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Questions tagged [parabolic-pde]

This tag is for questions relating to "Parabolic partial differential equation", are usually time dependent and represent diffusion-like processes. Solutions are smooth in space but may possess singularities. However, information travels at infinite speed in a parabolic system.

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Evans - existence of parabolic PDE, why does $B[u_m,v;t]\to B[u,v;t]$?

In Evans book, chapter 7.1, he establishes existence of weak solutions of $$\partial_t u + Lu = f$$ where $Lu:= -\nabla\cdot (A\nabla u) + b\cdot\nabla u + cu$. He first shows that for any $m$, the ...
l'étudiant's user avatar
1 vote
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semilinear transport equation

Good morning everyone! I have to solve the following transport equation for the temperature $T(x,t)$: $$ \frac{\partial T}{\partial t} + v \frac{\partial T}{\partial x} = k(T_a - T) $$ with boundary ...
Gregorio Coletti's user avatar
2 votes
0 answers
85 views

On fundamental solutions of parabolic PDEs

Let $U$ be an open bounded subset of $\mathbb{R}^n$ and consider the PDE in $U \times [0, T] \subset \mathbb{R}^n \times \mathbb{R}$, given by $$ \begin{cases} \partial_t \rho(x, t) = \frac{1}{\alpha} ...
mathusername's user avatar
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Convergence in comparison principles of parabolic pdes

I assume that the following equations all have sufficiently smooth strong solutions. I have demonstrated that the solution to the (imaginary-time) Schrödinger equation: \begin{equation*} \begin{...
J.J.Zou's user avatar
1 vote
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Theorem 2.12 in Lieberman's Second Order Parabolic Differential Equations

In Theorem 2.12 of Liberman's textbook the following result is established. Consider a space-time domain $\Omega\subseteq \mathbb{R}^n \times \mathbb{R}$ ($\Omega$ is not necessarily of cylinder shape ...
Tibeku's user avatar
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4 votes
2 answers
110 views

Does smoothness of solution to parabolic equation require smoothness of coefficients?

I have a function that solves a parabolic partial differential equation $$ \partial_tu - Lu = 0 $$ with a linear second order uniformly elliptic-in-space differential operator $L$, whose coefficients ...
Bananach's user avatar
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1 vote
0 answers
40 views

Time regularity vs space regularity for parabolic PDE

Suppose that there exist separable Hilbert spaces $V, H, X$ such that $V\hookrightarrow H\hookrightarrow X\hookrightarrow V'\,$ continuously, where $V'$ denotes the dual of the Hilbert space $V$. Let ...
Azam's user avatar
  • 31
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Variational calculus with constraints of boundary conditions. How to take into account

I am looking for references to understand the following. I've recently solved the thin plate functional minimization subjected to interpolation constraints. The calculations I did are mostly here: ...
user8469759's user avatar
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60 views

Viscosity solution $u$ and the equation of $e^t u$

The function $u_0$ is called the viscosity subsolution of the equation $$\partial_t u+\Delta u=0$$ with $u(x,0)=u^*(x)$ if for any $(x_0,t_0)\in\mathbb{R}^d\times[0,T]$ and any test function $\varphi \...
mnmn1993's user avatar
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1 answer
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Solution of the parabolic PDE using Green's function

Green's function for the parabolic PDE is defined as: $$\Delta G(\vec{x},t,\vec{\xi},\theta)=\delta(\vec{x}-\vec{\xi},t-\theta)$$ Where $G$ satifies the homogeneous initial and boundary conditions. ...
Krum Kutsarov's user avatar
1 vote
1 answer
128 views

Initial and boundary conditions for parabolic PDEs

Consider a parabolic PDE of the form \begin{align} \phi_t=f\left(\phi,\phi_x,\phi_{xx} \right), \end{align} with $f$ some reasonable function, and if needed for the argument below linear $\phi_{xx}$. ...
DanielKatzner's user avatar
4 votes
0 answers
119 views

Heat Equation : Can a singularity develop away from origin?

