I am working with the following nonlinear heat equation $$ \begin{cases} u_t - u_{xx} + \mu u = u f(u_x), & t \in \mathbb R, x \in (0,1) \\ u (x,0) = u_0 (x) \\ u (0,t) = u(1,t) = 0 \\ \end{cases} $$ where $f$ is a Lipschitz continuous function and $\mu$ can be taken as large as necessary, but constant. I usually work with mild solution and the variation of constant formulas, but here I can find no way of working with the operators $u f(u_x)$.
Any idea would be appreciated.
Thank you,
D
You can get existence in H^1 for small data using Garlerkin, right? There is a term $$ \int u f'(u_x)u_{xx}u_xdx\leq \|u(t)\|_{L^\infty}(L\|u_x\|_{L^\infty}+f(0))\|u_{xx}\|_{L^2}\|u_x\|_{L^2}, $$ if now $\|u_0\|_{L^\infty}L<1/C$ (where $C$ is the sobolev embedding's constant) you get the appropriate bound, isn't it?