Render multiple components with f:ajax
You can render multiple components with single f:ajax. Just make sure all individual components you want to update have an id. In your sample it would be something like:
<f:ajax execute="search_form" render="linear1 linear2"/>Where the IDs need to be separated by just whitespace like linear1 linear2 and not commaseparated like linear1, linear2 (that works only in p:ajax).
See also:
For a4j jsf use ',':<a4j:support event="onchange" reRender="parent,child1,child2" />