Questions tagged [parabolic-pde]
Questions about partial differential equations of parabolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
384
questions
0
votes
0
answers
93
views
+50
Prove comparison principle for $u_t + f(u)_x = g(t,x)$ with $g \in L^\infty(0,T; L^2(\mathbb R))$
Let us consider
$$u_t + f(u)_x = k u_{xx} + g(t,x),$$ with $k>0$, $g \in L^\infty(0,T; L^2(\mathbb R))$ and $f \in W^{1,\infty}(\mathbb R)$ and $f \not \equiv 0$ (possibly, we can also add the ...
- 63
0
votes
1
answer
71
views
FEM based solution to parabolic problem
Consider the problem
$$
\begin{cases}
u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega
\end{cases}
$$
...
0
votes
1
answer
50
views
Changing the system of PDE by diffeomorphism and differentiate a composition
This problem comes from the book Hamilton's Ricci flow.
Given a smooth functional $f$, and following system. $$\partial_t f=-(\Delta f+R)$$ If there exist a 1 parameter family of diffeomorphism $\Psi(...
- 103
1
vote
0
answers
27
views
Parabolic theory for singular coefficients on bounded domains (Reference Request)
In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems.
Is ...
- 111
3
votes
0
answers
74
views
On the derivatives of the solutions of the heat equations with Neumann boundary condition
Let $\Omega$ be a smooth domain with compact closure. More precisely, $\Omega$ is a $C^\infty$ domain. We write $\partial \Omega$ for the boundary of $\Omega$, and denote by $n$ the inward unit ...
- 703
0
votes
0
answers
29
views
How was this heat semigroup estimate made in a paper on reaction–diffusion systems?
In Yamauchi - Blow-up results for a reaction–diffusion system, in the proof of Lemma 3.3, there is the passage
$$S(t-s)|x|^{\sigma/1-k} \geq C_1(t-s)^{\sigma/2(1-k)}.$$
Here $S(t)$ denotes the heat ...
- 225
1
vote
0
answers
14
views
Stabilization of the second BVP solutions for nondivergence parabolic equations
Let $Q\subset \mathbb R^n$ be a bounded domain with smooth enough boundary $S$.
For a uniformly parabolic operator
$$
Lu=u_t-\sum_{i,j=1}^n a_{ij}(x)\partial_{ij}u-\sum_{i=1}^n b_{i}(x)\partial_{i}u
$...
- 2,515
1
vote
0
answers
66
views
Partial derivative of the Bessel's operator
Let $J^s = (I- \Delta)^{\frac{s}{2}}$ where $\Delta$ is the Laplacian, and $w(x,y) \in L^2(\mathbb{T}^2)$. During my study to the paper, https://arxiv.org/pdf/1809.02027.pdf, the author stated that
$$\...
- 223
2
votes
0
answers
60
views
parabolic schwarz lemma
Trying to follow the computation in https://arxiv.org/pdf/math/0602150.pdf, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they computed $\Delta \text{tr}_{g}h = g^{i \bar l}...
- 59
3
votes
1
answer
100
views
Looking for references to study $U^p$ and $V^p$ spaces
I am studying some papers in the analysis of nonlinear PDEs and I am encountering the $U^p$ and $V^p$ spaces for the first time. Where can I find references more detailed than papers?
Edited
The ...
- 388
1
vote
0
answers
51
views
Is there any class of initial data for which the heat semigroup is increasing in time?
Lee and Ni proved in the paper Global existence, large time behavior and life span of solutions of semi linear parabolic Cauchy problem that the heat semigroup decays as $t$ tends to infinity, that is
...
- 225
5
votes
0
answers
47
views
$L^p$ estimates for linear parabolic pdes
Let $u$ solve the linear parabolic equation
$$
u_t - \Delta u = f \text{ on } \Omega \times (0,T)
$$
with initial condition $u(0)=u_0$ and homogeneous Dirichlet boundary condition on $\partial \Omega ...
