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.
493
questions
1
vote
0
answers
101
views
Classical solution to logarithmic diffusion equation
Consider the one-dimensional logarithmic diffusion equation for $u: \mathbb R_+ \times [0,1]\ni (t,x)\to u(t,x)\in \mathbb R$ :
$$(\ast)\quad\quad
\begin{cases}
2u_t = \big(\log(u)\big)_{xx} & \...
- 975
3
votes
0
answers
116
views
Wellposedness of this parabolic PDE
Consider a terminal-boundary value problem for $v: (t,x,y)\in [0,T]\times \mathbb R^2_+\to \mathbb R\ni v(t,x,y)$:
$$
\begin{cases}
v_t + \max(v_x,v_y)+ \frac 1 2 (v_{xx}+v_{yy})=0, & \forall (t,...
- 975
0
votes
0
answers
44
views
This result of nonexistence of Hamilton–Jacobi equations in Lebesgue spaces is valid in domains?
In the article The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces Ben Artzi, Souplet, Weisller studies the equation
\begin{equation}
\left\{
\begin{array}{rll}
...
- 585
4
votes
1
answer
97
views
A formula in harmonic heat flow
Assume that $ (M,g) $ and $ (N,h) $ are two smooth closed manifold and $ N $ is embedded isometrically into $ \mathbb{R}^K $ for some $ K\in\mathbb{Z}_+ $. Assume that $ u\in C^{\infty}(M\times\mathbb{...
- 1,133
0
votes
0
answers
27
views
How to deal with discontinous problem with numerical method?
I would like to consider how to deal with the indicator function in a PDE. For example, for the PME with 𝑚=5, the initial condition is the two-Box solution with the same height, namely
$$
u_0(x)= 1 \...
- 1
4
votes
1
answer
176
views
On a non-linear PDE $p_t = e^{-p}p_{xx}$
Consider the PDE for $p:[0,T]\times [0,1]\to\mathbb R$ as follows:
$$
\begin{cases}
p_t = e^{-p}p_{xx}, & (t,x)\in (0,T)\times (0,1),\\
p(0,\cdot)\equiv -c &\text{for }x\in (0,1)\\
p(\cdot,0)\...
- 975
0
votes
0
answers
81
views
Maximum principle for nonlocal equation
I was reading this paper:
Luis Silvestre - On the differentiability of the solution to the
Hamilton-Jacobi equation with critical fractional diffusion (https://arxiv.org/pdf/0911.5147.pdf)
In Lemma ...
- 313
6
votes
1
answer
560
views
Nash–Moser–De Giorgi differences
The names of Nash, Moser and De Giorgi are associated to elliptic and parabolic regularity theory.
But what are the differences in the approach between the three contributions? Can you briefly sketch ...
- 73
2
votes
1
answer
112
views
Reference: Result of interior parabolic regularity theory for Hamilton–Jacobi equations
Does anyone know the parabolic regularity result that Ben-Artzi used in the article The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces used to prove that the solution to the ...
- 585
0
votes
0
answers
34
views
Solution for a non-linear parabolic pde
I want to know how to deal with the following non-linear parabolic pde
$$\begin{cases}
W_t(t,x)+W+W_x-W_{xx}-\mathrm{e}^xW_x^{-1}W_{xx}-\mathrm{e}^x=0, \quad (t,x)\in (0,T]\times(0,\infty)\\
W(0,x)=\...
- 105
1
vote
0
answers
52
views
Comparison principle for porous medium equation in Fourier variables
Let $V:[0,\infty) \to[0,\infty)$ be convex, $C^2$ with $V(0)=0$. Define $F(u): = uV'(u)-V(u) $. Let $v\in L^1 (\Bbb R^d)$, $v\geq0$ so that $F\circ v\in L^1 (\Bbb R^d)$. For fixed $\varepsilon>0$, ...
