Originally Posted by

**NoClue** Hello!

I am required to do a paper on the Semilinear Wave Equation as part of a Stochastic Partial Differential Equations seminar.

The problem is, that I've never taken a course on partial differential equations (only ordinary differential equations) or functional analysis.

Now I have the following in front of me:

Let $\displaystyle f(s,x,t)$ and $\displaystyle \sigma(s,x,t)$ be predictable random fields

depending on the parameters $\displaystyle s\in\mathbf{R}$ and $\displaystyle x\in D$. We consider

the initial-boundary value problems for a stochastic wave equation

as follows

$\displaystyle \frac{\partial^{2}u}{\partial t^{2}}=(\kappa\Delta-\alpha)u+f(u,x,t)+\dot{M}(u,x,t)$

$\displaystyle x\in D$, $\displaystyle t\in(0,T]$

$\displaystyle Bu$|$\displaystyle _{\partial D}=0$

$\displaystyle u(x,0)=g(x),$ $\displaystyle \frac{\partial u}{\partial t}(x,0)=h(x)$

where

$\displaystyle \dot{M}(s,x,t)=\sigma(s,x,t)\dot{W}(x,t)$

My problem is now, that I don't understand how this second order differential was constructed.

What are $\displaystyle (\kappa\Delta-\alpha)u$, $\displaystyle f(u,x,t)$, $\displaystyle \dot{M}(u,x,t)$ supposed to be here and what has this to do with waves?

Thank you for your time.

If you can pinpoint me to some resources, which will make my life easier in understanding this. I will be very thankful!