Questions tagged [cauchy-problem]

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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& ...
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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)...
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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} ...
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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) ...
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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 ...
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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 ...
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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 ...
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  • 183
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2 answers
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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 ...
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3 answers
287 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 ...
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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+...
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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} \...
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1 vote
0 answers
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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 \...
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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} ...
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2 votes
0 answers
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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 ...
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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 ...
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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 ...
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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 ...
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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 $$\frac{{\partial u}}{{\partial t}} = Au,$$ ...
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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 ...
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0 answers
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 ...
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2 votes
0 answers
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 ...
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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 ...
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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 \...
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