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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Elliptic regularity when the Lagrangian is possibly infinite

I want to solve variational problems of the form $$\inf_u \int_{-1}^1 \phi(u'(x)) \text{ with } u(-1)=u(1) = 0,$$ where $\phi(p)$ is convex and is allowed to take on the value $+\infty$ for some ...
Sébastien Loisel's user avatar
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Is the time of solution shorter as the initial data increases?

I'm reading the book superlinear parabolic problems and I came across the following situation twice: given two initial data $u_0$ and $\underline{u_0}$ with $u_0\geq \underline{u_0}$, $u_0\neq \...
Ilovemath's user avatar
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2 votes
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Continuity of the entropy of the solution of a parabolic PDE at $t=0$

Consider the following initial value problem for a parabolic PDE : $$\begin{cases} \textrm{div}\big(A\,\nabla u(t,x) + b(x)\, u(t,x)\big) \,=\,\partial_t u(t,x) \quad x\in\mathbb R^d\,,\ t>0 \\[4pt]...
tituf's user avatar
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2 votes
2 answers
297 views

Solution of parabolic partial differential equation using singular perturbation method

Consider the following parabolic partial differential equation (PDE) \begin{align} \label{eq:42} \left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \psi} + ...
GilbertDu's user avatar
4 votes
0 answers
173 views

$L^\infty$ solutions for parabolic Neumann problem (heat equation)

Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs: $$u_t - \Delta u = f$$ $$\partial_\nu u = 0$$ $$u|_{t=0} = u_0$$ where $f \in L^p(0,T;L^r(\Omega))$ and $...
soup's user avatar
  • 277
4 votes
1 answer
354 views

Contractivity of Neumann Laplacian

I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on semigroups of linear operators, I found on many places properties of the Neumann Laplacian. In ...
Bogdan's user avatar
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1 vote
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72 views

Hypoellipticity of a heat-like parabolic operator on Riemannian manifolds - reference request

Let $(M,g)$ be a Riemannian manifold and $L$ be a differential operator on $M$, with smooth coefficients, such that its symbol be $g$ (a "generalized Laplacian"). Where can I find proved ...
Alex M.'s user avatar
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3 votes
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Conditions of parameters to have bounded solution of Dynkin's equation in exit problem

Consider the following Dynkin’s equation in exit problem defined on unit disk $D_1(0)$ \begin{align} \left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma-1}{r} \sin\psi \frac{\partial}{\partial\...
GilbertDu's user avatar
3 votes
1 answer
420 views

Lax-Milgram and the existence of solution to parabolic equation

I think it is standard and common to use Lax-Milgram theorem to prove the existence of solution to elliptic equation. However, can we use it to establish the existence of parabolic equation? I do not ...
mnmn1993's user avatar
2 votes
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156 views

Existence and properties of the solution of a type of PDE

In doing optimal control of Parabolic PDE's we often have to solve a problem like this: $$\begin{cases} \dfrac{\partial y}{\partial t}-d\Delta y(t,x)=f(y(t,x),p(t,x)) & (t,x)\in (0,T)\times\Omega ...
Bogdan's user avatar
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89 views

Results about Schrödinger equations

Does anyone know any paper or book that deals with Schrodinger equations, specifically on asymptotic properties like blowup or limitation of solution when time goes to infinity using Schrödinger ...
Ilovemath's user avatar
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1 vote
0 answers
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Regularity of a Fokker-Planck PDE with unbounded coefficient

Let $A$ be a positive definite symmetric matrix, let $b\in C^1(\mathbb R^d\!\times\!(0,\infty))\cap C(\mathbb R^d\!\times\![0,\infty))$ taking values in $\mathbb R^d$. Consider the parabolic PDE $$ \...
tituf's user avatar
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3 votes
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Does the weak formulation of a parabolic PDE applies to a (good) non-test function?

