, formal and rigorous
Let $ (M,g) $ be a smooth Riemann manifold without boundary, $ S^{n-1} $ is an $ n $-dimensional sphere, and $ T>0 $. Consider a weak solution $ u:M\times[0,T]\to S^{n+1} $ of
$$
\partial_tu-\Delta ...
- 1,219
3
votes
0
answers
135
views
Holmgren's theorem on the boundary
Consider $\Omega$ a bounded Lipschitz domain, with $\gamma \subseteq \partial \Omega$ a $C^2$ manifold. I am interested in proving the following.
Let $u: \Omega\times [0,T]\rightarrow \mathbb{R}$ be ...
- 235
3
votes
0
answers
159
views
Does the weak formulation of a parabolic PDE applies to a (good) non-test function?
Let $\rho:\mathbb R^d\times[0,\infty)\to(0,\infty)$ such that $\int \rho_t(x)\,dx=1$ for all $t\geq0\,$, $\rho$ is Holder-continuous (in both variables) and $\rho_t\in W^{1,1}(\mathbb R^d)$ for a.e. $...
- 311
3
votes
0
answers
376
views
Existence and uniqueness for reaction-diffusion equations
I am interested in the following PDE on a $d$-dimensional torus $\mathbb{T}^d$
\begin{align*}
&\partial_tu(t,x) = \Delta u(t,x) +f(u(t,x),t,x),\\
& u(0)=u_0\in L_2
\end{align*}
where the ...
- 931
3
votes
0
answers
185
views
Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)
Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse.
Does there ...
- 93
3
votes
0
answers
163
views
Reference for existence results for 2D forced viscous Burgers equation
I am looking for results concering the following parabolic PDE
$$u\cdot\nabla u + \Delta u = F(x),$$
where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or ...
- 161
2
votes
1
answer
338
views
Why is this test function admissible? [Paper explanation]
Reading Non-linear Elliptic and Parabolic Equations Involving Measure Data by Boccardo$\&$Gallouet , I had trouble understanding the following:
Why is $\psi(u_n)\chi_{(0,t)}$ admissible as a ...
- 227
2
votes
1
answer
1k
views
Strong convergence in the Bochner space L^p([0,T],X)
Dear mathoverflowers, I have a question concerning the strong convergence in $L^p([0,T],X)$.
Let $X_1,X$ be two Banach spaces such that $X_1\subset X$ with compact embedding. Let $x_n(t)\in X_1$ be ...
- 843
2
votes
1
answer
79
views
There is some initial data such that the decay of the semigroup in it is faster than $t^{-n/2}$?
Lee and Ni show in their work Link Here that the heat semigroup $e^{t \Delta}u_0$ has decay as $t^{-\min \{a, n\} /2}$, $t \to \infty$ if $u_0 = C(1+|x|^2)^{a/2}$ if $a \neq n$. I'm trying to ...
- 677
2
votes
1
answer
449
views
Derivative of Yosida-Approximation
i have got a problem with some assumptions to solve a parabolic variational inequality. My Problem is: Find a function $u$ with
\begin{align}
u\in L^2(0,T;V),~ u' \in L^2(0,T;V') \\
(u'(t),v-u(t)) + ...
- 187
2
votes
1
answer
578
views
Is the Lopatinski-Shapiro condition invariant under diffeomorphism?
If a PDE (eg. the heat equation with Robin BCs, or the elliptic version) on a bounded smooth domain $U$ satisfies the Lopatinski-Shapiro condition (for a definition see eg. Wloka), and if $T:U \to W$ ...
- 21
2
votes
1
answer
3k
views
A comparison principle for parabolic equation
(Crossposted from https://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear)
Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for ...
- 266
2
votes
1
answer
171
views
Mean value formula for fractional heat equation
For the solution $u(z) = u(t,x)$ of the heat equation $u_t -\Delta u = 0$ we have
$$u(z_0) = \int_{\Omega_r(z_0)}u(z) K_r(z_0-z) dz,$$
where $$\Omega_r(z_0) = \left\{z \in \mathbb{R}^{N+1}: \Gamma(z_0-...
- 161
2
votes
1
answer
496
views
Some properties of fractional Dirichlet heat kernel
Let $\Omega\subset \mathbb{R}^n (n\geq 2)$ be a bounded open domain with smooth boundary $\partial\Omega$. Consider the fractional heat equation with Dirichlet boundary condition：
\begin{equation}
\...
- 287
2
votes
1
answer
179
views
Reference request: Parabolic Schauder estimates for the heat equation with $f \in L^\infty$
Let us consider the heat equation
$$\partial_t u - \Delta u = f(x, t) \quad \text{in }Q_R $$
where $Q_R = B_R \times (-R^2,0].$ I would like to know the kind of regularity we should expect of $u$ if ...
- 452
2
votes
0
answers
328
views
Conditions for an existence of smooth solution to a parabolic PDE
I'm interested to know the conditions of when the parabolic PDE ($U \subset \mathbb{R}^n$ is some bounded open subset):
\begin{equation*}
u_t - \sum_{i,j=1}^n(a^{ij}(x,t)u_{x_i})_{x_j} + \sum_{i=1}^nb^...
- 425
2
votes
0
answers
159
views
On Fredholm alternative for Neumann conditions
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation $$\begin{cases}
-\Delta ...
- 1,350