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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Why is density and separability needed for uniqueness of weak (time) derivatives?

Let $X,Y$ be Banach spaces with $X \subset Y$. Recall that $u \in L^1(0,T;X)$ has weak derivative $g \in L^1(0,T;Y)$ if $$\int_0^T u(t)\phi'(t) = -\int_0^T g(t)\phi(t) \qquad\forall \phi \in C_c^\...
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11 votes
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Modified energy method for transformed Fokker-Planck equation (tricky integration by parts…)

I came across Villani's paper titled "Hypocoercive diffusion operators" and couldn't figure out a computation that is skipped in that paper. Specifically, consider the following transformed ...
Fei Cao's user avatar
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1 answer
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Reference to log-transition-density of a diffusion process

Consider the scalar diffusion $X=(X_t)_{t\geq 0}$ given by $$\mathrm{d}X_t \ = \ b(X_t)\,\mathrm{d}t \ + \ \sigma(X_t)\,\mathrm{d}B_t, \qquad X_0=\xi,$$ with $b$, $\sigma$ smooth, $\xi$ absolutely ...
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2 votes
0 answers
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Uniqueness of solution to Cauchy problem with quadratic nonlinearity

Consider the non-linear differential operator $$\mathfrak{L}: \ C^2((0,T)\times\mathbb{R}^2)\ni\varphi\equiv\varphi(t,x,y) \, \mapsto \, \partial_x^2\varphi + (\partial_x\varphi)^2.$$ For $U\subset\...
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138 views

When linear strongly elliptic operators are invertible?

I am studying Pazy's book "Semigroups of Linear Operators and Applications to Partial Differential Equations" and when considering an operator like: A linear differential operator, $$A : W^{...
L.F. Cavenaghi's user avatar
1 vote
1 answer
140 views

first order derivative of the parabolic equation

Assume $b, \ell \in C_b^{1,2}(\mathbb R^2)$. We consider parabolic PDE $$(P1)\quad \partial_t v = b \partial_x v + \partial_{xx} v + \ell, \ \forall (t, x) \in \mathbb R^+\times \mathbb R; \quad v(0, ...
kenneth's user avatar
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Is there a better reference for existence/regularity for parabolic PDEs (and systems) than the book of Ladyzenskaja, Solonnikov, Uralceva?

The book of Ladyzenskaja, Solonnikov, Uralceva contains almost everything most people need yet the typesetting and notation is disgusting to the eye. Is there any better text that covers the same type ...
StopUsingFacebook's user avatar
2 votes
1 answer
102 views

Explicit solution of a free boundary problem for heat equation

Consider the free boundary problem $$ \min\{u_t - u_{xx} -1, u \} = 0 \qquad \text{ in } (0,T)\times (-1,1) \\ u(0,\cdot) = 0 \qquad \text{ in } (-1,1)\\ u(\cdot, -1) = u(\cdot, 1) = 0 \qquad \...
Zac's user avatar
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classical solution of nondegenerate HJB equation

Let $b\in C(\mathbb R)$ and $L \in C_b^2(\mathbb R)$. Consider an equation $$v_t (x, t) + \inf_{a\in A} \{b(a) v_x(x, t) + a^2 \} + v_{xx}(x, t) + L(x) = 0, \hbox{ on } \mathbb R \times (0, 1)$$ with ...
kenneth's user avatar
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improved regularization for $\lambda$-convex gradient flows

It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
leo monsaingeon's user avatar
1 vote
0 answers
82 views

Solution existence for two-dimensional parabolic PDEs

I am looking for a solution $(f,g) \in C^{1,2}([0;T]\times\mathbb R;\mathbb R^2)$ to the following PDE system $$ f_t(t,x) + a_1(t,x) f_x(t,x) + a_2(t,x) f_{xx}(t,x) - b_1 f(t,x)^2 + c(t) g(t,x) - d(t,...
Kolodez's user avatar
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437 views

Backward uniqueness for a heat equation with a drift

Consider heat equation with a drift (=reaction-diffusion equation) $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1] $$ with periodic or ...
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2 votes
0 answers
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Integral estimate for the solution of the heat equation

