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.

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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 ...
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FEM based solution to parabolic problem

Consider the problem $$ \begin{cases} u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega \end{cases} $$ ...
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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(...
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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 ...
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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 ...
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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 ...
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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 $...
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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 $$\...
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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}...
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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 ...
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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 ...
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$L^p$ 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 ...
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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(\...
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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\...
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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 ...
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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, ...
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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. $...
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4 votes
1 answer
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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 ...
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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,...
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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& ...
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3 votes
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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 \...
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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 ...
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1 vote
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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: $$\...
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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 ...
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1 vote
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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 $u(x,0)=x^{1/a}$ ...
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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'...
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2 votes
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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 $...
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1 vote
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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 \...
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4 votes
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$\mathcal{C}^1(\overline{\Omega})$ 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{...
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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 ...
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1 answer
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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 ...
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2 votes
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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, &...
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1 vote
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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}...
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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 $C^{-1}\le f(x)\le C$, ...
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1 vote
1 answer
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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,\...
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2 votes
1 answer
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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 $p(0,\cdot)=\rho$...
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2 votes
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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 ...
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5 votes
1 answer
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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\{\...
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3 votes
1 answer
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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{...
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2 votes
1 answer
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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 $$\...
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2 votes
1 answer
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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 ...
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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) :\...
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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 \...
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2 votes
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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]...
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2 votes
2 answers
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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} + ...
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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 $...
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3 votes
1 answer
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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....
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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 ...
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3 votes
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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\...
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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 ...
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