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Consider the parabolic PDE: $$\frac{\partial u}{\partial t} = u^2\frac{\partial^2u}{\partial x^2} + u^3$$ with some initial condition.

Apparently this is a straightforward parabolic PDE in which I can apply standard results to prove short term existence and uniquness. Can someone tell/refer me to these standard results please? The equation is non-linear and I haven't seen any theory for non-linear results.


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up vote 3 down vote accepted

A standard theory can be found in, e.g., Ladyženskaja, O. A.; Solonnikov, V. A.; N. N., Ural'ceva (1968), Linear and quasi-linear equations of parabolic type.

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Thanks a lot... – blahb Jun 7 '12 at 12:20

I will assume that the initial data is smooth. First we define the appropriate energy (this depends on oyur problem, but usually it is enough to take some high order Sobolev norm). For instance we take $E[u]=\|u\|_{H^1}$. Now we obtain an a priori bound for this energy. Multiplying the equation by u and integrating by parts we get $$ \frac{1}{2}\frac{d}{dt}\|u(t)\|_{L^2}^2=\int u u_t=\int u^3 u_{xx}+u^4=\int -3(u_x)^2u^2+u^4 $$ $$ \leq \|u\|_{L^2}^2\|u\|_{L^\infty}^2\leq C\|u\|_{H^1}^4 \text{ (by Sobolev embedding)}. $$ Now we take one derivative and follow the same way: $$ \frac{1}{2}\frac{d}{dt}\|u_x(t)\|_{L^2}^2=\int u_x u_{xt}=\int 2u(u_x)^2u_{xx}+u^2u_xu_{xxx}+3u^2(u_x)^2= $$ We integrate by parts in the second term and we get $$ \int u^2u_xu_{xxx}=-\int u^2(u_{xx})^2-2u(u_x)^2u_{xx}. $$ Inserting this in the previous equation we get $$ \frac{1}{2}\frac{d}{dt}\|u_x(t)\|_{L^2}^2\leq3\|u\|_{L^\infty}^2\|u_x\|_{L^2}^2\leq C\|u\|_{H^1}^4. $$ Collecting both estimates, we get $$ \frac{d}{dt}\|u\|_{H^1}\leq C\|u\|_{H^1}^3, $$ and we can use Gronwall inequality to get $$ \sup_{t\in[0,T)}\|u(t)\|_{H^1}\leq C(u_0), $$ for an explicit time $T=T(u_0)$.

To conclude the argument you should regularize the system. Typically you should take some convolutions with the usual mollifier to have that the derivative operator is bounded in this Sobolev spaces. If you take the correct regularized system the same bounds hold and you have enough compactness to ensure the existence of a smooth limit which is also a solution.

All these techniques are quite standart and I refer you the book of A.Majda and A. Bertozzi "Vorticity and incompressible flow" (Chapter 3).

I hope that this is useful for you.

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