Consider the following mixed problem for a radial, nonlinear, 2D-heat equation $$ \begin{cases} u_t = u_{rr} + \dfrac{1}{r}u_r + F(t,r,u,u_r), \quad (t,r) \in [0,+\infty) \times [0,1] \\ u(0,r) = f(r) ...
Desura's user avatar
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Newton forward and Forward Euler method

in the article "Numerical studies on 2-dimensional reaction-diffusion equations" (Tang/Qin/Weber 1993?) i found the following: we look for $u(x, y, t)$ satisfying $$ \frac{\partial u}{\...
IHEM's user avatar
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1 answer
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solution to a moving boundary heat equation

I was reading Rubinstein's book 'The Stefan Problem'(https://www.scirp.org/reference/referencespapers?referenceid=1620640) where in chapter2, page 98 he gave an analytical solution to the 1-d moving ...
Maskoff's user avatar
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1 answer
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Confusion about one initial/boundary value problem for heat equation

This is a follow-up question to this. The referenced question arose while I was trying to solve $$ \begin{cases} u_t = \frac{1}{2} \Delta u, & x \in X, \\ u ( 0, x ) = 1, & x \in X, \\ u ( t, ...
tsnao's user avatar
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Difference of boundary conditions between elliptic and parabolic PDEs

For elliptic PDEs we know that the boundary condition no matter Dirichlet, Neumann or mixed, needs to be defined over the whole boundary of the given volume $\partial V$. Only then we will get a ...
Krum Kutsarov's user avatar
0 votes
0 answers
11 views

Regularity of difference of fundamental solutions to heat equation

Let $K_t^{(1)}$ and $K_t^{(2)}$ be two fundamental solutions to the heat equation $\partial_t-\Delta$ on a non-compact Riemannian manifold $(M,g)$, i.e. $$(\partial_t -\Delta_x)K_t^{(i)}(x,y)=0\,,\...
crimsonmist's user avatar
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14 views

Linear differential expression $L(x,t,D,D_t)$

The following is taken from "Parabolic Boundary Value Problems" by Eidelman. Let \begin{equation}L(x,t,D,D_t)\hspace{4cm}(1)\end{equation} be a linear differential expression of an arbitrary ...
Seurat's user avatar
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1 vote
2 answers
78 views

Maximum principle of strong solution of linear parabolic equation in $\mathbb{R}\times [0,T]$

Suppose that there is a strong solution $u(x,t)\in W^{2,1}_{2,loc}(\mathbb{R}\times [0,T])$ solving the linear parabolic equation $$-\partial_t u +\partial_x^2 u +b(x,t)\partial_x u+c(x,t) u =0\quad\...
mnmn1993's user avatar
  • 395
4 votes
0 answers
99 views

Feynman-Kac theorem of the weak solution of parabolic PDEs

Is there any reference on the Feynman-Kac theorem of the weak solution of parabolic PDEs? So far I can only find the one for classical solution.
mnmn1993's user avatar
  • 395
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0 answers
53 views

Matlab code for solving inhomogeneous 1D heat equation using Crank-Nicolson

I'm trying to solve the 1D heat equation using Crank-Nicolson in Matlab. The problem is, my numerical solution is not the same as the exact solution. (I already checked the plots and their shapes are ...
Kim Juhee's user avatar
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33 views

Numerical solution of Perona-Malik equation: How to handle the boundary properly?

In the paper Perona-Malik equation and its numerical properties, the following PDE is considered: The $u_0$ I'm (and so is the author) interested in is given by an image and hence decomposes into ...
0xbadf00d's user avatar
  • 13.9k
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0 answers
19 views

Existence and regularity of solutions to parabolic equation on domain with disjoint boundary

Let $\Omega$ be an open set with boundary such that $\partial\Omega = \Gamma_1 \cup \Gamma_2$ where $\Gamma_1$ and $\Gamma_2$ are closed and smooth but disconnected, in particular think of $\Omega$ to ...
C_Al's user avatar
  • 670
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0 answers
29 views

Spatial discretization of the PDE $\partial_tu=\nabla\cdot\kappa(t,\nabla u)\nabla u$ for image processing

The question is basically in the title. I want to evolve an image $u_0$ (i.e. a $n_1\times n_2$ resolution set of discrete values in $[0,1)$, which is a discrete function on $\{0,\ldots,n_1-1\}\times\{...
0xbadf00d's user avatar
  • 13.9k
1 vote
0 answers
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Math-Bio: Reference/Example needed for a reaction-diffusion system coupled to a conservation law

DISCLAIMER: I also made a post about it in biology stack exchange but I'm not sure whether that forum is the most appropriate. Hence I thought to share my question here as well. While looking at this ...
kaithkolesidou's user avatar
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53 views

solve a parabolic PDE backwards in time

I am trying to solve the following equations: \begin{equation*} \begin{aligned} -\frac{\partial u}{\partial t}&=\Delta u+Cu+Z, (x,t)\in\Omega_T\\ \frac{\partial u}{\partial x}&=0, x\...
79999's user avatar
  • 167
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Is there an analytical solution for the three-dimensional time-dependent parabolic equation (using Green's functions)?

