Questions tagged [cauchy-problem]
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27
questions
0
votes
0
answers
39
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
65
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
$$\frac{d}{dt} f(t) = A f(t)$$
$$f(0)...
0
votes
0
answers
128
views
Feynman Kac representation for nonlinear heat equation
Consider the following Cauchy problem
\begin{align}
\begin{cases}
\partial_t u=\sigma(t)\partial_{xx} u+ b(u),\; (t,x)\in[0,T]\times \mathbb R\\
u(0,x)=u_0(x)=Ce^{-x^2/2},
\end{cases}
\end{align}
...
1
vote
1
answer
413
views
Forwards Feynman–Kac formula
This might be a simple question, but I'm having trouble with it.
Consider the Cauchy problem with final condition.
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
0
votes
0
answers
53
views
Global existence for one Cauchy problem based on global existence of other two auxiliary Cauchy problems
I have a Cauchy problem for the differential equation
\begin{equation}
y' = f(t, y),
\end{equation}
with initial condition $y(0) = y^0$; here, $y$ and $f$ are two-dimensional vector-functions. The ...
1
vote
0
answers
47
views
Do the solutions of parabolic PDE problems with different initial conditions converge to each other?
Let's say we have a parabolic PDE system:
$$
(PDE) \hspace{0.5cm} u_t+f(u)_x=\mu \cdot u_{xx},
$$
where $x \in A \subseteq \mathbb{R}$, $t \in [0,T]$ and $u \in \mathbb{R}^n, \: n\geq 2$. And let's ...
4
votes
1
answer
359
views
Local solvability and Cauchy-Kovalevskaya theorem for PDEs
I am trying to understand the exact implications between local solvability and a general version of the Cauchy-Kovalevskaya (CK) theorem, in the context of PDEs.
Let $\Delta(x,u^{(n)})=0$ be a system ...
1
vote
2
answers
290
views
Variational formulation of abstract Cauchy problem, possible?
Recently I have come across a method known as "variational method" in which we try to establish weak solutions of various boundary value problems involving ordinary derivatives, partial ...
6
votes
3
answers
377
views
$u_t=Au+F(u)$ where $A$ is the infinitesimal generator of $C_0$-semigroup
I asked this question on Mathematics Stackexchange, but got no answer.
In Pavel's book: Nonlinear Evolution Operators and Semigroups - Applications to Partial Differential Equations, we have the ...
2
votes
0
answers
47
views
Initial value problem with heterogeneous initial values
In all the references I checked the standard initial value problem for an ODE is stated as:
\begin{equation}
\begin{cases}
y'=F(y,t)\\
y(t_0)=y_0
\end{cases}
\end{equation}
for some $F:\mathbb{R}^{n+...
-2
votes
2
answers
607
views
Cauchy integral and residue theorem [closed]
What is the difference between the two sets of the following Cauchy integral,
$$
\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:
$$
(1) \hspace{0.5cm} u_t+f(u)_x=0
$$
$$
(2) \hspace{0.5cm} ...
2
votes
0
answers
75
views
Smooth Cauchy problem on a cylindrical manifold (or how to define the exponential of a differential operator)
Let $M$ be a manifold, $E \rightarrow M$ be a real or complex smooth vector bundle, and $D: \Gamma_c(M,E) \rightarrow \Gamma_c(M,E)$ be a (first order if necessary) differential operator on smooth ...
3
votes
0
answers
66
views
How to solve this linear Cauchy Problem
within my thesis, I am struggeling with the following PDE:
$u_t+a(x,y)u_{xx}+b(x,y)u_{xy}+c(x,y)u_{yy}+d(x,y)u_{x}+e(x,y)u_{y}+f(x,y)u=0$
$u(T,x,y)=1,$
where $a,b,c,d,e,f$ are polynomials and the ...
2
votes
2
answers
800
views
Method of characteristics for 2x2 systems
In the literature it is easy to find books that show, step by step, how to get the solution of partial differential equation using various techniques, but it is not so easy to do the same for PDE ...
3
votes
0
answers
197
views
Reference request on connection between PDE problems
I am trying to find references in the literature that connect solutions of any two of the problems given bellow. I study deterministic and stochastic conservation laws. Problems that I am interested ...
1
vote
1
answer
236
views
Infinitesimal generator of a semigroup with parameter
When we talk about evolution equation the first idea that comes to the mind is the semigroups theory. This theory deals with Cauchy problems of the form $$\frac{{\partial u}}{{\partial t}} = Au,$$
...
1
vote
0
answers
60
views
Supnorm problem involving kernel of Cauchy problem
Let $M$ be the $2$-dimensional hyperbolic manifold. Let $K(t,x,y)$ be the kernel appearing in the fundamental solution of the Cauchy problem
$$(\partial^2_t-\Delta_M)u=0,\text{ on }\mathbb{R}^+\times ...
0
votes
0
answers
513
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
146
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
461
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 \...