Let $u$ be a solution of $\partial_t u - \Delta u =f$ with initial data $u(0,x) = 0$ on $\mathbb R^N$. How do one prove the following inequality? $$ \int_0^T \int_{\mathbb R^N} \phi(f)(- \Delta) u(...
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1 vote
0 answers
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Regularity theory for parabolic PDEs in fractional Sobolev spaces

I am trying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya and the ...
George's user avatar
  • 435
0 votes
1 answer
235 views

Scaling argument for the heat equation in a bounded domain [closed]

We want to study the long time behavior of the heat equation $u_t - u_{xx} = 0$ in the domain $[-1,1]$. Now consider the rescaling $u^{\epsilon} = u(x/\epsilon, t/\epsilon^2)$. Then $u^\epsilon$ ...
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2 votes
0 answers
55 views

Stability of weak solutions to quasilinear parabolic equations

Consider the quasilinear operator $A(x,t,\nabla u)$ satisfying $$A(x,t,\nabla u).\nabla u \geq C_0 |\nabla u|^p$$ and $$|A(x,t,\nabla u)| \leq C_1 |\nabla u|^{p-1}$$ where $1<p<\infty$. Note ...
Adi's user avatar
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8 votes
1 answer
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Positive solutions for semilinear parabolic equations

Let $X$ be a Banach lattice. Consider the system $$y'(t)=Ay(t)+f(t,y(t)) \qquad \text{in } (0,T) , \qquad y(0)=y_0, \qquad (*)$$ where $T>0$, $A$ generates an analytic positive semigroup $S(t)$ on $...
S. Euler's user avatar
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1 vote
0 answers
142 views

Rigorous error estimate for semi-discrete heat equation in bounded domain

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^N$ and $u_h$ be a solution of $$ \begin{cases} \partial_t u_h -\Delta_h u_h = f(x) & \text{ in } \Omega_h\\ u_h=0 &\text{ in } \...
Kei's user avatar
  • 267
3 votes
2 answers
876 views

Gradient $L^p$ estimates for heat equation

I'm looking for a proof of the gradient estimate associated to the heat equation with Dirichlet boundary conditions, to see if I can express the constant $\color{red}{C}$. $$\|e^{t\Delta_d}f\|_{W^{1,...
Malik Amine's user avatar
1 vote
0 answers
74 views

Very weak solution to parabolic PDE (pointwise a.e. in time with time derivative on test function)

Consider the parabolic PDE $$u' + Au = 0$$ as an equality in $L^2(0,T;V^*)$ for some Hilbert space $V$ with $A\colon L^2(0,T;V) \to L^2(0,T;V^*)$ a coercive, bounded linear operator. Here $u'$ is the ...
MMML's user avatar
  • 107
2 votes
1 answer
357 views

Regularity on the boundary for the heat equation with linear source

This is probably a known problem but I was not able to find exactly what I am looking for. I have the following linear heat equation with zero-flux boundary conditions: \begin{equation} \begin{cases}...
MathMax's user avatar
  • 203
2 votes
1 answer
215 views

Continuity of solution of a parabolic PDE w.r.t. system parameters

If we have a system of PDE of the form: $$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)=D\Delta y+F(t,x,f(x),y) ,\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial \nu}(t,x)=0,\ (t,x)\in (...
Bogdan's user avatar
  • 1,330
2 votes
1 answer
449 views

Some properties of fractional Dirichlet heat kernel

Let $\Omega\subset \mathbb{R}^n (n\geq 2)$ be a bounded open domain with smooth boundary $\partial\Omega$. Consider the fractional heat equation with Dirichlet boundary condition: \begin{equation} \...
pxchg1200's user avatar
  • 265
4 votes
2 answers
357 views

Angle of analyticity of semigroup

Is there any known parabolic PDEs in the literature where the angle of analyticity of the associated semigroup is $<\pi/2$ ? For example, the angle of heat semigroup in $L^2$ is exactly $=\pi/2$. ...
Sigma's user avatar
  • 97
2 votes
1 answer
420 views

Existence of weak solutions of a parabolic PDE

Assume that $\Omega\subset\mathbb{R}^n(n\geq3)$ is a bounded open set with smooth boundary, $\varphi\in H^1_0(\Omega)$, $F(t)$ is differentiable in $\mathbb{R}$ and $F'$ is bounded. Given a PDE $$ \...
张旭辉's user avatar
2 votes
1 answer
284 views