- 1,033
1
vote
0
answers
64
views
A parametrix construction for heat boundary value problem using Fourier transformation
Let $\Omega$ be a smooth bounded open subset in $\mathcal{R}^{d}$, with $d \geqslant 3
$ and $T>0$. Consider the linear parabolic initial Dirichlet boundary value problem with $f\in H^{-1}(\Omega)$...
- 41
0
votes
0
answers
27
views
Is there a principle of comparison for mild-solutions?
We consider the equations of the form
\begin{equation}\label{Eq.un}
\left\{
\begin{array}{rll}
u_t - \Delta u &= |\cdot|^{\gamma}u^{p}& \mbox{ in } \mathbb{R}^n \times (0,T),...
- 585
7
votes
2
answers
409
views
Existence of solutions to the heat equation on nonsmooth domains
Let $\Omega \subset \mathbb{R}^n$ be a compact domain and for given functions $g: \partial \Omega \times [0,T] \to \mathbb{R}$ and $h: \Omega \to \mathbb{R}$ consider the heat equation
$$
\begin{cases}...
2
votes
0
answers
58
views
Well-posedness or existence for a Poisson problem in Orlicz spaces
I know that the problem
\begin{equation}
\Delta_p u = f
\end{equation}
make sense if $f \in L^q$ with $n/p<q<n$ and that is there a existence theory for
$$
u_t -\Delta_p u = f
$$
For a given ...
- 191
2
votes
0
answers
28
views
Free programs suggestions to simulate parabolic EDPs
I'm interested in learning how to computationally simulate the behavior of parabolic partial differential equations, but I don't know where to start, what are the best free programs to use and where ...
- 585
0
votes
0
answers
138
views
Convergence of Solutions of Integral Equations with Weakly Converging Forcing Terms
Let $\Omega$ be a bounded interval of $\mathbb{R}$ and let $y\in L^\infty(\Omega \times (0,T))$ be a mild solution of the integral equation
$$
y(\cdot,t)=S(t) y_0+\int_0^t S(t-s) \left[u(\cdot,s)y(...
- 55
1
vote
0
answers
179
views
For the solvability of the poisson equation $\Delta u = f$ on manifold with boundary
For poisson equation $\Delta u = f$ in bounded domain in $\mathbb{R}^n$, we can directly get the solution by Green function. For poisson equation $\Delta u = f$ on closed Riemannian manifold, the ...
- 729
0
votes
0
answers
61
views
Symmetric formulation for the heat equation
Originally posted on MSE
Consider the heat equation on a domain $U$: $$\partial_t u - \operatorname{div}(A\,\nabla u)=f$$
with $u(0)=0, u=0$ on the boundary of the domain of definition. Consider a ...
- 225
0
votes
0
answers
42
views
Boundary regularity for heat equation
Consider the heat equation $u_t - \Delta u=0$ with $u = u_0$ on $\partial B \times (0,T) \cup B \times \{t=0\}$. We consider weak solutions $u \in C^0(0,T;L^2(B)) \cap L^2(0,T;u_0 + W_0^{1,2}(B))$ ...
- 483
5
votes
1
answer
191
views
Convergence of heat flow on non-compact manifolds?
Consider the heat equation $\partial_t u= \Delta u+\lambda_1 u$ on a non-compact complete manifold $M$ (with nonpositive curvature) where $\lambda_1$ is the first eigenvalue and we start with some ...
- 653
1
vote
1
answer
252
views
Physical relevancy of two curious PDE's
My research has brought me to the following linear parabolic second order PDE:
$$ \frac{\partial^2}{\partial x^2}\Psi(t,x)=c(t,x)\frac{\partial}{\partial t}\Psi(t,x) $$
for $c(t,x)=-\frac{t}{x}$ and $...
- 171
1
vote
0
answers
42
views
Wellposdeness of some HJB equation
Consider the non-linear PDE for $u:[0,1]\times [-1,1]\to\mathbb R$ as follows:
$$u_t= \inf_{b\ge 1/e} \big(-b u_{xx} - \log b - 1\big), \quad \forall (t,x) \in (0,1) \times (-1,1),$$
together with the ...
