# $L^p$ estimate for perturbed heat equation

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 $$p >1$$?

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