I'm reading the book Partial Differential Equations of Parabolic Type by Friedman, and I have a question on Green's function defined in p.82 (Sec.4/Chap.3). Assume $b, \sigma: \mathbb R^2 \to \mathbb R$ are as 'nice' as possible. Let $G:(x,t;\xi,\tau)\to G(x,t;\xi,\tau)\in\mathbb R$ be defined as the function s.t. for any continuous function $f$ having a compact support, the function

$$u(t,x):=\int_0^\infty G(x,t;\xi,\tau)f(\xi)d\xi$$


$$\partial_t u(x,t)= \frac{\sigma^2(t,x)}{2}\partial^{2}_{xx}u(x,t) - b(t,x)\partial_xu(x,t),\quad \forall t>\tau\ge 0, x, \xi>0$$

and $\lim_{t \searrow \tau} u(x,t)=f(x)$ for all $x\ge 0$ and $u(0,t)=0$ for all $t>0$. Theorem 16 ensures the wellposedness of $G$. Compared to the fundamental solution, is there any reference on the upper bound of $\partial_x G$ and $\partial^{2}_{xx}G$? Namely, $\exists \phi_t:\mathbb R^2_+\to\mathbb R_+$ s.t.

$$|\partial_{x}G(x,t;\xi,0)|+|\partial^{2}_{xx}G(x,t;\xi,0)|\le \phi_t(x;\xi)\quad\mbox{ and } \quad \int_0^\infty \phi_t(x,\xi)dx<\infty?$$

It is known that the fundamental solution admits such an estimation using parametrix. Can we expect a similar result for Green's function?

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