- 975
1
vote
0
answers
59
views
Solution to $u_t = A(t)u + f(t)$ on bounded domain
I am dealing with the problem
\begin{align}u_t &= \nabla \cdot (a(x,t) \nabla u) + f(x,t) &\text{ on } \Omega \times (0,T)\\
\partial_{\nu} u &= 0 &\text{ on } \partial \Omega \...
- 11
2
votes
0
answers
167
views
Function is in $L^2$ . how to show that gradient is also in $L^2$?
I am dealing with diffusion-reaction equation with three species. I have $L^2$ bound of concentrations. Now I want $L^2$ bound of gradient of concentrations. Somehow if I get $L^4$ or $L^\infty$ bound ...
- 41
5
votes
1
answer
348
views
The decay rate of a degenerate heat equation in torus $\mathbb{T}^2$
Consider the degenerate heat equation on torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$:
$$ \frac{\partial}{\partial t} u(x,t)= \left( \sin^2(\pi x_1) \frac{\partial^2}{\partial^2 x_2} + \sin^2(\pi ...
- 51
4
votes
0
answers
106
views
Continuity of solutions of Elliptic PDE with respect to parameters
Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy
$$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$
where $f$ is a fast decaying smooth function.
I would like to know how the ...
4
votes
0
answers
100
views
Systems of parabolic equations -- Petrovskii's condition
Consider the flat torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$ and define the corresponding periodic-parabolic cylinder $Q_T:=(0,T)\times\mathbf{T}^d$.
Given a matrix field $A:Q_T\rightarrow\text{M}...
- 2,710
3
votes
0
answers
93
views
Comparison principle for Elliptic PDE with exponential nonlinearity
Suppose $\varphi$ is a radial (and radially decreasing) solution of
$$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$
with $ R>0$, and $\psi$ is a decreasing radial function ...
4
votes
1
answer
391
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
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$
...
- 41.8k
1
vote
0
answers
35
views
Singular asymptotic limits of mean-convex MCF
Let $(M_t \mid t \geq 0)$ be a mean-convex mean curvature flow of hypersurfaces in ambient Riemannian manifold $(N^{n+1},g)$. Brian White proved that this flow (defined 'weakly' as a level set flow ...
- 4,968
1
vote
1
answer
91
views
Fractional reaction-diffusion with Caputo derivative
I'm interested in the following Cauchy problem for a linear diffusion equation
$$
\begin{cases}
{^C}\!D^{a}_tu(t,x) = \sigma\Delta u(t,x),\\
u(0)=u_0\in X.
\end{cases}
$$
where ${^C}\!D^{\sigma}_t$
...
2
votes
1
answer
300
views
Property so that $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?
Let $f:[0, \infty)\to [0, \infty)$ be non-decreasing and satisfy for all $t>t_{0}$, $$f(t)+C\int_{t_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t_{0}}\int_{t_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ ...
- 449
2
votes
0
answers
77
views
Convergence of Green's function of fractional heat equation
For the fractional heat equation
\begin{equation} \partial_t u + (-\Delta^s)u=0 \text{ in } \mathbb{R}^d \times (0,\infty),
\end{equation} where $s \in (0,1)$ where the fractional laplacian is the ...
- 21
1
vote
0
answers
84
views
Hölder regularity in a quantitative manner
Why the investigation pertaining to sharp estimates in diffusion pde is important? In what scenarios the quantitative way reveals important nuances of a problem and play a decisive role in a finer ...
- 371
0
votes
1
answer
167
views
Looking for English version of a paper of Jean Ginibre
I am in serious need of an English translation for the following paper:
Séminaire Bourbaki by Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires
périodiques en variables d’espace, d'après ...
- 159
1
vote
0
answers
237
views
Maximal regularity heat equation
Considering the heat equation on the flat torus $\mathbf{T}^d$, we have the maximal regularity estimate
\begin{align*}
\forall \varphi\in\mathscr{D}(\mathbf{R}\times\mathbf{T}^d),\quad \|\Delta \...
