On the Cauchy problem for the equation \(u'-\Delta u=u| u| ^{\gamma -1}\) in the supercritical case
We prove global existence of classical solutions to the above problem whenever the initial function is continuous and can be enclosed between a sub- and a supersolution of the corresponding stationary problem, where the latter are assumed to be smooth bounded functions lying in some \(L^ p\). (In the title, ``supercritical'' means \(\gamma \geq (n+2)/(n-2)\) where n is space dimension.) The method of proof combines local existence results from semilinear parabolic theory with results on the corresponding stationary equation via the use of the Phragmén-Lindelöf principle for parabolic equations in unbounded domains. Examples are provided, illustrating the extension of previously known global existence results.
scientific article; zbMATH DE number 4115131
scientific article