Making the accusation of willfulness is trivial; proving it at a trial is more difficult. Simple failure to search is not sufficient basis for a finding of willfulness. Some showing of actual knowledge of infringement or likely infringement is required. Notification (whether by the patent holder or any other party) is evidence of knowledge.
As with so many of the answers you will see on this site, even this explanation comes with a large helping of "it depends." For example, one may avoid searching because he knows (somehow) that relevant patented technology is out there. If that can be shown, then the failure to search supports an argument for willfulness. But even then, if the infringer was utterly ignorant of the patent claims being asserted, a jury might well find that the specific infringement was not willful.
The determination of willfulness is up to the finder of fact - either the judge (in a bench trial) or a jury. The finding will always depend on the specific set of facts presented at trial. "Facts" as reported in the news or assumed by the public are not the same sort of creature as a "fact" admitted into evidence in a trial. All facts are weighed by the finder of fact to arrive at a verdict. Different juries (and judges) could well reach different conclusions based on the same sets of facts. Litigation is very expensive, so "slam dunk" questions rarely make it into a trial.
Something else to be aware of: a jury's (or judge's) finding of willfulness is one condition under which the judge can enhance a monetary award, but the judge is not required to do so. The willfulness premium is limited at the top end, and the lower bound is zero. The judge has discretion to select an enhanced value that seems most appropriate in all of the circumstances - which may include mitigating factors that reduce or preclude additional compensation even when the infringer is fully aware of the patent before initiating the infringement.