- 2,710
0
votes
1
answer
89
views
Reference and hint for L^p estimates of the gradient of solutions to parabolic equation in divergence form
Considering a weak solution $u\in L^2(0,1;H^1(B_1))$ with $\partial_t u \in L^2(0,1;H^{-1}(B_1))$ to
$$\partial_t u-\operatorname{div}(A(x,t)\nabla u)=f+\operatorname{div}(F) \hspace0.5cm \text{in} \...
- 61
2
votes
0
answers
52
views
Are there results that relate the restriction of the heat semigroup in $\mathbb{R}^n$ and the semigroup in a domain?
I'm thinking about the following situation:0 suppose that
$$
S_{\Omega}(t)f = \int_{\Omega} K_{\Omega}(x,y,t)f(y)dy
$$ where $K_\Omega(x,y ,t)$ is the Dirichlet heat kernel in the domain $\Omega$. It ...
- 585
3
votes
0
answers
189
views
A non-linear PDE $v^2v_t=v_{xx}v-v_{x}^2$
PS : Indeed, there is a typo in my equation. Thanks to Zachary's observation.
Consider a PDE for $v: [0,1]^2\to (-\infty,0]$ satisfying
$$v_t(t,x) = \frac{v_{xx}(t,x)v(t,x)-v_{x}(t,x)^2}{v(t,x)^2},\...
- 975
1
vote
0
answers
65
views
Parabolic PDE: Zero now means zero anytime before
Studying some mathematical models I came across a simple-looking question that I do not know how to handle.
If we have the following problem:
$$\begin{cases} \dfrac{\partial Z}{\partial t}-\Delta Z=aZ-...
- 1,330
3
votes
0
answers
101
views
Uniqueness continuation property for parabolic equation
Consider the following parabolic equation:
$$\DeclareMathOperator{\Div}{div}
\begin{cases}
\dfrac{\partial \rho }{\partial t}-\Div\left( a\left( x\right) \nabla
\rho \right) +p(x)\rho = 0 & \...
0
votes
1
answer
155
views
Distance function to mean curvature flow
In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made. (Just for reference, this is on page 60.) For the proof—apparently standard calculations—the ...
- 4,968
4
votes
1
answer
131
views
Interior Sobolev regularity of parabolic solutions
In Evans book (and many others) there are a classic result about interior regularity in Sobolev spaces for solutions to uniformly elliptic problem (Theorem 1, p. 309). That is, let $\Omega\subset\...
- 61
2
votes
1
answer
442
views
Reference for representation of heat equation with Neumann boundary condition on smooth domain using reflected Brownian motion
We know that the solution of the heat equation $\partial_tu=\frac 12\Delta u$ with Dirichlet boundary condition $u\rvert_{\partial\Omega}=g$ is $u(t,x)=\mathbb{E}[g(B_\tau)\mid B_t=x]$, with $\tau$ ...
- 665
6
votes
0
answers
109
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
1
answer
201
views
Two dimensional oscillatory integral
I am having a little confusion in verifying the two dimensional oscillatory integral in Lemma 2.1 in This paper, namely
$$I_t (x,y) = \int_{\mathbb{R}^2} |\xi|^{\epsilon + i \beta} e^{i t(\xi^3 + \xi \...
- 159
2
votes
0
answers
121
views
On Fredholm alternative for Neumann conditions
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation $$\begin{cases}
-\Delta ...
- 1,320
2
votes
1
answer
220
views
Periodic solution for linear parabolic equation - existence, regularity
I am interested in proving the existence and regularity of solution to the following problem:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-\Delta y(t,x)+c(t,x)y(t,x)=f(t,x), & (t,x)\in (0,T)...
- 1,330
2
votes
0
answers
183
views
Boundedness for singular parabolic p-Laplace equation
Local boundedness of singular parabolic $p$-Laplace equation
$$\partial_t u - \operatorname{div}(|\nabla u|^{p-2}\nabla u)=0,\,1<p<2,$$
requires additional integrability assumption for the ...
