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
1answer
50 views
Explicit solution of a free boundary problem for heat equation
Consider the free boundary problem
$$
\min\{u_t - u_{xx} -1, u \} = 0 \qquad \text{ in } (0,T)\times (-1,1) \\
u(0,\cdot) = 0 \qquad \text{ in } (-1,1)\\
u(\cdot, -1) = u(\cdot, 1) = 0 \qquad \...
1
vote
0answers
26 views
classical solution of nondegenerate HJB equation
Let $b\in C(\mathbb R)$ and $L \in C_b^2(\mathbb R)$. Consider an equation
$$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)$$
with ...
2
votes
0answers
42 views
improved regularization for $\lambda$-convex gradient flows
It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
1
vote
0answers
61 views
Solution existence for two-dimensional parabolic PDEs
I am looking for a solution $(f,g) \in C^{1,2}([0;T]\times\mathbb R;\mathbb R^2)$ to the following PDE system
$$
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
1answer
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
0answers
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
0answers
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?
$$
\int_0^T \int_{\mathbb R^N} \phi(f)(- \Delta) u(...
1
vote
0answers
48 views
Regularity theory for parabolic PDEs in fractional Sobolev spaces
I am trying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya and the existence ...
0
votes
1answer
71 views
Scaling argument for the heat equation in a bounded domain [closed]
We want to study the long time behavior of the heat equation $u_t - u_{xx} = 0$ in the domain $[-1,1]$. Now consider the rescaling $u^{\epsilon} = u(x/\epsilon, t/\epsilon^2)$. Then
$u^\epsilon$ ...
2
votes
0answers
34 views
Stability of weak solutions to quasilinear parabolic equations
Consider the quasilinear operator $A(x,t,\nabla u)$ satisfying $$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}$$ where $1<p<\infty$. Note ...
8
votes
1answer
173 views
Positive solutions for semilinear parabolic equations
Let $X$ be a Banach lattice. Consider the system
$$y'(t)=Ay(t)+f(t,y(t)) \qquad \text{in } (0,T) , \qquad y(0)=y_0, \qquad (*)$$
where $T>0$, $A$ generates an analytic positive semigroup $S(t)$ on $...
1
vote
0answers
117 views
Rigorous error estimate for semi-discrete heat equation in bounded domain
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^N$ and $u_h$ be a solution of
$$
\begin{cases}
\partial_t u_h -\Delta_h u_h = f(x) & \text{ in } \Omega_h\\
u_h=0 &\text{ in } \...
3
votes
2answers
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 $\color{red}{C}$.
$$\|e^{t\Delta_d}f\|_{W^{1,...
1
vote
0answers
30 views
Very weak solution to parabolic PDE (pointwise a.e. in time with time derivative on test function)
Consider the parabolic PDE
$$u' + Au = 0$$
as an equality in $L^2(0,T;V^*)$ for some Hilbert space $V$ with $A\colon L^2(0,T;V) \to L^2(0,T;V^*)$ a coercive, bounded linear operator. Here $u'$ is the ...
2
votes
1answer
202 views
Regularity on the boundary for the heat equation with linear source
This is probably a known problem but I was not able to find exactly what I am looking for.
I have the following linear heat equation with zero-flux boundary conditions:
\begin{equation}
\begin{cases}...
2
votes
1answer
102 views
Continuity of solution of a parabolic PDE w.r.t. system parameters
If we have a system of PDE of the form:
$$\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
1answer
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
2answers
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
1answer
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
$$
\...
2
votes
1answer
267 views
An inequality for abstract Cauchy problem
Consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x_0$, where $A$ generates an analytic $C_0$-semigroup on a Banach space $X$. How we can prove an inequality of ...
2
votes
0answers
79 views
Confusion optimal control abuse notation
I'm currently reading this paper describing a numerical scheme for the approximating optimal policy of a stochastic control problem. However, I run into a confusion directly on the first page where ...
3
votes
1answer
215 views
Inequality for initial data
I'm wondering if there is any idea to bound the norm of the initial data by the norm of corresponding solution? To make it clear, consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,...
2
votes
1answer
109 views
Reference request: Schauder estimate in the space variable for parabolic equations
Setting: Let $(M,g)$ be a compact Riemannian manifold without boundary. Let $\Delta_g$ be the Laplacian and $L=\Delta_g-\partial_t$ the heat operator. Let $0<\alpha<1$, $0<t_0<T$.
Let $$u\...
1
vote
1answer
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:
$$\frac{\partial{u}}{\partial{t}}= \alpha\frac{\partial^2{u}}{\partial{x^2}}-ku$$
$$...
1
vote
1answer
129 views
Leray projector in $L^{\infty}$ and negative order Besov spaces for the Navier-Stokes equations
I was reading the paper "Norm inflation for the generalized Navier-Stokes equations" which can be found here: https://arxiv.org/abs/1212.3801.
In Lemma 2.1, the authors said for any $\phi \in L^{\...
4
votes
0answers
75 views
Biharmonic heat flow on compact manifolds
Consider $\partial _t u (t,x) = -\partial _x ^4 u$ on a compact manifold, or even a special specific one like the torus.
Are there any estimates on the Green function (bihamornic heat kernel), for ...
4
votes
1answer
283 views
Lack of exponential $L^2_{t,x}$ decay for a heat equation with growing coefficients
Edit: I have changed the nature of the question, but in order to have a better idea of what I can expect for the original problem (see below).
Given $T>0$ and $n \in \bf Z$, consider the following ...
5
votes
0answers
99 views
Reference request: Existence and regularity for parabolic PDEs with smooth coefficients on compact manifolds with boundary
I'm looking for a reference for a statement like:
Let $M$ be a $n$-dimensional smooth compact manifold with smooth boundary $\partial M$. In coordinates, let $\mathcal L$ have the form
$\mathcal L ...
1
vote
1answer
72 views
A symmetric parabolic second order PDE
Here I want to solve a second order PDE symmetrically depending on two variables
$$
3(\partial_{\alpha\alpha}f+\partial_{\beta\beta}f-2\partial_{\alpha\beta}f)+2(\cot\alpha)\partial_\alpha f +2(\cot\...
5
votes
3answers
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
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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:
$$
\...
3
votes
0answers
35 views
Reference request: existence of strong solutions to a linear parabolic problem with mixed boundary conditions
on a domain $\Omega \subset \mathbb{R}^d$ with smooth boundary $\partial\Omega$ subdivided into two parts $\Gamma_D$ and $\Gamma_N$ I am considering the parabolic problem
$$
\partial_t u = \Delta u + ...
1
vote
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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.
