ON TWO CRITICAL EXPONENTS FOR SOLUTIONS OF THE CAUCHY PROBLEM FOR A MODEL NONLINEAR COMPOSITE TYPE EQUATION WITH GRADIENT NONLINEARITY

The report considers the Cauchy problem for a nonlinear composite type equation with nonlinearity of the form the modulus of the gradient raised to the power q>1. It is proved that there exist two critical exponents q_1 and q_2 such that for $1<q<=q_1$ there is no weak local solution in time, for $q_1<q<=q_2$ a unique weak local solution in time exists, but any non-trivial solution collapses in finite time. Finally, for $q>q_2$ there exists a unique weak global solution in time for a sufficiently small initial function.