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-linearityof the form $|u|^q$, where $u=u(x,t)$ for $x\in\mathbb{R}^3$ and $t\ge 0$. We construct a fundamental solution for thelinear part of the equation and use it to obtain analogues of Green's third formula for elliptic operators, firstin a bounded domain and then in unbounded domains. We derive an integral equation for classical solutions of theCauchy problem. A separate study of this equation yields that it has a unique inextensible-in-time solutionin 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'snon-linear capacity method to show that the Cauchy problem has no local-in-time weak solutions for a large class ofinitial functions. When $q\in(3,4]$, this method enables us to prove that the Cauchy problemhas no global-in-time weak solutions for a large class of initial functions.

blow-up, non-linear equations of Sobolev type, blow-up, local solubility, non-linear capacity, bounds for the blow-up time