**0**

votes

**0**answers

63 views

### Estimate in nonlinear heat quations

We consider nonlinear heat equation: \begin{eqnarray}u_t=u_{xx}+f(x,t, u(x,t),u_x(x,t))\end{eqnarray} where $(x,t)\in \Omega =(0,\pi)\times (0,T)$ and $|f(x,t,u_1,v_1)-f(x,t,u_2, v_2)|\le K(|u_1-u_2|+|...

**0**

votes

**0**answers

49 views

### $L^\infty$ bounds for pseudo-differential equations of parabolic type

It is well-known that if the solution of $u_t=u_{xx}$, with $t>0$ and $x\in\mathbb R$, is bounded, then $a(t)=\sup_{x\in \mathbb R}u(x,t)$ is non-increasing, while
$b(t)=\inf_{x\in \mathbb R}u(x,t)$...

**1**

vote

**1**answer

99 views

### Is there a way to solve this integral equation?

I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem.
For $\xi = (\alpha\theta)^{1/\alpha}$ and for ...

**0**

votes

**0**answers

48 views

### stokes-equation estimate in $L^2(0,T,L^\frac{3}{2}(\Omega))$

I'm interested in the default Stokes-system, e.g.
$ \frac{\partial}{\partial t} u - \Delta u + \nabla p = f \; \text{in} \; \Omega$
$ \nabla \cdot u = 0 \; \text{in} \; \Omega$
$ u = 0 \; \text{on} ...

**2**

votes

**1**answer

104 views

### Is the Lopatinski-Shapiro condition invariant under diffeomorphism?

If a PDE (eg. the heat equation with Robin BCs, or the elliptic version) on a bounded smooth domain $U$ satisfies the Lopatinski-Shapiro condition (for a definition see eg. Wloka), and if $T:U \to W$ ...

**0**

votes

**1**answer

68 views

### Equivalent ways to study a semilinear parabolic equation as a perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$:
\begin{cases}
u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\
u(0)=x_0,
\end{cases}
Suppose $A$ generates an analyitc ...

**1**

vote

**0**answers

76 views

### $C^{1,2}$ regularity of (weak) solutions to the heat equation

Let $\Omega$ be a bounded Lipschitz domain (smoother if needed), and consider the heat equation
$$u_t - \Delta u = 0$$
$$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$
$$u(0) = u_0$$...

**0**

votes

**0**answers

65 views

### Find a estimate for quasilinear parabolic equation

I am studying quasilinear parabolic operator:
$Pu=-u_t+u_{xx}+a(x;t;u,u_x)$ where $(x,t)\in \Omega=(0,\pi)\times(0,T)$ and $za(x,t,z,0)\le kz^2+b$
Suppose that $u$ satisfies: $P(u)=0$ in $\Omega.$
...

**5**

votes

**1**answer

120 views

### $L^\infty$ estimate on heat equation with a lower order term

Let $u$ be the weak solution on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ (for $n \leq 3$) of
$$u_t - \Delta u = f$$
$$u(0) = u_0$$
$$\partial_\nu u = 0 \quad\text{on $\partial\Omega$}$$
...

**1**

vote

**0**answers

39 views

### About the “method of lines”: when are such solutions good approximations for **all** future time?

This question is about approximate solutions to some classes of PDEs obtained using the "method of lines".
For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...

**2**

votes

**4**answers

186 views

### Ill-posedness of a generalized heat equation

Suppose we have the following one-dimensional generalized heat equation:
$$u_t(x,t)=g(x,t)\Delta u(x,t), \quad x\in \mathbb{R},t\in(0,\infty).$$
I need to prove that this equation is ill-posed, for ...

**1**

vote

**0**answers

36 views

### Functional Derivative estimate

Recently I've conisdered a functional derivative estimate on the first spatial derivative of bounded classical solutions $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ to
$$ u_t - u_{xx} - f(u) = 0 \ \ \ (x,...

**1**

vote

**1**answer

112 views

### $L^p-L^q$ estimates for heat equation - regularizing effect

Where can I find a proof of the following estimate
$$\|S(t)v\|_{L^p(\Omega)}\leq C_{N,p,q} t^{-\frac{N}{2}\left(\frac{1}{q}-\frac{1}{p}\right)}\|v\|_{L^q(\Omega)}, $$
where $1\leq p<q<+\infty$, $...

**0**

votes

**1**answer

74 views

### Proving short time existence for semi-linear parabolic PDE

I am following up on the answer of Denis Serre to this same question here Short time existence on nonlinear parabolic PDE
I have tried to generalise the proof of the Picard-Lindelof theorem, as ...

**3**

votes

**0**answers

85 views

### an inverse problem related to gaussian integral

Suppose we have a function $\rho(x,t)$ defined over $[-1,1]\times[0,1]$.
Define the integral
$
f_T(x)=\int_{[0,1]} (\rho(.,t)*K_{T-t})(x) dt
$
for $x\in R$ and $T>1$, where $*$ is the convolution, ...

