On bounded and unbounded global solutions of a supercritical semilinear heat equation (Q1434234)

The authors consider the Cauchy problem for the equation \(u_ t = \Delta u +| u| ^ {p-1} u\) with initial data in \(C_0(\mathbb R^n)\) when \(n\geq 11\) and \(p \geq [(n-2)^2-4n+8\sqrt{n-1}]/(n-2)(n-10)\). It is known that positive steady-state solutions of this problem are stable (in a suitable sense) if and only if these conditions are satisfied. One of the many interesting results in this work is that, for suitable initial data, the solution of this problem approaches a continuum of steady-states. In addition, the authors prove the existence of unbounded global solutions.