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If I consider the standard Cauchy Problem in one-dimension of the form:

\begin{equation} \begin{cases} a(x,t) \frac{\partial^2u}{\partial^2x} + b(x,t)\frac{\partial u}{\partial x} + c(x,t) u - \frac{\partial u}{\partial t} = f(x,t) \quad \text{in}\quad \mathbb{R}\times (0, T]\\ u(x,0) = g(x) \quad \text{on} \quad \mathbb{R} \end{cases} \end{equation}

We can find many different theorems according to properties of solution, e.g. in Frieman's books about PDEs or SDEs.

My question is:

What if I consider such a problem, but for my purpose the domain is of the form $\mathbb{R}^{+}\times[0, T]$ instead of $\mathbb{R}\times[0, T]$. Can I apply all of these theorems about existence of solution, uniquenes of solution and so on.

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