# Error bound for a PDE with a perturbation term

In ODE theory, if we have $$y' = f(t, y),\ y(0) = c$$ and $$x' = f(t, x)+g(t, x),\ x(0) = c+d$$ we are able to say that $$|x(t)-y(t)|.

Is there an analogy theorem in PDE? Suppose I have a second-order linear parabolic pde $$u_t = Lu,\ u(x, 0) = u_0(x)$$

Consider a similar pde with a perturbation term $$\hat u_t = L\hat u+f(x,t),\ \hat u(x, 0) = u_0(x)+g(x)$$

Are we able to solve the bound of $$|u-\hat u|$$?

Update:

My target pde is black-scholes equation. Its domain is $$[0,\infty]\times [0, T]$$ where T is a constant. And the $$[0,\infty]$$ can be truncated to $$[0, S_{Max}]$$.

• What you are looking for a standard estimates for parabolic equations. Usually you have some restrictions on the operator (especially a lower bound on the differential part) and the domain. – maxmilgram Sep 20 '19 at 17:37
• @maxmilgram Sorry I didn't realize that. My target pde is black-scholes equation. Its domain is $[0,\infty]\times [0, T]$ where T is a constant. And the $[0,\infty]$ can be truncated to $[0, S_{Max}]$. – Wen Sep 20 '19 at 17:40