Questions tagged [cauchy-problem]

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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{...
Zydragon's user avatar
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 ...
B.Hueber's user avatar
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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 ...
Ilya Kossovskiy's user avatar
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 ...
Isaac's user avatar
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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& ...
mnmn1993's user avatar
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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)...
Overflowian's user avatar
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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} ...
Chaos's user avatar
  • 485
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) ...
Paulo Rocha's user avatar
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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 ...
StaTik's user avatar
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1 vote
0 answers
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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 ...
Mark's user avatar
  • 647
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 ...
Michele's user avatar
  • 313
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 ...
avg_ali's user avatar
  • 59
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 ...
Math's user avatar
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2 votes
0 answers
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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+...
Matteo's user avatar
  • 43
-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} \...
BeeTiau's user avatar
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 \...
Tirregs's user avatar
  • 11
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} ...
Mark's user avatar
  • 647
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 ...
user209029's user avatar
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 ...
utcyp's user avatar
  • 31
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 ...
Mark's user avatar
  • 647
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 ...
Mark's user avatar
  • 647
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 $$\frac{{\partial u}}{{\partial t}} = Au,$$ ...
Gustave's user avatar
  • 545
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 ...
Subhajit Jana's user avatar
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 ...
Amir Sagiv's user avatar
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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 ...
asm's user avatar
  • 31
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 ...
Marcelo Forets's user avatar
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 \...
Andrew's user avatar
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