# Questions tagged [cauchy-problem]

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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 ...
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 ...
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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 ...