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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Short time existence on nonlinear parabolic PDE

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fact, the number of ...
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4 votes
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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 ...
leo monsaingeon's user avatar
12 votes
3 answers
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Alternative proof of Varadhan's formula on Riemann manifolds

Consider Varadhan's famous formula for the kernel of the heat equation on a manifold: $$ \lim_{t \rightarrow 0} t \log h(t,x,y) = - \frac{d(x,y)^2}{4} .$$ I do not have access to his 1967 two papers,...
Alex M.'s user avatar
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11 votes
1 answer
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Nash's paper on parabolic equations

I am currently studying the paper "CONTINUITY OF SOLUTIONS OF PARABOLIC AND ELLIPTIC EQUATIONS" by John Nash (cf. American Journal of Mathematics, Vol. 80, 1958, https://doi.org/2372841). ...
nicolas's user avatar
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9 votes
3 answers
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Real analyticity of solution of heat equation

Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
SMS's user avatar
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7 votes
1 answer
868 views

Weak parabolic maximum principle on Riemannian manifolds

I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken, specifically, I'm ...
George's user avatar
  • 435
6 votes
1 answer
372 views

Does $E^{x,t}(f(X_T))$ solve a PDE if $f$ is not continuous?

Many books [see below for references] explore the connections between partial differential equations and expectation values. Assume $X$ is a diffusion with generator $A$, then they conclude, that ...
JSG's user avatar
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5 votes
1 answer
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Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$: $$ \begin{cases} \partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\ u(0,x)=u_0(x). \end{cases} $$ ...
Juhana Siljander's user avatar
5 votes
2 answers
344 views

Functional decaying under the heat flow (?)

Let $(M, g)$ be a compact Riemannian manifold and let $a$, $p$ be two real numbers greater than $1$. For any positive function $v$, I set $$ J(v) = \int_M \left|\nabla(v^a)\right|^p d\mu^g. $$ ...
Romain Gicquaud's user avatar
5 votes
3 answers
451 views

Structure of sign changes under the heat flow

Let $f$ be a smooth function on $R^2$, and define $N_f$ to be the set of points $p$ such that the nodal set of $f$ ($\{x\in R^2: f(x)=0\}$) divided every neighborhood of $p$ into four regions. Indeed, ...
Matchmaticians's user avatar
4 votes
1 answer
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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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4 votes
2 answers
438 views

Heat equation and evolution of number of critical points

Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...
Matchmaticians's user avatar
4 votes
0 answers
141 views

Uniqueness of the "weak solution" to Fokker-Plank PDE

Let $C_b^2(\mathbb R_+)$ be the set of functions $f: \mathbb R_+\to\mathbb R$ s.t. $f, f' ,f''$ are bounded and $f(0)=0$. Consider a measurable function $p: \mathbb R_+^2\to\mathbb R_+$ satisfying $$\...
GJC20's user avatar
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4 votes
1 answer
334 views

Caffarelli-Kohn-Nirenberg-type inequality with nonradial weight

The Caffarelli-Kohn-Nirenberg inequalities are a set of inequalities generalizing the Gagliardo-Nirenberg inequalities and are of the form $$\||x|^\gamma u\|_{L^p} \leq C\||x|^\alpha \nabla u\|_{L^q}^...
Keefer Rowan's user avatar
4 votes
0 answers
304 views

Banach's fixed point theorem for quasilinear parabolic PDEs

I have recently started reading into PDE theory, and came across the following question. Consider the PDE $$ \begin{cases} \partial_t \rho = \Delta (\rho + \rho^2) & \text{ on } (0,T) \times \...
Peter Koepernik's user avatar
3 votes
1 answer
928 views

Where to learn about parabolic Hölder spaces and when to use them

Is there a good resource from where I can learn about parabolic Hölder spaces? I see quite a few different definitions of this space in different papers. I am clueless about why, for example, one may ...
user24394's user avatar
3 votes
0 answers
243 views

Existence of solutions to a reaction-diffusion problem.

