Why don't I get the p-value doing an ANOVA test? I want to regress two series (one big series divided in half) with the mean of the big series. 
I do that because I would like to "investigate" the relationship between those two subseries and the mean.
Does this make any sense for you?
When running the code below, I don't understand why I don't get p-value:
> x  = rnorm(200)
> m  = mean(x) 
> anova(lm(rep(m, 100) ~ x[1:100]), lm(rep(m, 100) ~ x[101:200])) 
Analysis of Variance Table

Model 1: rep(m, 100) ~ x[1:100]
Model 2: rep(m, 100) ~ x[101:200]
  Res.Df RSS Df Sum of Sq F Pr(>F)
1     98   0                      
2     98   0  0         0   

 A: There is no variance in the response $y$, so nothing to explain using the covariate $x$. If you look at the two individual models, the covariate x explains no variance once you estimate an intercept:
> summary(lm(rep(m, 100) ~ x[1:100]))

Call:
lm(formula = rep(m, 100) ~ x[1:100])

Residuals:
   Min     1Q Median     3Q    Max 
     0      0      0      0      0 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.02633    0.00000     Inf   <2e-16 ***
x[1:100]     0.00000    0.00000      NA       NA    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 0 on 98 degrees of freedom
Multiple R-squared:   NaN,  Adjusted R-squared:   NaN 
F-statistic:   NaN on 1 and 98 DF,  p-value: NA

As both models contain a covariate that explains nothing in the response there is no difference in the residual sums of squares of the two models, hence no ratio (no $F$) and hence no $p$-value.
Note also that you are abusing the $F$-ratio test here, which would assume that the two models are nested and fitted to the same data, which your models aren't.
