# 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.

330
questions

**3**

votes

**0**answers

34 views

### How to prove the existence of weak solutions of parabolic PDEs using Rothe's method?

I am reading a textbook on PDE (Introduction to Elliptic and Parabolic Partial Differential Equations by Z. Wu, J. Yin and C. Wang, written in Chinese) where a proof of the existence of weak solutions ...

**1**

vote

**0**answers

26 views

### Pointwise estimate of solutions to the parabolic equation with a monotonic drift

I wonder for a parabolic equation
$$u_t+(a(t,x)u)_x= u_{xx},$$
if we know that $a(t,x)$ is monotonic decreasing in $x$ with $a(t,-\infty)=C_L, a(t,+\infty)=C_R$, $C_L>C_R\geq 0$, are there results ...

**3**

votes

**1**answer

79 views

### Reaction-diffusion systems treated as dynamical systems

I wonder if there is a good reference on reaction-diffusion systems on $\mathbb{R}^N$, that treats them as dynamical systems.
I have the book of Alain Haraux – Systèmes dynamiques dissipatifs et ...

**0**

votes

**1**answer

54 views

### Interchange of integration and supremum

Let $u \in C^0(-T,T; L^2(B_R))$ be a measurable function, then is the following true?
$$
\int_0^R \sup_{-T<t<T} \int_{S_r} |u(\sigma ,t)|^2 \ d \sigma \ dr = \sup_{-T<t<T}\int_0^R \int_{...

**2**

votes

**0**answers

57 views

### Decay rate of transition density of a SDE system

Consider the following SDE system
$$dx_t = b(y_t)dt + dw^1_t, \quad dy_t = dw^2_t.$$
Here the drift $b(\cdot)$ is a smooth function that may decay slowly. For example, $|b(x)| \le C/|x|^\sigma$ for ...

**2**

votes

**1**answer

59 views

### Generalized Fokker-Planck equation

Consider the diffusion process
$$
d X = \mu(X, t) dt + \sigma(X, t) dY.
$$
When $Y$ is a Brownian motion, we know that the density follows the Fokker-Planck equation. Here I'm considering the general ...

**0**

votes

**0**answers

41 views

### Fractional heat equation and analyticity

Consider the problem
$$
\begin{cases}
u_t + (-\Delta)^s u = 0 & \text{ in } \Omega \times (0,\infty) \\
u(x,t) = 0 & \text{ in } (\mathbb R^n \setminus \Omega ) \times (0,\infty) \\
u(\cdot,0) ...

**1**

vote

**0**answers

57 views

### While doing Lp estimates, is the constant C monotonically increasing with respect to the parameters it depends on?

For example, consider the third boundary value problem:
\begin{align}
&\frac{\partial u}{\partial t}-\sum_{i,j=1}^n a_{ij}(x,t) \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum_{i=1}^n ...

**1**

vote

**0**answers

28 views

### Existence and uniqueness for fractional parabolic equation with transport term

Let us consider the problem
\begin{equation}
\begin{cases}
u_t+(-\Delta)^{\sigma}u+\mathrm{div}(a(t,x)u) = 0 & \text{in } \mathbb{R}^n \times [0, T) \\
u(x,0)=u_0(x) & \text{in } \...

**5**

votes

**1**answer

67 views

### Spectrum of an elliptic operator in divergence form with a reflecting boundary condition

Let $\Omega$ be a bounded open domain and $v:\Omega\to\mathbb{R}^n$. Consider the following elliptic operator in divergence form, defined on smooth functions $u: \Omega \to \mathbb{R}$
\begin{align}
L ...

**0**

votes

**0**answers

42 views

### Rigorous energy estimate for advection-diffusion equation

Let $a \in L^q([0,T];L^p(\mathbb R^N))$ with $2/q + N/p \le 1$ and
$q \in [2,\infty), p \in (N,\infty) \text{ if } N \ge 2$
$q \in [2,4], p \in [2,\infty] \text{ if } N = 1$
and consider the ...

**2**

votes

**1**answer

71 views

### Lower Gaussian estimates for Dirichlet heat kernel on manifolds

Let $(M,g)$ be a Riemannian $n$-manifold with $Ric_g\ge -Kg$, $\Omega\subset M$ be an open subset. We can define Dirichlet heat kernel on $\Omega$, $p_{\Omega}(y,t,y',t')$ as the minimal fundamental ...

**1**

vote

**1**answer

147 views

### Forwards Feynman–Kac formula

This might be a simple question, but I'm having trouble with it.
Consider the Cauchy problem with final condition.
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...