problem: $\frac{\partial w}{\partial t}=a_1(x,t) \frac{\partial^2 w}{\partial x^2}+\Phi(x, y, z, t)$ with $w=f(x, y, z) \quad$ at $\quad t=0$.(initial condition) $\frac{\partial w}{\partial x}+k(\...
Yilin Cheng's user avatar
1 vote
0 answers
20 views

Pointwise convergence of heat kernels on perturbed Riemannian manifolds

Thank you in advance for your comments! Let $(M,g)$ be a Riemannian manifold (in general non-compact, connected). Then the associated Dirichlet Laplacian $\Delta_g$ generates the heat semigroup $(e^{s\...
crimsonmist's user avatar
3 votes
1 answer
39 views

How does the mean value theorem with Hölder seminorms depend on dimension?

I am reading through Krylov's Lectures on Elliptic and Parabolic Equations in Hölder Spaces. In an attempt to prove the interpolation inequalities for parabolic PDEs, I've stepped back in the text to ...
IdenticallyEulerian's user avatar
2 votes
0 answers
72 views

Is the solution of the stochastic heat equation a Gaussian process?

We know that the solution of a linear SDE is a Gaussian process. I wonder whether the same is true for a SPDE casted to an infinite-dimensional SDE on a Hilbert space $H$ like $${\rm d}X_t=f(t)AX_t+\...
0xbadf00d's user avatar
  • 13.9k
1 vote
1 answer
33 views

How do we show that on a parabolic Hölder space, the polynomial and Hölder seminorms are equivalent?

I am currently working through Krylov's Lectures on Elliptic and Parabolic Equations in Holder Spaces. One of the key points of chapter 8 is that the two seminorms $$[u]_{1+\delta/2,2+\delta;U} := \...
IdenticallyEulerian's user avatar
0 votes
1 answer
30 views

Spatial analyticity for solutions of linear parabolic PDEs

I believe that the following result is true, but am having a hard time finding a suitable reference to convince myself: Suppose $X:\mathbb{R}^n\to \mathbb{R}^n$ is analytic. If the smooth function $u:(...
Timothy Buttsworth's user avatar
0 votes
1 answer
66 views

Heat semigroup is self-adjoint

Consider a closed Riemannian manifold $(M,g)$ and the heat equation $\partial_tu = \Delta_g u$ on it. Let $P_t$ be the heat semigroup generated by the equation. Of course the laplacian is symmetric ...
theflame's user avatar
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0 answers
25 views

Odd extension on two-dimensional parabolic PDE with boundary conditions

When solving a parabolic PDE with boundary conditions (e.g. the reflection method of the barrier option), we can change the parabolic PDE into a heat equation by applying some substitution. For ...
sssssoku's user avatar
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0 answers
29 views

Higher regularity for linear parabolic equation with time depndent coefficient

I am looking for a higher regularity result for the solution of the problem $$\partial_t u+div(-A(t,x)\nabla u)=f$$ in a bounded smooth domain $\Omega$ with ...
user1288096's user avatar
0 votes
0 answers
20 views

Phase speed of backward-time, central-space scheme

I'm studying the backward-time, central-space (BTCS) scheme $$u_{j}^{n+1} + \frac{k}{2h}[u_{j+1}^{n+1} - u_{j-1}^{n+1}] = u_j^n,$$ for $k$ the step size in time and $h$ the step size in space. I have ...
jackyooo's user avatar
  • 149
3 votes
1 answer
119 views

Showing the integral of the solution to a heat equation is constant

I'm trying to prove the fowllowing problem: Let $u(x, t)$ be the solution to the heat equation $$ \begin{aligned} u_t - \Delta u &= 0\;\;\;\; \text{in} \;\mathbb{R}^d\times\mathbb{R}_+\,,\\ u(x, 0)...
Ayanamiprpr's user avatar
0 votes
0 answers
18 views

Well-posedness of parabolic differential equations with Robin boundary condition

I'm considering the following parabolic PDE \begin{equation} \frac{\partial u}{\partial t}+\frac{1}{2} \sum_{i, j=1}^d a^{i j}(t, x) \frac{\partial u}{\partial x^i \partial x^j}+\sum_{i=1}^d b^i(t, x) ...
Xuz's user avatar
  • 1
0 votes
0 answers
109 views

Solving heat equation with Lax-Milgram theory

I'm interested in proving the existence of parabolic equations using Lax-Milgram theory, and I saw this question: Lax-Milgram and the existence of solution to parabolic equation. I have read the book ...
Zhang Yuhan's user avatar
1 vote
1 answer
85 views

