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Let us consider the heat equation $$ \begin{cases} u_t + f(u)_x - u_{xx} = 0 & x \in (-1,1), \quad t >0\\ u(t,-1) = a(t), \\ u(t,1) = b(t), \\ u(0,x) = u_0(x) \end{cases} $$ where $f \in C^\infty(\mathbb R)$. How can I estimate the quantity $$ \partial_t \int_{-1}^1 |u|^{p-1} dx =- \int_{-1}^1 p f(u)_x |u|^{p-1} dx + \int_{-1}^1 p|u|^{p-1} u_{xx}dx $$ for $p >1$?

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  • $\begingroup$ When you say $f\in C^\infty(\mathbb{R})$, does that implicitly mean that $f'$ is uniformly bounded? $\endgroup$ Dec 7, 2020 at 5:04
  • $\begingroup$ @WillieWong Sure, we can assume that too. $\endgroup$
    – Hiro
    Dec 7, 2020 at 11:40

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