Problem set 4, PDE

Problem 1

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


u(x,0) = u_0(x) \qquad \text{on } \R^n.

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

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

Problem 2

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

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

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