# Questions tagged [cauchy-problem]

The tag has no usage guidance.

19 questions
Filter by
Sorted by
Tagged with
33 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 ...
37 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 ...
172 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 ...
123 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 ...