- 256
6
votes
1
answer
102
views
Short time existence for fully nonlinear parabolic equations
I am trying to assert short time existence for a fully nonlinear equation of the general form
\begin{equation}
\begin{cases}
u_t = F(x,u,Du,D^2u) & \text{in }(0,T)\times\Omega\\
u(\cdot,0) = u_0(\...
- 61
2
votes
0
answers
64
views
Gradient $L^\infty$-estimate for heat equation with homogeneous Dirichlet boundary condition
$\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Consider the homogeneous heat equation with zero boundary condition in $\Omega$
\begin{cases}
\partial_t u-\Delta u=0 \quad(x,t)\in \Omega\...
- 121
0
votes
0
answers
69
views
Compact embedding of anisotropic Sobolev space
I am wondering if the embedding from $W^{2,1}_p(\Omega \times [0,T])$ to $C^{\alpha,\alpha/2}(\Omega \times [0,T])$ is compact, for some suitable domain, $p$ and $\alpha$. I have found some results. I ...
- 168
1
vote
0
answers
60
views
Reference for unique classical solution to quasilinear uniformly parabolic PDEs
In this post, the author mentioned that "we know there is a unique classical solution (see the references below, for example)". I have tried to read the two references the author provided, ...
- 11
1
vote
0
answers
96
views
Do zeroes of $f(t)= \sum_{k\in \mathbb{Z}} e^{\lambda_k t} c_k$, have zero Lebesgue measure ?$\{\lambda_k\}_k$ eigenvalues of elliptic s.a. operator
This question is inspired by the zeroes of solutions to parabolic PDEs (interpret the $\lambda_k$ above as eigenvalues of an elliptic operator), even though I abstracted it from its original context.
$...
4
votes
1
answer
159
views
Reference request: continuity of the derivatives of the (fundamental) solution to a parabolic equation
Consider the parabolic equation in $p: \mathbb R^2\to\mathbb R$
$$\partial_t p + b(t)\partial_x p + D(t,x)\partial^2_{xx}p=0,$$
where $b$, $D$ are nice enough functions. I look for the continuity of ...
- 804
1
vote
0
answers
84
views
Uniqueness of the solution to some parabolic PDE
Consider the system
$$
\begin{eqnarray}
\partial_t p(t,x) + b(t)\partial_x p(t,x) - \partial_{xx}^2\left(\frac{\sigma(t,x)^2p(t,x)}{2\big(1+\alpha(t)\big)^2}\right) &=& 0, & \forall t>0,...
- 804
1
vote
1
answer
83
views
Local boundedness for Cauchy problem
Consider the Cauchy problem
$$\left\{\hspace{5pt}\begin{aligned}
&-\dfrac{\partial u }{\partial t}
+a\dfrac{\partial^2 u}{\partial x^2}
+b \dfrac{\partial u }{\partial x}
+c u
= f(u) \leq 0& ...
- 168
3
votes
0
answers
147
views
Banach's fixed point theorem for quasilinear parabolic PDEs
I have recently started reading into PDE theory, and came across the following question. Consider the PDE $$
\begin{cases}
\partial_t \rho = \Delta (\rho + \rho^2) & \text{ on } (0,T) \times \...
- 141
0
votes
0
answers
29
views
Dependence of the density on the coefficients
Consider a parametric SDEs
$$dX_t = b\big(t,X_t,\alpha(t)\big)dt + dW_t,\quad \forall t\ge 0,\quad \quad \quad \quad (\ast)$$
where $\alpha=(\alpha(t))_{t\ge 0}$ be a parameter taking values in some ...
- 804
1
vote
0
answers
22
views
Examples of reaction-diffusion systems with analytical solutions
I want to study how some numerical schemes work on $2$-dimensional reaction-diffusion systems on rectangles with Neumann Boundary conditions and I search for a while for a problem of the form:
$$\...
- 892
1
vote
0
answers
118
views
Is this generalization of differentiable manifolds to mixed dimensions a known object?
Suppose you are studying the evolution of some electromagnetic quantity in a conductor consisting of objects of several dimensions, i.e. wires, plates and balls.
This would amount to studying the ...
