# Questions tagged [cauchy-problem]

The cauchy-problem tag has no usage guidance.

27
questions

0
votes

0
answers

45
views

### Attenuation estimation of the solution of the $n$-dimensional wave equation Cauchy problem

This is the equation given ($n\geq2$)
$$
\begin{cases}
u_{tt}=a^{2}\left(\Delta u\right), \\
\left.u\right|_{t=0}=\varphi(x_1,\cdots,x_n ),\\
\left.u_{t}\right|_{t=0}=\psi(x_1,\cdots,x_n ) .
\end{...

2
votes

0
answers

82
views

### Question on Cauchy problem on manifolds

Let $(M,g)$ be a globally-hyperbolic manifold, i.e. $M=\mathbb{R}\times\Sigma$ and $g=-\beta^2 \, dt^2+h_{t}$. Furthermore, let $E$ be a vector bundle over $M$ and $\square$ a normally-hyperbolic ...

3
votes

2
answers

125
views

### $C^{k,\alpha}$ dependence of ODE solutions on initial data

I faced such a question. Consider the Cauchy problem for an ODE:
$$
\begin{cases}
y'=F(t,y)\\ y(0)=y_0.
\end{cases}
$$ Assume $F$ has the regularity $C^{k,\alpha}$ (i.e. it has partial derivatives of ...

1
vote

1
answer

106
views

### Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data

I already asked the question on MSE, and have tried to figure it out myself.
But the problem seems trickier than expected, so I guess MO is a better place to ask..
For the sake of completeness, I ...

1
vote

1
answer

102
views

### Local boundedness for Cauchy problem

Consider the Cauchy problem
$$\left\{\hspace{5pt}\begin{aligned}
&-\dfrac{\partial u }{\partial t}
+a\dfrac{\partial^2 u}{\partial x^2}
+b \dfrac{\partial u }{\partial x}
+c u
= f(u) \leq 0& ...

0
votes

0
answers

66
views

### Regularity of solution to Cauchy problem given regular initial data

Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus)
be a solution of a Cauchy problem
$$\frac{d}{dt} f(t) = A f(t)$$
$$f(0)...

0
votes

0
answers

131
views

### Feynman Kac representation for nonlinear heat equation

Consider the following Cauchy problem
\begin{align}
\begin{cases}
\partial_t u=\sigma(t)\partial_{xx} u+ b(u),\; (t,x)\in[0,T]\times \mathbb R\\
u(0,x)=u_0(x)=Ce^{-x^2/2},
\end{cases}
\end{align}
...

1
vote

1
answer

426
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) ...

0
votes

0
answers

53
views

### Global existence for one Cauchy problem based on global existence of other two auxiliary Cauchy problems

I have a Cauchy problem for the differential equation
\begin{equation}
y' = f(t, y),
\end{equation}
with initial condition $y(0) = y^0$; here, $y$ and $f$ are two-dimensional vector-functions. The ...

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 ...

4
votes

1
answer

366
views

### Local solvability and Cauchy-Kovalevskaya theorem for PDEs

I am trying to understand the exact implications between local solvability and a general version of the Cauchy-Kovalevskaya (CK) theorem, in the context of PDEs.
Let $\Delta(x,u^{(n)})=0$ be a system ...

1
vote

2
answers

292
views

### Variational formulation of abstract Cauchy problem, possible?

Recently I have come across a method known as "variational method" in which we try to establish weak solutions of various boundary value problems involving ordinary derivatives, partial ...

6
votes

3
answers

380
views

### $u_t=Au+F(u)$ where $A$ is the infinitesimal generator of $C_0$-semigroup

I asked this question on Mathematics Stackexchange, but got no answer.
In Pavel's book: Nonlinear Evolution Operators and Semigroups - Applications to Partial Differential Equations, we have the ...

2
votes

0
answers

47
views

### Initial value problem with heterogeneous initial values

In all the references I checked the standard initial value problem for an ODE is stated as:
\begin{equation}
\begin{cases}
y'=F(y,t)\\
y(t_0)=y_0
\end{cases}
\end{equation}
for some $F:\mathbb{R}^{n+...