Let $\rho:\mathbb R^d\times[0,\infty)\to(0,\infty)$ such that $\int \rho_t(x)\,dx=1$ for all $t\geq0\,$, $\rho$ is Holder-continuous (in both variables) and $\rho_t\in W^{1,1}(\mathbb R^d)$ for a.e. $...
tituf's user avatar
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5 votes
1 answer
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Connecting PDE notions for functions $[0,T] \to (\Omega \to \mathbb{R})$ to related notions for functions $[0,T] \times \Omega \to \mathbb{R}$

Fix $\Omega \subseteq \mathbb{R}^N$ a bounded domain (of whatever smoothness you end up needing, let's say $C^1$ domain for definiteness) and fix some $0 <T < \infty$. In considering evolution ...
Keefer Rowan's user avatar
2 votes
1 answer
136 views

Mean value formula for fractional heat equation

For the solution $u(z) = u(t,x)$ of the heat equation $u_t -\Delta u = 0$ we have $$u(z_0) = \int_{\Omega_r(z_0)}u(z) K_r(z_0-z) dz,$$ where $$\Omega_r(z_0) = \left\{z \in \mathbb{R}^{N+1}: \Gamma(z_0-...
Zac's user avatar
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0 answers
124 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 ...
Wentao Hu's user avatar
1 vote
0 answers
33 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 ...
jean's user avatar
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4 votes
1 answer
239 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 books of Alain Haraux – Systèmes dynamiques dissipatifs et ...
Bogdan's user avatar
  • 1,330
0 votes
1 answer
428 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_{...
Adi's user avatar
  • 483
2 votes
0 answers
79 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 ...
Jacob Lu's user avatar
  • 903
2 votes
1 answer
413 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 ...
John Wong's user avatar
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1 vote
0 answers
73 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 ...
Wentao Hu's user avatar
1 vote
0 answers
35 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 } \...
user173196's user avatar
5 votes
1 answer
154 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 ...
Mr_Rabbit's user avatar
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0 answers
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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 ...
user175203's user avatar
2 votes
1 answer
196 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 ...
WhiteDwarf's user avatar
1 vote
1 answer
413 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) ...
Paulo Rocha's user avatar
2 votes
2 answers
236 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 (...
Dang Zheng's user avatar
2 votes
0 answers
63 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 ...
Alex M.'s user avatar
  • 5,207
3 votes
1 answer
326 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}\ \...
Bogdan's user avatar
  • 1,330
1 vote
1 answer
581 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 ...
Jay's user avatar
  • 109
1 vote
0 answers
54 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{...
Mat's user avatar
  • 11
0 votes
1 answer
88 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 $$\Vert u(t,\cdot) \Vert_{L^2(\mathbb R)} + 2\int_0^t \Vert u_x(s,\cdot) \Vert_{L^2} ds \le \Vert u(0,\cdot) \Vert_{L^2(\mathbb R)}.$$...
user avatar
6 votes
1 answer
215 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 ...
User's user avatar
  • 402
0 votes
0 answers
118 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 \\ \...
Bogdan's user avatar
  • 1,330
3 votes
1 answer
199 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}^...
Guohuan Zhao's user avatar
4 votes
2 answers
218 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 }\...
Guy Fsone's user avatar
  • 1,033
0 votes
0 answers
52 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^\...
Hiro's user avatar
  • 131
1 vote
0 answers
52 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, ...
Keba's user avatar
  • 303
6 votes
0 answers
200 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, \...
Keba's user avatar
  • 303
1 vote
0 answers
92 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 $\...
Asanovic Tomas's user avatar
4 votes
0 answers
100 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{...
Rooibos's user avatar
  • 101
1 vote
0 answers
47 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 ...
Mark's user avatar
  • 647
2 votes
0 answers
83 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 ...
Mark's user avatar
  • 647
1 vote
0 answers
85 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{...
Eddy's user avatar
  • 111
2 votes
0 answers
51 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 ...
Ali's user avatar
  • 4,077
2 votes
0 answers
357 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}\...
Yuval's user avatar
  • 637
0 votes
0 answers
94 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 ...
Riku's user avatar
  • 819
2 votes
0 answers
86 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 ...
Riku's user avatar
  • 819
1 vote
0 answers
117 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(...
Ma Joad's user avatar
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