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Hello.

I want to prove the existence of a weak solution to:

Find $u:S^1 \times [0,T) \to \mathbb{R}$ such that $$\frac{\partial u}{\partial t} = u^{n_1}\frac{\partial^2 u}{\partial x^2} + u^{n_2}$$ with $u(x, 0) = u_0(x)$ where $u_0 \in C^{1+\alpha}(S^1)$ (and is non-negative) and $n_1$ and $n_2$ are fixed integers (eg. $n_1 = 2$, $n_2 = 3$, which I fix for now).

Also, $u$ should lie in the space $C^{2+\alpha, 1+\alpha}(S^1 \times [0, T-\epsilon])$ for $\epsilon > 0$.

How to prove short-time existence for this PDE? Every book (Michael Taylor's Nonlinear.., Ladyzhenskaya) I look at has existence results that require strong parabolicity (i.e., require $u^2\eta^2 \geq C\eta^2$ for all non-zero $\eta$, which I cannot say I have).

A reference to where this kind of PDE is proven would be appreciated or a detail of how to prove such PDEs.

PS: This question is on Math.SE for a while but it has not received much attention, so I hope it is OK to post it here as otherwise I would be very stuck. Thanks.

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This is weakly parabolic, so the trick is to pretend that it's hyperbolic rather than parabolic. Look up how to prove local time existence of a nonlinear wave equation using energy estimates. The idea is to prove $L^2$ energy estimates. The parabolic term contributes a negative term to the evolution inequality for the $L^2$ energy, which can be tossed away. –  Deane Yang Jun 12 '12 at 15:52
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2 Answers

up vote 3 down vote accepted

Use the references on strongly parabolic PDE's to show that for each $\epsilon > 0$, you can solve $$ \partial_t u_\epsilon = (\epsilon + |u_\epsilon|^{n_1})\partial_x^2u_\epsilon + |u_\epsilon|^{n_2}. $$ Using energy estimates, get estimates for the time of existence and the $L^2$ Sobolev norms of $u$ that are independent of $\epsilon$. Let $\epsilon \rightarrow 0$.

By the maximum principle, $u$ will be strictly positive for small positive time. Therefore, the equation is strictly parabolic for small positive time. This will imply that $u$ is smooth for small positive time.

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Thanks a lot. Unfortunately I can't even show that that strongly parabolic PDE can be solved. Michael Taylor's theorem requires $u_0 \in H^2$ while I only have only have $C^1$. The same is true for existence results in Ladyzhenskaya (in addition, the results require a lot of requisities.. mainly horrible inequalities). Do you know of any better references? Also, if I may ask, from where did you learn this "trick?" It's difficult to find much literature on this topic. –  user24394 Jun 13 '12 at 20:22
    
A $C^1$ function on a compact manifold is in $H^2$ (which I presume means the first derivative is in $L^2$). –  Deane Yang Jun 13 '12 at 20:49
    
The model case is not a strongly parabolic equation. The model case is, say, $\partial_t u = a(x, u)\partial_x u + b(x,u)$. –  Deane Yang Jun 13 '12 at 20:51
    
I apologise for any misunderstanding. By $H^2$ I refer to the $L^2$ functions with second derivatives also (and not just the first) in $L^2$. So knowing $u_0 \in C^1$ on a compact set only means $u_0 \in H^1$ (first derivative in $L^2$). I was just referring to the PDE you suggested when I said "strongly parabolic". –  user24394 Jun 13 '12 at 21:40
    
I accepted your answer. Still not sure how to do this question but I'll read up on your comment on the original post. –  user24394 Jun 14 '12 at 11:53
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Some years ago we wrote a paper on degenerate parabolic equations (and, actually, systems) which you might find of help. We worked in Sobolev classes there, but I think you can adapt the techniques there to work under your assumptions. The paper is here.

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Thanks for the reply. I will take a look at your paper more closely but on the first glance it appears to use too many things I don't know anything about. –  user24394 Jun 13 '12 at 20:23
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