# 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 ...
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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 ...
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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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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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 $(e^{s\... 3 votes 1 answer 39 views ### How does the mean value theorem with Hölder seminorms depend on dimension? I am reading through Krylov's Lectures on Elliptic and Parabolic Equations in Hölder Spaces. In an attempt to prove the interpolation inequalities for parabolic PDEs, I've stepped back in the text to ... 2 votes 0 answers 72 views ### Is the solution of the stochastic heat equation a Gaussian process? We know that the solution of a linear SDE is a Gaussian process. I wonder whether the same is true for a SPDE casted to an infinite-dimensional SDE on a Hilbert space$H$like $${\rm d}X_t=f(t)AX_t+\... • 13.9k 1 vote 1 answer 33 views ### How do we show that on a parabolic Hölder space, the polynomial and Hölder seminorms are equivalent? I am currently working through Krylov's Lectures on Elliptic and Parabolic Equations in Holder Spaces. One of the key points of chapter 8 is that the two seminorms$$[u]_{1+\delta/2,2+\delta;U} := \... 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:(...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 \partial_t u_n -\Delta u_n = f_n \text{ on } [0,T] \times \mathbb{T}^4 \text{ ...
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### 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'...
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### 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 ...
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### 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 ...
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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 $$\max _{\bar{U_T}} ... • 800 2 votes 1 answer 98 views ### Reference to a Theorem (or book) about parabolic PDEs I am studying free-boundary problems, and I see that many authors face the following standard problem from the theory of parabolic PDEs: Given a domain [x_1, x_2] \times [t_0, t_1), find a function ... • 198 0 votes 0 answers 104 views ### Exponential decay for a smooth solution to a parabolic PDE Consider the following problems. Problem 1. Let \Omega \subset \mathbb{R}^n be open and bounded with smooth boundary, L an elliptic operator defined by$$Lu=-D_j(a^{ij}D_iu) where $a^{ij}=a^{ji}$ ...
Let us consider the linear parabolic PDE: $$\partial_t u - \Delta u=f \text{ and }u(0)=u_0$$ where $u_0 \in L^2(\mathbb{R})$ and $f \in L^2_t([0,T], L^2_x(\mathbb{R}))$. ...