**1**

vote

**1**answer

81 views

### $L^\infty(0,T;X) \cap C([0,T];Y) \subset C([0,T];X)$ for $X \subset Y$ dense?

is the Inclusion stated in the title true? In my case the spaces (essentially) are $X = H^1(\Omega)$ and $Y = L^2(\Omega)$ for $\Omega \subset \mathbb{R}$ bounded. My first try was to show
$\lim_{t_1 ...

**1**

vote

**0**answers

74 views

### Boundary regularity of solution to partial differential equation

I am conducting research on partial differential equations and I need a short-time existence result from the literature which I can not find at the moment. More precisely I would like to know the ...

**0**

votes

**0**answers

54 views

### Definiteness and infinite divisibility of kernels including heat semigroup

Let $P_{t}$ be the usual heat semigroup. Can one show (preferably) or disprove that for arbitrary $k \in \mathbb{R}_{>0}$ and $n \in \mathbb{N}$
we have
\begin{equation}
\sum_{i,j=1}^{n}a_{i}a_{j}\...

**0**

votes

**0**answers

89 views

### An Estimation on the Heat Kernel

I am reading Jost's Partial Differential Equations and meet an estimation( only stated in the book ) which I cannot verify by myself. In p.111, the book says that iteratively, we get
$$|S_{n}(x_{0},y,...

**5**

votes

**1**answer

185 views

### Reference for a Heat Process in a Wedge

I would like to ask about an explicit suggestion/reference for the following type of heat processes:
Roughly, assume we have a "wedge" $W$ of the following form - a domain in $\mathbb{R}^n$ with a ...

**0**

votes

**0**answers

49 views

### Weak solutions of linear parabolic PDEs and corresponding SDEs

It is well known that for an Stochastic differential equation (on the real line) of the form:
$dX_t = \mu(X_t)dt + \sigma(X_t)dW$
where $W$ is the standard Wiener process, the transition probability ...

**0**

votes

**0**answers

135 views

### Regularity of the heat equation: Neumann boundary conditions

I am looking for references to regularity estimates for the solution of a heat equation with homogeneous Neumann boundary conditions on $[0,T]\times D$ for some smooth domain $D\subseteq \mathbb{R}^3$ ...

**5**

votes

**1**answer

76 views

### Difference stencils approximating Laplacian

Let $\Delta$ be the Laplace operator on the interval $[0,1]\subset \mathbb{R}$.
Divide $[0,1]$ into small intervals of size $h$ to get an equidistant grid. One can approminate $-\Delta$ on this grid ...

**4**

votes

**0**answers

358 views

### Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention.
Note 2: This is my research problem, but the original problem ...

**1**

vote

**0**answers

46 views

### Another proof of the comparison principle for PDEs by the theory of viscosity solutions

In my research I study (fully) nonlinear PDEs with fractional derivatives. For an analysis of these, I use the theory of viscosity solutions.
So I am now at the end of my tether becasuse I can not ...

**0**

votes

**0**answers

79 views

### A heat equation approach to the perturbation of vector field with center

Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it.
We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)=...

**7**

votes

**2**answers

304 views

### 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 ...

**3**

votes

**1**answer

162 views

### Scaling properties of the Hölder estimate for heat equation

Lately, I have been interested in scaling properties of parabolic equations, and this question is related to an earlier one I asked about Harnack constants.
Let $Q(R) := Q(R^2,R) = B(0, R) \times [-R^...

**2**

votes

**1**answer

122 views

### Heat equation: impact of the diffusion coefficient on the Harnack constant

Consider the heat equation
$$
u_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1}
$$
for a Hölder continuous coefficient $a(x,t)$ satisfying
$$
0<C_o \le a(x,...

**2**

votes

**0**answers

64 views

### Reference/proof for parabolic Holder spaces property

Suppose that a function $u:[0,1]\times[0,T]\to\mathbb{R}$ belongs to the parabolic Holder space $C^{2+\alpha,1+\alpha/2}$, for $\alpha\in(0,1)$.
What can be said about $u_x=\partial_x u$?
I am not ...

**1**

vote

**0**answers

40 views

### Behavior of fundamental solution for parabolic equation on non compact complete riemannian manifold

Suppose that M is a complete noncompact Riemannian manifold. What is the necessary and sufficient condition that an operator on $L^{2}(M)$ comes from a smooth kernel that itself and all of the its ...

**1**

vote

**0**answers

61 views

### Is there any theory of Hamilton-Jacobi system?

I am curious that is there any theory for (time-dependent) HJ system? I know for HJ equation, we have viscosity solution, which depends heavily on Maximal principle. However, for systems, this seems ...

