Consider the PDE $$\partial_t u = \partial_x (u^2 \partial_x u)$$ with some initial condition $u_0(x)$ and homogeneous Dirichlet boundary conditions for $x \in (0,1)$
I want to study existence and uniqueness of the solution via Lax-Milgram lemma: the continuity is not a problem, my concern is about coercivity.
More specifically, I need to show that the bilinear form $$a(u,v)=-(u^2 \partial_x u,\partial_x v)$$ is weakly coercive, i.e. there are $\lambda\geq0$ and $\alpha>0$ s.t. $$a(u,u) + \lambda ||u||_{L^2}^2 \geq \alpha ||u||_{H_0^1}^2$$
So I start with $a(u,u)=-\int_0^1 u^2 u_x^2dx $ but I don't know how to derive that inequality above. How should I move?