**1**

vote

**1**answer

106 views

### Heat flow derivative of entropy

In a 1966 paper (Speed of Approach to Equilibrium for Kac's Caricature of a Maxwellian Gas, Arch. Rational Mech. Anal., Vol. 21), McKean seems to suggest that the successive derivatives of entropy $H (...

**2**

votes

**0**answers

40 views

### Hypoellipticity or parabolic regularity for vector bundles

Let $E \to M$ be a Hermitian vector bundle (of finite rank) over a Riemannian manifold (not necessarily compact). Let $H : \Gamma(E) \to \Gamma(E)$ be a differential operator with smooth coefficients ...

**3**

votes

**1**answer

109 views

### Neumann/Robin Laplacian semigroup well-known estimate

Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:
$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \...

**1**

vote

**1**answer

98 views

### Estimates of fractional heat kernel

Is there any estimate available for the derivatives of the fractional heat kernel? Estimates on the kernel itself are at this link.
Also is any estimate available if we consider the problem with ...

**1**

vote

**0**answers

39 views

### Regularity and existence linear parabolic fractional equation

\begin{equation}
\begin{cases}
a(x, t)u_t+(-\Delta)^{\sigma}u+b(x,t)u=f(x,t), & \text{in } \mathbb{R}^n \times [0, T) \\
u(x,0)=u_0(x), & \text{in } \mathbb{R}^n
\end{cases}
\end{...

**0**

votes

**1**answer

73 views

### Does $\int_0^t \Vert u_x(s,\cdot) \Vert_{L^2} ds \le C$ imply $\Vert u_x (t,\cdot) \Vert_{L^2(\mathbb R)} \le C$ in the heat equation?

For the parabolic equation
$$u_t + f(u)_x - u_{xx} = 0$$
one has
$$\Vert u(t,\cdot) \Vert_{L^2(\mathbb R)} + 2\int_0^t \Vert u_x(s,\cdot) \Vert_{L^2} ds \le \Vert u(0,\cdot) \Vert_{L^2(\mathbb R)}.$$...

**4**

votes

**0**answers

60 views

### Intersection of self-shrinkers

I have a problem regarding a statement in the paper Smooth compactness of self-shrinkers by Colding and Minicozzi.
In the article, they define a surface $\Sigma$ in $\mathbb R^3$ to be a self-shrinker ...

**0**

votes

**0**answers

98 views

### Positivity of solution for Fisher-Kolmogorov Equation

How can we prove that if $y=y(t,x)$ is the solution of the problem:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-d\Delta y(t,x)=r(x)y(t,x)-\rho(x) y^2(t,x),\ (t,x)\in (0,T)\times \Omega \\ \...

**1**

vote

**0**answers

35 views

### Parabolic Sobolev inequality in Sobolev mixed norm spaces

Assume $p,q\in (1,\infty)$, $r\in [p,\infty)$, $s\in [q,\infty)$ and
$$
1<\frac{d}{p}+\frac{2}{q}=1+\frac{d}{r}+\frac{2}{s}.
$$
Let $u\in C_c^\infty((0,1)\times B_1)$, where $B_1=\{x\in \mathbb{R}^...

**3**

votes

**2**answers

134 views

### Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight

Let $\zeta, u_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded.
\begin{equation}\label{Star-3.7}
\begin{cases}
\partial_t u -\Delta u + \zeta u=0 &\mbox{ in }\...

**0**

votes

**0**answers

82 views

### Wellposedness of semilinear fractional heat equation

Do you have a reference on the wellposedness of the problem
\begin{align*}
\begin{cases}
u_t + (-\Delta)^su = f(u) \qquad & (t,x) \in (0,+\infty) \times (0,1) \\
u(0,\cdot) = u_0 & x \in (0,1)...

**0**

votes

**0**answers

45 views

### $L^p$ estimate for perturbed heat equation

Let us consider the heat equation
$$
\begin{cases}
u_t + f(u)_x - u_{xx} = 0 & x \in (-1,1), \quad t >0\\
u(t,-1) = a(t), \\
u(t,1) = b(t), \\
u(0,x) = u_0(x)
\end{cases}
$$
where $f \in C^\...

**0**

votes

**0**answers

15 views

### Bound for the time derivative of a parabolic equation ith Neumann boundary condition

Dears,
I was wondering how to get an upper bound for the time derivative of the following PDE, (parabolic with Neumann boundary condition), for example in the segmant $[0,a]$:
$$\partial_tu- \sigma(t,...

