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The question of finite time singularity formation vs. global existence forsolutions to the generalized Constantin-Lax-Majda equation is studied, withparticular emphasis on the influence of a parameter $a$ which controls thestrength of advection. For solutions on the infinite domain we find a newcritical value $a_c=0.6890665337007457\ldots$ below which there is finite timesingularity formation % if we write a=a_c=0.6890665337007457\ldots here then\ldots doesn't fit into the line that has a form of self-similar collapse, withthe spatial extent of blow-up shrinking to zero. We find a new exact analyticalcollapsing solution at $a=1/2$ as well as prove the existence of a leadingorder complex singularity for general values of $a$ in the analyticalcontinuation of the solution from the real spatial coordinate into the complexplane. This singularity controls the leading order behaviour of the collapsingsolution. For $a_c<a\leq1$, we find a blow-up solution in which the spatialextent of the blow-up region expands infinitely fast at the singularity time.For $a \gtrsim 1.3$, we find that the solution exists globally withexponential-like growth of the solution amplitude in time. We also consider thecase of periodic boundary conditions. We identify collapsing solutions for$a<a_c$ which are similar to the real line case. For $a_c<a\le0.95$, we findnew blow-up solutions which are neither expanding nor collapsing. For $ a\ge1,$ we identify a global existence of solutions.

Collapse vs. blow up and global existence in the generalized Constantin-Lax-Majda equation