-2
votes

2
answers

610
views

### Cauchy integral and residue theorem [closed]

What is the difference between the two sets of the following Cauchy integral,
$$
\begin{split}
\int_c t^k \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &=4\pi i \zeta^k\\[8pt]
\int_c \frac{1}{t^k} \...

1
vote

0
answers

91
views

### Equivalence of Cauchy Type Problems and Volterra Integral Equations

The following Theorem is from the book Theory and Applications of Fractional Differential Equations by Kilbas, Srivastava and Trujillo. It's Theorem 3.10 from page 163.
Theorem.
Let $ \alpha \in \...

2
votes

1
answer

290
views

### Comparing solutions of PDE problem with different initial conditions

My question(s) is about what happens with the solution of the problem if we change initial conditions.
Let's say we have a PDE problem:
$$
(1) \hspace{0.5cm} u_t+f(u)_x=0
$$
$$
(2) \hspace{0.5cm} ...

2
votes

0
answers

75
views

### Smooth Cauchy problem on a cylindrical manifold (or how to define the exponential of a differential operator)

Let $M$ be a manifold, $E \rightarrow M$ be a real or complex smooth vector bundle, and $D: \Gamma_c(M,E) \rightarrow \Gamma_c(M,E)$ be a (first order if necessary) differential operator on smooth ...

3
votes

0
answers

66
views

### How to solve this linear Cauchy Problem

within my thesis, I am struggeling with the following PDE:
$u_t+a(x,y)u_{xx}+b(x,y)u_{xy}+c(x,y)u_{yy}+d(x,y)u_{x}+e(x,y)u_{y}+f(x,y)u=0$
$u(T,x,y)=1,$
where $a,b,c,d,e,f$ are polynomials and the ...

2
votes

2
answers

811
views

### Method of characteristics for 2x2 systems

In the literature it is easy to find books that show, step by step, how to get the solution of partial differential equation using various techniques, but it is not so easy to do the same for PDE ...

3
votes

0
answers

197
views

### Reference request on connection between PDE problems

I am trying to find references in the literature that connect solutions of any two of the problems given bellow. I study deterministic and stochastic conservation laws. Problems that I am interested ...

1
vote

1
answer

236
views

### Infinitesimal generator of a semigroup with parameter

When we talk about evolution equation the first idea that comes to the mind is the semigroups theory. This theory deals with Cauchy problems of the form $$\frac{{\partial u}}{{\partial t}} = Au,$$
...

1
vote

0
answers

61
views

### Supnorm problem involving kernel of Cauchy problem

Let $M$ be the $2$-dimensional hyperbolic manifold. Let $K(t,x,y)$ be the kernel appearing in the fundamental solution of the Cauchy problem
$$(\partial^2_t-\Delta_M)u=0,\text{ on }\mathbb{R}^+\times ...

0
votes

0
answers

516
views

### Domain of dependence for Hyperbolic system of PDES

In Leveque's "Numerical Methods for Conservation Laws", Ch. 3.1.1., he says that given a system of Hyperbolic PDE's and a point $(\bar{t},\bar{x} )$, its domain of dependence $D(\bar{t},\bar{x} )$ is ...

2
votes

0
answers

160
views

### An question about Cauchy Problem in General Relativity [closed]

Yesterday, in Brazilian School on Differential Geometry, a friend asked me the question:
Given an (non-trivial) initial data set $(M,g,k)$ for the Cauchy problem in General Relativity. Is there ...

3
votes

2
answers

148
views

### An English version Borok's work on finite-infinite systems of ordinary differential equations

I am looking for the English translation of the paper by V. M. Borok (originally in Russian)
The Cauchy problem for finite-infinite systems of linear differential equations. This work is about the ...

1
vote

2
answers

463
views

### Integral representation of the Cauchy problem solution for the heat equation

Consider the Cauchy problem for the heat equation
$u_t=\Delta u$, $u|_{t=0}=\varphi$.
S. Täcklind showed its solution $u$ is unique in the class $|u|\le e^{|x|h(|x|)}$, $|x|>1$, iff $\int_1^\infty \...