**1**

vote

**0**answers

37 views

### When is a solution concept sensible?

For many parabolic PDE (systems), one has to weaken the solution concept in order to obtain global solutions. Apart from classical solutions, the most known concept is surely that of weak solutions, ...

**4**

votes

**0**answers

48 views

### Interior regularity for parabolic systems in divergence form

Let $\Omega \subset \mathbb R^n$, $n \in \mathbb N$, be a smooth, bounded domain. Suppose $N \in \mathbb N$, $D \subset \mathbb R^N$ and that $a_{ij} : D \to \mathbb R$ are smooth for $i, j \in \{1, \...

**1**

vote

**0**answers

45 views

### Time dependent reaction-diffusion semigroup

I'm interested in the following linear reaction-diffusion equation
\begin{align*}
&\partial_tu(t,x) = \sigma(t)\Delta u(t,x),\\
& u(0)=u_0\in X
\end{align*}
where $X$ is a Banach space and $\...

**4**

votes

**0**answers

75 views

### Global existence of $L^p$-solutions to a quasilinear diffusion equation

We consider the diffusion problem
$$\begin{cases}
\partial_t u = \nabla \cdot (a(u)\nabla u), \quad t>0, x \in \mathbb{R}^n \\
u(0) = u_0
\end{cases}$$
for functions $u \colon [0,T] \times \mathbb{...

**1**

vote

**0**answers

38 views

### Do the solutions of parabolic PDE problems with different initial conditions converge to each other?

Let's say we have a parabolic PDE system:
$$
(PDE) \hspace{0.5cm} u_t+f(u)_x=\mu \cdot u_{xx},
$$
where $x \in A \subseteq \mathbb{R}$, $t \in [0,T]$ and $u \in \mathbb{R}^n, \: n\geq 2$. And let's ...

**2**

votes

**0**answers

63 views

### Changing a little assumptions in famous paper Vanishing viscosity solutions of nonlinear hyperbolic systems?

The question that I hope to find some answer here is: do the results from
Bianchini, Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, 2005
paper still apply if we change a ...

**1**

vote

**0**answers

45 views

### Reference request: existence/uniqueness of solutions to convection diffusion equations

I am looking for a reference wherein existence and uniqueness results are proven for a system of PDEs of the form
$$
\frac{\partial Q}{\partial t} + A \frac{\partial Q}{\partial x} = f(Q,x,t) + \frac{...

**2**

votes

**0**answers

41 views

### Mixed boundary value problems for Heat equation

This might be a very simple question, but basically I am looking for a good reference for studying the heat equation on Riemannian manifolds with boundary, specifically when data is put on lateral ...

**2**

votes

**0**answers

123 views

### A question in Sobolev spaces involving time

Let $X$ be a Banach space, we understand $L^1(0, T, X)$ is the space of strongly measurable functions from $[0, T]$ valued in $X$, that is integrable. Assume ${\bf u}\in L^1(0, T, X)$, we say ${\bf v}\...

**0**

votes

**0**answers

77 views

### Equation $u_t - u_{tx} - u_{xx} = 0$

Consider the following heat equation with a perturbation given by a second order mixed derivative:
$$u_t - u_{tx} - u_{xx} = 0$$
Does this equation have a name? How can one prove a wellposedness ...

**2**

votes

**0**answers

73 views

### Estimate in vanishing viscosity for the difference $\Vert u^\epsilon - u^\eta \Vert_{L^2(\mathbb R^N)} $

Consider the following advection-diffusion equation
$$
\begin{cases}
u^\epsilon_t + f(u^\epsilon)_x = \epsilon \Delta u^\epsilon\\
u^\epsilon(0,\cdot) = u_0,
\end{cases}
$$
How can one prove an ...

**1**

vote

**0**answers

73 views

### Perturbation in the equation $u_t=\epsilon Pu$, where $P$ is an elliptic partial differential operator

Let $Pu=\sum_{ij} \partial_j(a_{ij}(x) \partial_{i} u)$ be an elliptic operator. Consider the equation
$$
(u=u_\epsilon)\\
\partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\
u(0,x)=u_0(...

**5**

votes

**1**answer

111 views

### 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^\...

**12**

votes

**1**answer

643 views

### 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 ...

**1**

vote

**0**answers

54 views

### 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 ...

**2**

votes

**0**answers

58 views

### 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\...

**0**

votes

**0**answers

89 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^{...

**1**

vote

**1**answer

116 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, ...

**3**

votes

**0**answers

78 views

### 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 ...

**2**

votes

**1**answer

82 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

32 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

68 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

68 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

232 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 ...