1
vote
0
answers
59
views
Parabolic/Elliptic equation with nonlinear gradient term
Let $a\in (0,1)$ and $(0,1) \subset \mathbb{R}$, we consider the below equation in $(0,1) \times (0,T)$
$$ \partial_t u -\partial_x^2 u - \dfrac{1}{|u|^a} \partial_x u =f(x).$$
And $u(x,0)=x^{1/a}$ ...
- 168
0
votes
1
answer
48
views
Set invariance for differential inclusions applied to PDES?
This question is somewhat related to this one that I posted a while back on MSE, but the context has slightly changed since then. My question here relates to the consequences of a result in Weinberger'...
- 121
2
votes
0
answers
184
views
Dependency of fundamental solution on coefficients of heat equation
Let $b: \mathbb R_+\to\mathbb R_+$ and $\sigma: \mathbb R_+\times \mathbb R\to\mathbb R_+$ be Lipschitz and bounded. Assume further $\sigma$ is elliptic, i.e. $\inf_{(t,x)}\sigma(t,x)>0$. For each $...
- 804
1
vote
0
answers
34
views
Time evolution of Wigner transform
I am studying the Hartree equation for N-particles for the first time and things are not clear to me. Given the density matrix
$$\gamma_{N, t}^{(n)} (x_1,..,n_n; y_1,...,y_n) =
\begin{cases}
\int \...
- 388
4
votes
0
answers
82
views
$\mathcal{C}^1(\overline{\Omega})$ gradient bounds for the Dirichlet problem of the heat equation on general domains
I am studying the heat equation on a general bounded domain $\Omega \subset \mathbb{R}^+ \times \mathbb{R}^n$ with continuously differentiable Dirichlet data $\phi$ on the boundary,
$$
\left\{
\begin{...
- 173
0
votes
0
answers
48
views
Parabolic systems with gradient terms, what has been studied so far?
I'm interested to know what happens with systems
(in the sense of knowing in which spaces the solution exists, if it is global or blow-up in finite time) like
$$
\begin{cases}
u_t - \Delta u = |\nabla ...
- 225
0
votes
1
answer
94
views
Proof of vanishing viscosity error rate
Consider the initial value problem associated to the parabolic equation $u^\epsilon_t + u^\epsilon_x = \epsilon u^\epsilon_{xx}$ and the corresponding hyperbolic problem $u_t + u_x = 0$.
What is a ...
- 63
2
votes
0
answers
76
views
Solution verification of some PDE with an additional condition
Provided a probability density $\rho:\mathbb R_+\to\mathbb R_+$ (as nice as possible), consider the PDE
$$
\begin{cases}
\partial_t p = \dfrac{\partial_{xx}p}{2\big(1+m(t)\big)^2} - \partial_x p, &...
- 804
1
vote
0
answers
46
views
Self-similar solutions for a parabolic system
Formally, how can one find a self-similar solution for the parabolic system
\begin{align}
\begin{cases}
u_t - \Delta\Big((a_1 + a_{11} u + a_{12} v) u\Big) = 0\\
v_t - \Delta\Big((a_2 + a_{22}...
- 63
1
vote
0
answers
50
views
Regularity of solutions to heat equations
Let $d$ denote a positive integer. Let $f$ be a positive function on $\mathbb{R}^d$.
We also assume that $f$ is bounded above and below. That is, there exists $C>0$ such that $C^{-1}\le f(x)\le C$, ...
- 703
1
vote
1
answer
105
views
PDE interpretation of some properties of the solution to Fokker–Planck equations
Consider
$$X_t=X_0 + \int_0^t b(s)ds+ \int_0^t \sigma(s)dW_s,\quad \forall t\ge 0,$$
where $X_0\ge 0$ is a random variable of density $\rho$, $(W_t)_{t\ge 0}$ is an independent Brownian motion and $b,\...
- 804
2
votes
1
answer
218
views
A variant to the Fokker–Planck equation
Consider the PDE of $p(t,x)\ge 0$ given as
$$\partial_t p = \frac{\partial^2_{xx}p}{(1+m(t))^2} - \partial_x p,\quad \forall t,~x \in (0,\infty)$$
with initial and boundary conditions $p(0,\cdot)=\rho$...
