Well posedness question

MathematicalPhysicist

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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
[tex]u_t= \alpha u_{xx}+ \beta u_{xxxx}[/tex] 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:
[tex]E(t)=\int_{-\infty}^{\infty} |u(w,0)|^2 e^{2t(-\alpha w^2 +\beta w^4} dw[/tex]

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?
 
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