Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity

Abstract

We are concerned with blow-up mechanisms in a semilinear heat equation: \\begin{document}$ u_t = \\Delta u + |x|^{2a} u^p , \\quad x \\in \\textbf{R}^N , \\, t>0, $\\end{document} where \\begin{document}$ p>1 $\\end{document} and \\begin{document}$ a>-1 $\\end{document} are constants. As for the Fujita equation, which corresponds to \\begin{document}$ a = 0 $\\end{document} , a well-known result due to M. A. Herrero and J. J. L. Velazquez, C. R. Acad. Sci. Paris Ser. I Math. (1994), states that if \\begin{document}$ N\\geq 11 $\\end{document} and \\begin{document}$ p> 1 + 4/(N-4-2\\sqrt{N-1}) $\\end{document} , then there exist radial blow-up solutions \\begin{document}$ u_{\\ell, {\\rm HV}}(x, t) $\\end{document} , \\begin{document}$ \\ell \\in \\bf{N} $\\end{document} , such that \\begin{document}$ \\lim\\limits_{t\\to T} \\left( T-t \\right)^{1/(p-1)} \\| u_{\\ell, {\\rm HV}}(\\cdot, t) \\|_{L^{\\infty}(\\textbf{R}^N )} = \\infty, $\\end{document} where \\begin{document}$ T $\\end{document} is the blow-up time. We revisit the idea of their construction and obtain refined estimates for such solutions by the techniques developed in recent works and elaborate estimates of the heat semigroup in backward similarity variables. Our method is naturally extended to the case \\begin{document}$ a\\not = 0 $\\end{document} . As a consequence, we obtain an example of solutions that blow up at \\begin{document}$ x = 0 $\\end{document} , the zero point of potential \\begin{document}$ |x|^{2a} $\\end{document} with \\begin{document}$ a>0 $\\end{document} , and behave in non-self-similar manner for \\begin{document}$ N > 10 + 8a $\\end{document} . This last result is in contrast to backward self-similar solutions previously obtained for \\begin{document}$ N , which blow up at \\begin{document}$ x = 0 $\\end{document} .