Well posedness question

1. Jul 19, 2010

MathematicalPhysicist

I have this question I got in the exam, I am pretty much sure I did it right, but I guess something got wrong.

We have the next PDE
$$u_t= \alpha u_{xx}+ \beta u_{xxxx}$$ for t>0 and x is on the whole plane.
the question asks to analyse when the above PDE is well posed (strong or weakly)
for the next cases:
1. alpha and beta >=0
2. alpha <=0 beta >0
3. alpha >0 and beta <=0
4. alpha and beta <0

after using fourier tranform, I get that the energy functional is of the form:
$$E(t)=\int_{-\infty}^{\infty} |u(w,0)|^2 e^{2t(-\alpha w^2 +\beta w^4} dw$$

Now in 3 when alpha is greater than 0 and beta is negative we have that E(t)<=E(0) which means it's well posed, but what of the other options?

Anyone?

Last edited: Jul 20, 2010