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 $$ \partial_t \int_{1}^1 u^{p1} dx = \int_{1}^1 p f(u)_x u^{p1} dx + \int_{1}^1 pu^{p1} u_{xx}dx $$ 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? $\endgroup$– Willie WongDec 7, 2020 at 5:04

$\begingroup$ @WillieWong Sure, we can assume that too. $\endgroup$– HiroDec 7, 2020 at 11:40
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