Questions tagged [cauchy-problem]
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23
questions
1
vote
1
answer
84
views
Local boundedness for Cauchy problem
Consider the Cauchy problem
$$\left\{\hspace{5pt}\begin{aligned}
&-\dfrac{\partial u }{\partial t}
+a\dfrac{\partial^2 u}{\partial x^2}
+b \dfrac{\partial u }{\partial x}
+c u
= f(u) \leq 0& ...
- 198
0
votes
0
answers
49
views
Regularity of solution to Cauchy problem given regular initial data
Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus)
be a solution of a Cauchy problem
$$\frac{d}{dt} f(t) = A f(t)$$
$$f(0)...
0
votes
0
answers
70
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}
...
- 475
1
vote
1
answer
201
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) ...
- 13
0
votes
0
answers
38
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 ...
- 19
1
vote
0
answers
42
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 ...
- 627
4
votes
1
answer
236
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 ...
- 183
1
vote
2
answers
222
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 ...
- 59
4
votes
3
answers
287
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$u_t=Au+F(u)$ 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 ...
- 395
2
votes
0
answers
42
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+...
- 43
-2
votes
2
answers
534
views
Cauchy integral and residue theorem [closed]
What is the difference between the two sets of the following Cauchy integral,
$$
\begin{split}
\int_c t^k \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &=4\pi i \zeta^k\\[8pt]
\int_c \frac{1}{t^k} \...
- 7
1
vote
0
answers
86
views
Equivalence of Cauchy Type Problems and Volterra Integral Equations
The following Theorem is from the book Theory and Applications of Fractional Differential Equations by Kilbas, Srivastava and Trujillo. It's Theorem 3.10 from page 163.
Theorem.
Let $ \alpha \in \...
- 11
2
votes
1
answer
231
views
Comparing solutions of PDE problem with different initial conditions
My question(s) is about what happens with the solution of the problem if we change initial conditions.
Let's say we have a PDE problem:
$$
(1) \hspace{0.5cm} u_t+f(u)_x=0
$$
$$
(2) \hspace{0.5cm} ...
- 627
2
votes
0
answers
70
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 ...
- 21
3
votes
0
answers
63
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 ...
- 31
2
votes
2
answers
512
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 ...
- 627
3
votes
0
answers
185
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 ...
- 627
1
vote
1
answer
230
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 $$\frac{{\partial u}}{{\partial t}} = Au,$$
...
- 525
1
vote
0
answers
53
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 ...
- 1,568
0
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0
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415
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 ...
- 3,200
2
votes
0
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150
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 ...
- 31
3
votes
2
answers
139
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
362
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 \...
- 2,515