Use of Poincaré Inequality in Parabolic Equations a Priori Estimates

I am working through a a priori estimates related to parabolic equations, and I've encountered a conceptual challenge regarding the use of the Poincaré inequality. The context and steps I'm referring ...
Luigi's user avatar
  • 75
0 votes
1 answer
41 views

Reference request: fractional heat equation $\partial_t u=(-\Delta)^s u$ with $s>1$

I was wondering if there are any good reference on the fractional heat equation $\partial_t u=(-\Delta)^s u$ with the fractional exponent $s>1$. I have found many references in the case $0<s<...
DDG's user avatar
  • 23
0 votes
0 answers
24 views

How to find numerically $\int_{0}^{1} w \langle \mathbf{p}, \mathbf{p}'\rangle \ du$ such $\nabla^2 w=0$ without solving the PDE for $w$ first?

Description Let $w(x, \ y)$ be a function that satisfies laplace's equation $$\nabla^2 w = 0 \ \ \ \ \ \ \text{on} \ \Omega$$ $$\dfrac{\partial w}{\partial n} = \langle \mathbf{p}, \ \mathbf{t}\...
Carlos Adir's user avatar
  • 1,300
0 votes
0 answers
26 views

If $\partial_t u_n -\Delta u_n = f_n$ , $\partial_t u -\Delta u =f$ and $f_n \to f$ in $L_t^p L_x^q$, do we have $u_n \to u$ in $W^{1,p}_t L^q_x$?

The question is summarized in the title. For each $n \in \mathbb{N}$, let us consider the Cauchy problem \begin{equation} \partial_t u_n -\Delta u_n = f_n \text{ on } [0,T] \times \mathbb{T}^4 \text{ ...
Keith's user avatar
  • 7,829
0 votes
0 answers
18 views

Existence of solution to a parabolic bvp

I should study some parabolic PDEs, but I'm not an expert, so I would like to ask your advice. First, could you give me some useful references concerning PARABOLIC PDES? I started reading DiBenedetto'...
mathmagic's user avatar
0 votes
1 answer
68 views

The exact, formal definition of a prabolic problem/equation [closed]

While searching for information about properties of parabolic problems, I stumbled upon a publication titled "Study of Nonlinear Parabolic Problems". This made me wonder why the writers ...
Ada Az's user avatar
  • 89
1 vote
0 answers
50 views

numerically compute eigenfunctions of $a(u,v)=\langle f(\nabla u)\nabla u,\nabla v\rangle_{L^2}$

Let $D:=(0,1)^2$ and consider the nonnegative form $$a(u,v):=\langle f(\nabla u)\nabla u,\nabla v\rangle_{L^2(D;\:\mathbb R^2)}$$ for $u,v\in L^2(D)$ where $f:\mathbb R^2\to(0,\infty)$ is a smooth ...
0xbadf00d's user avatar
  • 13.9k
0 votes
0 answers
117 views

Strong Maximum Principle for the heat Equation - Evans - Theorem 4 - Why do we need U to be bounded.

I had a doubt in following theorem of Evans THEOREM 4 (Strong maximum principle). Suppose $u \in C^2_1(U) \cap C(\bar{U})$ is satisfies $u_t + \Delta u = 0$ within $U_T$. (i) Then $$ \max _{\bar{U_T}} ...
pde's user avatar
  • 800
2 votes
1 answer
98 views

Reference to a Theorem (or book) about parabolic PDEs

I am studying free-boundary problems, and I see that many authors face the following standard problem from the theory of parabolic PDEs: Given a domain $[x_1, x_2] \times [t_0, t_1)$, find a function $...
george's user avatar
  • 198
0 votes
0 answers
104 views

Exponential decay for a smooth solution to a parabolic PDE

Consider the following problems. Problem 1. Let $\Omega \subset \mathbb{R}^n$ be open and bounded with smooth boundary, L an elliptic operator defined by $$Lu=-D_j(a^{ij}D_iu)$$ where $a^{ij}=a^{ji}$ ...
TryingToLearn's user avatar
0 votes
0 answers
43 views

Tricky confusion with the definition of a weak solution for the linear parabolic equations

Let us consider the linear parabolic PDE: \begin{equation} \partial_t u - \Delta u=f \text{ and }u(0)=u_0 \end{equation} where $u_0 \in L^2(\mathbb{R})$ and $f \in L^2_t([0,T], L^2_x(\mathbb{R}))$. ...
Keith's user avatar
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