An inequality for abstract Cauchy problem

Consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x_0$, where $A$ generates an analytic $C_0$-semigroup on a Banach space $X$. How we can prove an inequality of ...
Sigma's user avatar
  • 97
1 vote
0 answers
99 views

Confusion optimal control abuse notation

I'm currently reading this paper describing a numerical scheme for the approximating optimal policy of a stochastic control problem. However, I run into a confusion directly on the first page where ...
ABIM's user avatar
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3 votes
1 answer
228 views

Inequality for initial data

I'm wondering if there is any idea to bound the norm of the initial data by the norm of corresponding solution? To make it clear, consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,...
Sigma's user avatar
  • 97
3 votes
1 answer
659 views

Reference request: Schauder estimate in the space variable for parabolic equations

Setting: Let $(M,g)$ be a compact Riemannian manifold without boundary. Let $\Delta_g$ be the Laplacian and $L=\Delta_g-\partial_t$ the heat operator. Let $0<\alpha<1$, $0<t_0<T$. Let $$u\...
Yuxiao Xie's user avatar
1 vote
1 answer
197 views

Analytical solution to inhomogeneous parabolic PDE [closed]

I would be thankful to anyone who can present an analytical solution to the following inhomogeneous PDE equation: $$\frac{\partial{u}}{\partial{t}}= \alpha\frac{\partial^2{u}}{\partial{x^2}}-ku$$ $$...
Behzad Lachini's user avatar
2 votes
1 answer
386 views

Leray projector in $L^{\infty}$ and negative order Besov spaces for the Navier-Stokes equations

I was reading the paper "Norm inflation for the generalized Navier-Stokes equations" which can be found here: https://arxiv.org/abs/1212.3801. In Lemma 2.1, the authors said for any $\phi \in L^{\...
Oguz's user avatar
  • 43
4 votes
0 answers
107 views

Biharmonic heat flow on compact manifolds

Consider $\partial _t u (t,x) = -\partial _x ^4 u$ on a compact manifold, or even a special specific one like the torus. Are there any estimates on the Green function (bihamornic heat kernel), for ...
Amir Sagiv's user avatar
  • 3,544
5 votes
1 answer
466 views

Lack of exponential $L^2_{t,x}$ decay for a heat equation with growing coefficients

Edit: I have changed the nature of the question, but in order to have a better idea of what I can expect for the original problem (see below). Given $T>0$ and $n \in \bf Z$, consider the following ...
char's user avatar
  • 309
6 votes
0 answers
174 views

Reference request: Existence and regularity for parabolic PDEs with smooth coefficients on compact manifolds with boundary

I'm looking for a reference for a statement like: Let $M$ be a $n$-dimensional smooth compact manifold with smooth boundary $\partial M$. In coordinates, let $\mathcal L$ have the form $\mathcal L ...
Nathanael Schilling's user avatar
1 vote
1 answer
86 views

A symmetric parabolic second order PDE

Here I want to solve a second order PDE symmetrically depending on two variables $$ 3(\partial_{\alpha\alpha}f+\partial_{\beta\beta}f-2\partial_{\alpha\beta}f)+2(\cot\alpha)\partial_\alpha f +2(\cot\...
Iew's user avatar
  • 121
5 votes
3 answers
514 views

Reference request: Long-term behaviour of the heat equation for bounded initial data

Let us consider the heat equation \begin{align*} \frac{\partial}{\partial t}u(t,x) & = \Delta u(t,x), \\ u(0,x) & = f(x) \end{align*} on the whole space $\mathbb{R^d}$. If $f \in L^p := L^...
Jochen Glueck's user avatar
2 votes
0 answers
33 views

Discrete maximum priniciple for parabolic operators

While reading a paper on the topic 'Numerical solutions for generalized Black-Scholes equation', It is given that their numerical scheme can be executed explicitly by solving a linear system $\mathbf ...
RIYASUDHEEN TK's user avatar
4 votes
0 answers
602 views

Maximum Principles in Parabolic PDE with Neumann Condition

I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in ...
Bogdan's user avatar
  • 1,330
4 votes
1 answer
212 views

