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Problem set 4, PDE

Problem 1

Let $latex a = (a_1, \ldots, a_n), b, f, u_0$ continuous functions with $latex |a|\le 1/\kappa$, and $latex u\in C^1(\bar\R^{n+1}_+)$ a solution to the IVP

$latex \displaystyle \partial_t u + \sum_{i=1}^n a_i \partial_{x_i}u + bu = f \qquad \text{in  }\R^{n+1}_+,$
$latex u(x,0) = u_0(x) \qquad \text{on } \R^n.$

Then, for any $latex P = (X,T)\in\R^{n+1}_+$,

$latex \displaystyle \sup_{C_\kappa}(P)|e^{-\alpha t}u| \le \sup_{\partial_{-}C_\kappa(P)}|u_0| + \frac{1}{\alpha + \inf_{C_\kappa(P)}b} \sup_{C_\kappa(P)}|e^{-\alpha t}f|,$

where $latex \alpha>0$ is a constant such that $latex \displaystyle \alpha + \inf_{C_\kappa(P)}b > 0.$

Problem 2

Let $latex a = (a_1, \ldots,a_n), b, f, u_0$ be $latex C^1$ functions with $latex |a|\le 1/\kappa$, and $latex u$ a $latex C^2(\bar\R^{n+1}_+)$ solution of the IVP of the previous problem. Then, for any $latex P=(X,T)\in\R^{n+1}_+$,

$latex \displaystyle ||u||_{C^1(C_\kappa(P))} \le C \big( ||u_0||_{C^1(\partial_-C_\kappa(P))} + ||f||_{C^1(C_\kappa(P))}\big),$

where $latex C>0$ is a constant that depends on T and the $latex C^1$ norms of the $latex a_i$ and b in $latex C_\kappa(P)$.