# Parabolic PDE existence/uniqueness

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.

Thanks

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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 $T=T(u_0)$.