On critical exponents for weak solutions of the Cauchy problem for a non-linear equation of composite type

We consider the Cauchy problem for a model partial differential equation of third order with non-linearity of the form $|u|^q$, where $u=u(x,t)$ for $x\in\mathbb{R}^3$ and $t\geqslant 0$. We construct a fundamental solution for the linear part of the equation and use it to obtain analogues of Green's third formula for elliptic operators, first in a bounded domain and then in unbounded domains. We derive an integral equation for classical solutions of the Cauchy problem. A separate study of this equation yields that it has a unique inextensible-in-time solution in weighted spaces of bounded and continuous functions. We prove that every solution of the integral equation is a local-in-time weak solution of the Cauchy problem provided that $q>3$. When $q\in(1,3]$, we use Pokhozhaev's non-linear capacity method to show that the Cauchy problem has no local-in-time weak solutions for a large class of initial functions. When $q\in(3,4]$, this method enables us to prove that the Cauchy problem has no global-in-time weak solutions for a large class of initial functions.