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on $\mathbb{R}\...
JCM's user avatar
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3 votes
1 answer
322 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.
Math_tourist's user avatar
2 votes
0 answers
198 views

Dependency of fundamental solution on coefficients of heat equation

Let $b: \mathbb R_+\to\mathbb R_+$ and $\sigma: \mathbb R_+\times \mathbb R\to\mathbb R_+$ be Lipschitz and bounded. Assume further $\sigma$ is elliptic, i.e. $\inf_{(t,x)}\sigma(t,x)>0$. For each $...
GJC20's user avatar
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2 votes
1 answer
300 views

Property so that $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?

Let $f:[0, \infty)\to [0, \infty)$ be non-decreasing and satisfy for all $t>t_{0}$, $$f(t)+C\int_{t_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t_{0}}\int_{t_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ ...
Shaq155's user avatar
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2 votes
1 answer
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Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum

How does one show directly that the solution following parabolic partial differential equation (PDE) of $p(t,v)$ approaches its stationary solution which is a solution of an elliptic partial ...
Hans's user avatar
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2 votes
1 answer
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A comparison principle for parabolic equation

(Crossposted from https://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear) Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for ...
riem's user avatar
  • 256
2 votes
1 answer
314 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 ...
kaithkolesidou's user avatar
1 vote
1 answer
169 views

Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum II

This is a follow-up on a previous question. Now the parabolic PDE of $P(t,x,v)$ has two spatial dimensions. $$ \partial_t P = L^* P \tag1 $$ $$L^*P = \frac12\left(\kappa^2\frac{\partial^2}{\partial v^...
Hans's user avatar
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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
99 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 ...
ABIM's user avatar
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1 vote
0 answers
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Classical solution to logarithmic diffusion equation

Consider the one-dimensional logarithmic diffusion equation for $u: \mathbb R_+ \times [0,1]\ni (t,x)\to u(t,x)\in \mathbb R$ : $$(\ast)\quad\quad \begin{cases} 2u_t = \big(\log(u)\big)_{xx} & \...
Fawen90's user avatar
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1 vote
0 answers
103 views

Uniqueness of the solution to some parabolic PDE

Consider the system $$ \begin{eqnarray} \partial_t p(t,x) + b(t)\partial_x p(t,x) - \partial_{xx}^2\left(\frac{\sigma(t,x)^2p(t,x)}{2\big(1+\alpha(t)\big)^2}\right) &=& 0, & \forall t>0,...
GJC20's user avatar
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1 vote
0 answers
67 views

Existence of a classical solution to some linear parabolic PDE with Dirichlet condition

Consider the following parabolic PDE (Fokker-Planck equation) for $u: \mathbb R_+\times\mathbb R_+ \to\mathbb R$: $$\partial_t u(t,x) = \frac{1}{2(1+q(t))}\partial^2_{xx}\big(a(t,x)u(t,x)\big)-\frac{1}...
GJC20's user avatar
  • 1,220
0 votes
1 answer
101 views

Estimative for Hessian of Heat Kernel in $\mathbb{R}^d$

We consider the heat kernel $$ g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ). $$ The inequality (2.3) in this ...
Ilovemath's user avatar
  • 585
0 votes
0 answers
478 views

Solving a parabolic PDE with boundary conditions given over ranges

How can one solve a Parabolic PDE (like the wave or diffusion equations) if the boundary conditions were given over ranges? Here is an example: How to solve the equation $u_{xx}+u_{yy}-\alpha^{2}u_{t}...
user135626's user avatar
0 votes
1 answer
624 views

Green's functions/fundamental solution to a non-constant coefficients pde

We already know the relationship between Green's function and solution to elliptic partial differential equation, i.e $$u(y)=\int_{\partial \Omega}u\frac{\partial G}{\partial n} ds+\int_\Omega G\Delta ...
chengcheng ling's user avatar