Interesting questions for inverse parabolic problems

I'm looking for some interesting questions and maybe open problems in inverse problems theory, especially in the framework of parabolic PDEs (basically the heat equation). As key words here we can ...
S. Maths's user avatar
  • 561
1 vote
1 answer
121 views

Uniform Hopf Inequality

There is a Uniform Hopf Inequality as follow: Let $\Omega \subset \mathbb{R}^n$, $n \geq 1$ denote a smoothly bounded domain. Also let $\rho(x)=\mathrm{dist}(x,\partial \Omega)$, the distance ...
Hheepp's user avatar
  • 361
4 votes
3 answers
608 views

Question about the regularity of fractional Heat equation

Let $\Omega$ be a bounded smooth domain of $\mathbb{R}^n$, $0<s<1$ and $(-\Delta)^s$ denotes the restricted fractional Laplacian. Let consider the following fractional Heat equation: ‎$‎‎$‎ ‎\...
Hheepp's user avatar
  • 361
3 votes
0 answers
41 views

Reference request: existence of strong solutions to a linear parabolic problem with mixed boundary conditions

on a domain $\Omega \subset \mathbb{R}^d$ with smooth boundary $\partial\Omega$ subdivided into two parts $\Gamma_D$ and $\Gamma_N$ I am considering the parabolic problem $$ \partial_t u = \Delta u + ...
jfp's user avatar
  • 51
2 votes
0 answers
102 views

Energy functional continuous with respect of time $t$

I am studying a paper of Liu Yacheng which named "On potential wells and applications to semilinear hyperbolic equations and parabolic equations" it considers a nonlinear parabolic equation \begin{...
pxchg1200's user avatar
  • 265
2 votes
0 answers
59 views

Existence and uniqueness for semilinear problem

Consider the following problem: $$-\Delta u + [(u)^+]^\alpha = 0,$$ where $(\cdot)^+$ is the positive part function and $\alpha >0$. How does the theory of monotone operators provide existence ...
user avatar
1 vote
1 answer
246 views

PDE's : diffusion equation : polynomial diffusion coefficient

I'd like to find analytical solutions of that kind of differential equations : $$\partial_t c = \partial_x (D(c)\partial_x c) $$ with $D(c)$ a polynomial. The trivial cas $D(c)=a$ with $a$ a ...
J.A's user avatar
  • 121
5 votes
1 answer
330 views

Exponential decay of solution in $L^p$ with $p>2$

Consider the following evolution equation $$u_t=\Delta u$$ in a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$, with smooth initial conditions $u_0\geq 0$ and homogeneous Dirichlet ...
David Lingard's user avatar
1 vote
2 answers
112 views

Question on Parabolic PDEs: Improvement of the bound ${\vert \vert v(\cdot,t)-v_0(\cdot) \vert \vert}_{L^p}\le \hat C t^{1/q}$

Let $M$ be a $C^3-$compact manifold and $v \in W^{2,1}_p(M\times [0,T])$ ($1\le p<\infty$) be the solution of: $\begin{cases} \partial_t v-\Delta_{M} v=f(v), \quad M\times [0,T]\\ v(x,0)=v_0,...
kaithkolesidou's user avatar
2 votes
1 answer
593 views

Continuous embedding between parabolic Sobolev spaces

I was wondering whether the following continuous embedding theorem for parabolic Sobolev space is correct? Let $I=[0,T]$ and $\Omega$ be a sufficiently smooth domain in $\mathbb{R}^n$, we consider ...
John's user avatar
  • 483
1 vote
0 answers
90 views

Name for a Particular (Parabolic) PDE

This is a cross post from MSE. The original question can be found here:https://math.stackexchange.com/questions/3248114/name-for-a-particular-parabolic-pde Consider the following initial value ...
rubikscube09's user avatar
2 votes
0 answers
58 views

Time derivative in parabolic Hölder spaces

Let $\Omega$ be a regular open set in $\mathbb{R}^n$ and $T>0$. Let $C^{\frac{1+\alpha}{2};1+\alpha}([0,T]\times \overline{\Omega})$ be the space of functions $f$ which are $\frac{1+\alpha}{2}$-...
foo90's user avatar
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