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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3
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0answers
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
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0answers
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
1answer
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
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1answer
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
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0answers
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
1answer
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
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0answers
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
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0answers
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
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0answers
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
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1answer
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
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0answers
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
1answer
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
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1answer
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
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1answer
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
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0answers
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
1answer
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
1answer
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
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0answers
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
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1answer
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
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0answers
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
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0answers
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 \\ \...
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0answers
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
2answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
0answers
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
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0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
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0answers
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
1answer
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 ...

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