# 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.

444
questions

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### 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 ...

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### Traveling wave solutions for epidemic mathematical models

We know that traveling wave solutions are studied in order to obtain the speed of dengue dissemination in partial differential equation systems related to epidemic diseases such as Dengue. On the ...

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### 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)=\...

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### 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 $\varepsilon>0$, ...

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### 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 $f\in H^{-1}(\Omega)$...

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0
answers

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### 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),...

6
votes

2
answers

217
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### 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}...

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### 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 ...

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0
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### 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 ...

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### 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(...

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0
answers

83
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### 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 ...

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54
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### 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 ...

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0
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### 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 $u \in C^0(0,T;L^2(B)) \cap L^2(0,T;u_0 + W_0^{1,2}(B))$ ...

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### 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 ...

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1
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226
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### 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 $...

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0
answers

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### 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 ...

1
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0
answers

54
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### 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 \...

2
votes

0
answers

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### 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 ...

5
votes

1
answer

293
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### 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 ...

4
votes

0
answers

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### 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
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0
answers

58
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### 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}...

3
votes

0
answers

86
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

355
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
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$
...

1
vote

0
answers

24
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 ...

1
vote

1
answer

82
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

284
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### 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 $0<\gamma<1$ ...

2
votes

0
answers

69
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 ...

1
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0
answers

80
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 ...

0
votes

1
answer

159
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### 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 ...

1
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0
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157
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### 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 \...

0
votes

1
answer

53
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### 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} \...

2
votes

0
answers

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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 ...

3
votes

0
answers

184
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},\...

1
vote

0
answers

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### 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-...

3
votes

0
answers

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### 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

0
answers

39
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### Evolution operator pointwise continuity

For any $f\in L^{\infty}(\Omega)$ denote by $U(\tau, t)f$ the evolution operator, i.e. the solution of the following problem (I choose this logistic equation one because it has bounded solution on $(0,...

0
votes

1
answer

108
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### 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
votes

1
answer

84
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### 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\...

1
vote

1
answer

146
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### 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 $\tau$ ...

6
votes

0
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100
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### 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

174
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### 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 \...

2
votes

0
answers

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### 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 ...

2
votes

1
answer

116
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### 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)...

2
votes

0
answers

165
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### 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 ...

4
votes

1
answer

248
views

### Strong positivity of Neumann Laplacian

There are many places in the literature where the positivity of some semigroups is treated. However I did not know anyone which states and proves the strong positivity even for the basic semigroups ...

4
votes

1
answer

139
views

### $L^2$ norm for solutions of evolution equations driven by different elliptic operators

Let $u$ be a solution of the heat equation
$$u_t - \Delta u = 0, \qquad t >0, \ x \in \mathbb T^d$$
and $v$ be a solution of the bi-harmonic heat equation
$$v_t +\Delta^2 v = 0, \qquad t >0, \ x ...

1
vote

0
answers

48
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### Elliptic principal eigenfunction analysis for Langevin dynamics with a varying source term

Consider the Kolmogorov forward equation for a Langevin dynamic:
$$\DeclareMathOperator{\Div}{div}
\begin{cases}
\dfrac{\partial}{\partial t} f = \Delta f + \Div(f\nabla V)\\
\\
\displaystyle\int_{\...

4
votes

0
answers

92
views

### The logistic elliptic equation

Studying the Fisher-KPP evolution equation I came across the steady state elliptic problem which can be written in the following form:
$$
\begin{cases} -d\Delta Y(x)=r(x)Y(x)\left (1-\dfrac{Y(x)}{K(x)}...

3
votes

2
answers

214
views

### Change of variables for obtaining a unitary group

Consider the following NLS:
$$i u_t + \Delta u- 2 \operatorname{Re} u = F(u),$$
where $F(u):=(u + \bar{u} + |u|^2)u.$
In Scattering for the Gross–Pitaevskii equation, the authors S. Gustafson, K. ...

0
votes

0
answers

45
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### What is the deep logic for the resonance function of dispersive nonlinear PDEs

I have been studying some nonlinear dispersive PDEs since some months and I was able to understand some results related to well-posedness. However, I do not feel like I am fully understand the logic ...