Can I prove a curvilinear relationship when the linear independent variable is not significant I am investigating a curvilinear effect between X and Y by using a hierarchic regression analysis. To test for curvilinear effects, the squared term for X was computed (I mean center also variable X).
In model 1, the control variables were entered. In Model 2, X (linear) was entered. In Model 3, X (quadratic) was entered. 
In Model 2, X linear is significant. When the squared term is entered in Model 3, the quadratic term is significant but the linear term is not.  Does this prove a curvilinear effect? Or is it essential that in Model 3 both (linear and quadratic) are significant? 
When I do not mean center the independent variable, Model 3 stated X linear and X quadratic significant. The problem here is multicollinearity issues.
 A: No, it is not essential that both the linear and quadratic terms be significant. Only the quadratic term need be significant.
In fact, it is important to note that the linear term takes on a somewhat different interpretation in the context of a model that also includes the quadratic term. In such a model, the linear term now represents the slope of the line tangent to x at the y-intercept, that is, the predicted slope of x when and only when x = 0. So a test of the linear term in a model like this is not in general testing the same thing as in a model that just includes the linear term without the quadratic.
A: Think about what significance means. A relationship of the form you suggest can be characterized as 
$Y = a_1X^2+a_2X+b$
and empirically estimated as $\hat{Y}=\alpha_1\hat{X}^2+\alpha_2\hat{X}+\beta+\epsilon$.
What does the significance of an estimate - say, $\alpha_2$ - mean? The significance is Pr(data|H0), and given a probability that is "not significant", what you really do not reject, is the possibility that the coefficient might really be zero.
Does this invalidate the assumption of a curvilinear relationship? Not in my opinion. Rather, it seems to suggest that $a_2$ is really zero.
Consider the following example (written in Stata). 
First we generate some data:
set obs 20000
gen x = uniform()
gen control_one = uniform()
gen control_two = uniform()
drawnorm e, m(0) sd(0.5)

We then specify a new variable X = x^2 and a relationship for an outcome variable Y
gen Y = control_one+control_two+X+e

(This corresponds to a multidimensional curvilinear model in x with coefficient of the linear and constant term equal to zero).
We then run some regressions:
reg Y control_one control_two
reg Y control_one control_two x
reg Y control_one control_two X x

The x term is significant in the second model, but not in the third. As far as I understand, this reflects your experience with real data. 
A: It is actually not essential that either term be significant, but you never prove anything with just a model. 
The given estimates of coefficients are estimates, and they provide evidence. A large coefficient on the quadratic term provides a lot of evidence, a small coefficient provides a little evidence, of a curvilinear relationship. The linear term is irrelevant. It can be positive, negative, near 0 or whatever.
A plot of the data will also provide evidence of a curvilinear relationship. 
Statistical significance means a very precise thing: If, in the population from which this sample was drawn, the effect was really 0, is there a 5% chance that, in a sample of the size that is available, a test statistic this far or farther from 0 would be gotten.
A: As noted, the significance of the curvilinear term stands by itself, regardless of the significance of the linear term in the regression. If the linear term is near zero, then the curve is a U or inverted U if it is significant.  If both terms are significant, the resulting line is more like a hill with an accelerating (or decelerating) slope.