**0**

votes

**0**answers

48 views

### Is the heat kernel satisfies the heat equation in viscosity sense?

Let us see the heat kernel
\begin{equation}
k(x,t)=
\begin{cases}
(4\pi t)^{-\frac{n}{2}}\exp\{-\frac{1}{4}t^{-1}|x|^{2}\},t>0\\
0,t\leq0.
\end{cases}
\end{equation}
It is easy to see that $k\in C^{...

**0**

votes

**3**answers

169 views

### Fundamental solution for a parabolic PDE with constant coefficents

[Cross posting http://math.stackexchange.com/questions/1374384/fundamental-solution-for-a-parabolic-pde-with-costant-coefficents ]
I don't know if this question is more appropriate in Mathematics and ...

**2**

votes

**2**answers

150 views

### References for non-zero boundary value problem

I studied linear elliptic, parabolic and hyperbolic PDEs (boundary/initial value problem) in terms of existence, uniqueness and regularity.
I studied always, following Evans book "PDE", the case with ...

**0**

votes

**1**answer

91 views

### The hypoellipticity of a heat-like operator

I am aware that the heat operator (on a smooth manifold) is hypoelliptic. I am also aware that there are manifolds on which the Schrödinger's operator (with a $\Bbb i = \sqrt {-1}$ multiplying $\frac {...

**0**

votes

**2**answers

74 views

### Maximum principle for the heat equation with Dirichlet conditions

Let us consider the Laplacian operator in a domain $\Omega\subset \mathbb{R}^n$, with Dirichlet boundary conditions.
For all $f\in L^2(\Omega)$, we denote by $S(t)f$ the solution of the equation
$$
...

**9**

votes

**2**answers

245 views

### Evolution operator for a linear parabolic equation

Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator
$$D:= \frac{d}{dt}+A(t)$$
and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and ...

**0**

votes

**0**answers

99 views

### Existence of Solution steady navier stokes with do nothing outflow condition

We consider the stationary navier stokes equation with mixed boundary conditions
$$
\begin{align}
-\nu\Delta u +(u\cdot\nabla) u + \nabla p &= 0\ \textrm{in}\ \Omega \\
div\ u&=0\ \textrm{in}\ ...

**2**

votes

**1**answer

387 views

### Stochastic interpretation of heat kernel on fiber bundle

I'm looking for a stochastic interpretation of the heat equation for vector valued function.
The classical set up is the following :
If $(M,g)$ is a riemannian manifold then we could consider the ...

**5**

votes

**2**answers

295 views

### Well-posedness of Fokker-Planck equation

Consider the following equation on $[0,T]\times\mathbb{R}^n$
\begin{eqnarray}
&\partial_t\rho=\mathrm{div}(\rho\nabla V)+\Delta\rho\\
&\rho|_{t=0}=\rho^0,
\end{eqnarray}
where $V\in C^2(\...

**3**

votes

**1**answer

220 views

### When does the cumulative distribution function solve the Kolmogorov backward equation?

For a diffusion $X$ define the cumulative distribution function for $X_T$ started with $X_t=x$:
$$u(t,x):=E^{t,x}(1_{X_T\ge y})$$ Under what conditions does $u$ solve $X$'s Kolmogorov backward ...

**1**

vote

**0**answers

71 views

### Cauchy Problem and stochastic representation for discontinuous initial data

Where can I read more about the Cauchy problem, i.e. solutions to
$$ \frac{\partial u}{\partial t}+Lu=0 \text{ and } u(0,x)=f(x)$$
for some elliptic differential operator $L$ where $f$ is not ...

**3**

votes

**1**answer

163 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 ...

**5**

votes

**3**answers

370 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, ...

**1**

vote

**1**answer

105 views

### Nodal sets under the heat flow

Let $u(t,X)$ be a smooth solution of the heat equation on $R^2$
$u_t=\Delta u,$
where $(t,X)\in R \times R^2$. Suppose $\lim_{t \rightarrow 0} u(t,x,y)=x^2-y^2$. Can we prove that the nodal set of $...

**1**

vote

**1**answer

135 views

### Strong maximum principle for the heat equation in non-cylindrical domains

let $u(t,x)$ be a bounded smooth solution of the heat equation $u_t=\Delta u$, $(t,x) \in R \times R^2$, and let $V \subset (R \times R^2)$ be an open connected component of $\{(t,x) \in R \times R^2: ...

**7**

votes

**2**answers

433 views

### 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,...

**7**

votes

**1**answer

211 views

### Mountain Pass theorem for minimization problems with constraints

Let $I[u]$ be a functional on a (possibly infinite dimensional) Hilbert space. Then, under some conditions, the Mountain Pass theorem guarantees the existence of a saddle point (see http://en....

**1**

vote

**0**answers

78 views

### solution to a parabolic PDE

I'm reading a paper where the following parabolic PDE is considered:
$u_t(x,t)=u_{xx}(x,t)+b(x)u_x(x,t)+\lambda(x)u(x,t)$, with boundary conditions
$u_x(0,t)=qu(0,t) \text{ and } u(1,t)=\int_0^1 k(\...