6

Suppose $u : [0,1] \times [0,T] \to \mathbb{R}$, we consider the problem \begin{equation} \left\{\hspace{5pt}\begin{aligned} &\dfrac{\partial u}{\partial t} - a(x,t)x^2\partial_x^2 u - b(x,t)x\partial_x u - c(x,t) u= f(x,t)& \hspace{10pt} &\text{for $(x,t) \in \big(0,1\big) \times (0,T]$} ;\\ &u(x,0) = g(x) & \hspace{10pt} &\text{for $x \in \big(0,1\big)$.}\\ &u(0,t) = p(t) & \hspace{10pt} &\text{for $t \in [0,T]$.}\\ &u(1,t) = q(t) & \hspace{10pt} &\text{for $t \in [0,T]$.} \end{aligned}\right. \end{equation} Here $f,g,p,q$ are some suitable functions. $a(x,t) \geq c_0 >0$ in the domain, $a$, $b$ and $c$ are continuous.

I would like to have some gradient estimate results for this. May I have some reference for this? I have searched for "degenerate parabolic equation, Gradient estimate". But I do not have any result, there are only results for mean curvature flow.

mnmn1993
  • 458
  • 3
    This reminds me about the Black-Scholes equation. Note that through the change of variable $y=\ln x$ and $v(y,t)=u(x,t)$ the $x^2$ and $x$ disappear and the PDE is no more degenerate (but set on $\mathbb{R}^-$). Otherwise a good book for gradient estimate of parabolic equations is the one by Gary Liebermann "Second Order Parabolic Differential Equations" which is the companion book of Gilbarg-Trudinger for Elliptic equations. – Emmanuel Mar 10 '22 at 21:15
  • Thanks. I am exactly studying Black-Scholes equation. – mnmn1993 Mar 12 '22 at 11:18

0 Answers0