Energy estimates to show existence of a PDE I want to prove that there is a solution to the problem
$$u_t = u^{n}u_{xx} + u^m$$
$$u|_{t=0} = u_0 \in C^{1+\alpha}$$
in the domain $S^1 \times (0,T)$ and $n$ and $m$ are positive integers, and $u_0 > 0$. Take $n$ to be even and $m$ to be odd ($n = 2$ and $m = 3$ for example).
I found a thread on Mathoverflow in which it was suggested that one looks at 
$$\partial_t u_a = (a+|u_a|^n)\partial^2_x u_a + |u_a|^m$$
and we know the solution to this exists since it is strongly parabolic and so there are references to show it exists. I guess I need to find energy estimates involving $u_a$ that are independent of $a$ in a Hilbert space so that there is a weakly-convergent subsequence (as $a \to 0$) and I want to show that the limit of this subsequence solves the original PDE involving just $u$, right?
How do I find these energy estimates? I only know one method of multiplying by a test function and integrating by parts but that doesn't look good here. Any ideas?
All I have at the moment (from a book) is that $$\sup_{(0,T)}\lVert u_a \rVert_{H^l} < \infty$$ which doesn't help since I don't know if there is a dependence on $a$.
Also, passing to the limit in the equation involving $u_a$ doesn't look easy either.
 A: Actually, one can get a uniform growth rate on the $H^1$ norm as follows. Multiply the equation by $u$ and integrating by parts you get
$$ \frac12 \partial_t \|u\|_2^2 = \int (au + u^3)u_{xx} \mathrm{d}x + \|u\|_4^4 \leq \|u\|_4^4 \leq C \|u\|^2_{H^1} \|u\|_2^2 $$
where we used the fact that the first integral after the equality sign is negative semidefinite, and the 1-dimensional Sobolev inequality for the last implication. 
Similarly since
$$ u_{xt} = \partial_x( (a + u^2) u_{xx}) + 3 u^2 u_x $$
we multiply by $u_x$ and integrate to get
$$ \frac12 \partial_t \|u_x\|_2^2 = - \int (a+u^2) u_{xx} u_{xx}  + 3 u^2 u_x^2 \mathrm{d}x \leq 3 \|u\|_{\infty}^2 \|u_x\|_2^2 \leq C \|u\|_{H^1}^4 $$
Combining the two we get that
$$ \partial_t \|u\|_{H^1}^2 \leq C \|u\|_{H^1}^4 $$
which is a Ordinary differential inequality which you can solve. Note that the constant $C$ above only depends on the Sobolev constant in 1 dimension on the circle and not on $a$. This gives you an uniform upperbound to the growth rate of $u_a$ in $H^1$ (at least for small times: there can still be blowup in finite time, but the time of existence is bounded below by something depending only on the $H^1$ norm of the initial data). From this you can grab a weakly converging subsequence. 