- 804
2
votes
1
answer
58
views
Parabolic system with coupling in the diffusion
Let's consider the parabolic system
$$
\begin{cases}
u_t - \Delta u -a\Delta(uv) = 0 \\
v_t - \Delta v - b\Delta(uv) = 0
\end{cases}
$$
with $a,b >0$. What is the name of this system? Are there ...
- 63
5
votes
1
answer
144
views
Parabolic equation with Cauchy boundary condition
Consider the domain $[0,1] \times [0,T]$ and the uniformly parabolic operator $L -\partial_t$ with smooth coefficient. I would like to obtain the existence of the problem
\begin{equation}
\left\{\...
- 168
3
votes
1
answer
103
views
Gluing of two solutions to the same parabolic equation
Consider the domain $[0,1] \times [0,T]$ and the uniformly parabolic operator $L -\partial_t$ with smooth coefficient. Suppose I have $u_1(x,t) \in C^\infty([0,1] \times [0,T])$ solving
\begin{...
- 168
2
votes
1
answer
213
views
Compactness for initial-to-final map for heat equation
Let $M$ be a compact smooth manifold without boundary. Let $T>0$ and let $g$ be a smooth Riemannian metric on $M$. Given any $f \in L^2(M)$ let $u$ be the unique solution to the equation
$$\...
- 2,858
2
votes
1
answer
106
views
Elliptic regularity when the Lagrangian is possibly infinite
I want to solve variational problems of the form
$$\inf_u \int_{-1}^1 \phi(u'(x)) \text{ with } u(-1)=u(1) = 0,$$
where $\phi(p)$ is convex and is allowed to take on the value $+\infty$ for some ...
- 441
0
votes
0
answers
23
views
Parabolic equation in non-cylindrical domain with cone
Let $d_1(t)$ and $d_2(t)$ be smooth functions from $[0,T]$ to $\mathbb{R}$ such that $d_1(t) <d_2(t)$ for $t \in (0,T]$ and $d_1(0)=d_2(0)$. Suppose $L$ is a uniform elliptic operator and $u(x,t) :\...
- 168
0
votes
0
answers
28
views
Is the time of solution shorter as the initial data increases?
I'm reading the book superlinear parabolic problems and I came across the following situation twice: given two initial data $u_0$ and $\underline{u_0}$ with $u_0\geq \underline{u_0}$, $u_0\neq \...
- 225
2
votes
0
answers
83
views
Continuity of the entropy of the solution of a parabolic PDE at $t=0$
Consider the following initial value problem for a parabolic PDE :
$$\begin{cases}
\textrm{div}\big(A\,\nabla u(t,x) + b(x)\, u(t,x)\big) \,=\,\partial_t u(t,x) \quad x\in\mathbb R^d\,,\ t>0 \\[4pt]...
- 301
2
votes
2
answers
168
views
Solution of parabolic partial differential equation using singular perturbation method
Consider the following parabolic partial differential equation (PDE)
\begin{align}
\label{eq:42}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \psi} + ...
- 81
4
votes
0
answers
113
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 $...
- 277
3
votes
1
answer
126
views
Contractivity of Neumann Laplacean
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 Laplacean.
In W....
- 892
1
vote
0
answers
57
views
Hypoellipticity of a heat-like parabolic operator on Riemannian manifolds - reference request
Let $(M,g)$ be a Riemannian manifold and $L$ be a differential operator on $M$, with smooth coefficients, such that its symbol be $g$ (a "generalized Laplacian").
Where can I find proved ...
- 4,670
3
votes
0
answers
47
views
Conditions of parameters to have bounded solution of Dynkin's equation in exit problem
Consider the following Dynkin’s equation in exit problem defined on unit disk $D_1(0)$
\begin{align}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma-1}{r} \sin\psi \frac{\partial}{\partial\...
- 81
2
votes
1
answer
150
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 ...
- 168