# 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.

286
questions

**2**

votes

**1**answer

50 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 \...

**1**

vote

**0**answers

26 views

### 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 ...

**2**

votes

**0**answers

42 views

### 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 ...

**1**

vote

**0**answers

61 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,...

**5**

votes

**1**answer

128 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 ...

**0**

votes

**0**answers

112 views

### continuity with respect to the diffusion coefficient of the solution of a semilinear parabolic equation

Let $\Omega \subset
%TCIMACRO{\U{211d} }
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n},n\geq 1$ be an open bounded subset has a boundary $\Gamma $ of class $%
C^{2}$, $Q=\Omega \times \left( 0,T\...

**2**

votes

**0**answers

127 views

### 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(...

**1**

vote

**0**answers

48 views

### 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 existence ...

**0**

votes

**1**answer

71 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
$u^\epsilon$ ...

**2**

votes

**0**answers

34 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 ...

**8**

votes

**1**answer

173 views

### 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 $...

**1**

vote

**0**answers

117 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 } \...

**3**

votes

**2**answers

256 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,...

**1**

vote

**0**answers

30 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 ...

**2**

votes

**1**answer

202 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}...

**2**

votes

**1**answer

102 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 (...

**1**

vote

**1**answer

129 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}
\...

**4**

votes

**2**answers

297 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 $=\pi/2$. ...

**2**

votes

**1**answer

121 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
$$
\...

**2**

votes

**1**answer

267 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 ...

**2**

votes

**0**answers

79 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 ...

**3**

votes

**1**answer

215 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,...

**2**

votes

**1**answer

109 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\...

**1**

vote

**1**answer

87 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:
$$\frac{\partial{u}}{\partial{t}}= \alpha\frac{\partial^2{u}}{\partial{x^2}}-ku$$
$$...

**1**

vote

**1**answer

129 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^{\...

**4**

votes

**0**answers

75 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 ...

**4**

votes

**1**answer

283 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 ...

**5**

votes

**0**answers

99 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 ...

**1**

vote

**1**answer

72 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\...

**5**

votes

**3**answers

202 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^...

**1**

vote

**0**answers

24 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 ...

**4**

votes

**0**answers

119 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 ...

**4**

votes

**1**answer

138 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 ...

**1**

vote

**1**answer

96 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 ...

**3**

votes

**2**answers

204 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:
$$
\...

**3**

votes

**0**answers

35 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 + ...

**1**

vote

**0**answers

77 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{...

**1**

vote

**0**answers

47 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 ...

**1**

vote

**1**answer

117 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 ...

**3**

votes

**1**answer

159 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 ...

**1**

vote

**2**answers

96 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,...

**2**

votes

**1**answer

146 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 ...

**1**

vote

**0**answers

81 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 ...

**2**

votes

**0**answers

37 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 $\frac{1+\alpha}{2}$-...

**0**

votes

**0**answers

124 views

### Exercise 5.9. of the book A Basic Course in Partial Differential Equations, by Qing Han, about application of strong maximum principle

I have a question about exercise 5.9. of the book A Basic Course in Partial Differential Equations, by Qing Han.
Let assume $\Omega \subset \mathbb{R}^n$ is a bounded domain and $f$ and $u_0$ ...

**1**

vote

**1**answer

75 views

### A question about positivity preserving property of semigroup of Laplacian

Sry this the second question from the following article, I am asking in this week.
At page 6 (126), 3th line, of the following article.
THE HEAT EQUATION WITH A SINGULAR POTENTIAL
the authors say ...

**2**

votes

**1**answer

148 views

### Why is this test function admissible? [Paper explanation]

Reading Non-linear Elliptic and Parabolic Equations Involving Measure Data by Boccardo$\&$Gallouet , I had trouble understanding the following:
Why is $\psi(u_n)\chi_{(0,t)}$ admissible as a ...

**7**

votes

**0**answers

237 views

### Regularity result for the boundary value problem for the heat equation

Let $\Omega$ be an open bounded subset of $\mathbb R^N$.
Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$
Consider the following boundary value problem for the heat equation:
...

**3**

votes

**0**answers

236 views

### Existence and uniqueness for reaction-diffusion equations

I am interested in the following PDE on a $d$-dimensional torus $\mathbb{T}^d$
\begin{align*}
&\partial_tu(t,x) = \Delta u(t,x) +f(u(t,x),t,x),\\
& u(0)=u_0\in L_2
\end{align*}
where the ...

**2**

votes

**1**answer

150 views

### Reference request: Schauder estimates for parabolic equations

Where can I find Schauder estimates for second order linear parabolic equations (in divergence form with potential)?
Any reference would be highly appreciated.