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

We are concerned with blow-up mechanisms in a semilinear heat equation: document uₜ = u + |x|^2a uᵖ, x RN, \, t>0, document where document p>1 document and document a>-1 document are constants. As for the Fujita equation, which corresponds to document a = 0 document, a well-known result due to M. A. Herrero and J. J. L. Velázquez, C. R. Acad. Sci. Paris Sér. I Math. (1994), states that if document N 11 document and document p> 1 + 4/ (N-4-2N-1) document, then there exist radial blow-up solutions document u, ₇ₕ (x, t) document, document N document, such that document ₓ ₓ (T-t) ^1/ (p-1) \| u, ₇ₕ (, t) \|₋^ (RN) =, document where document T 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 document a = 0 document. As a consequence, we obtain an example of solutions that blow up at document x = 0 document, the zero point of potential document |x|^2a document with document a>0 document, and behave in non-self-similar manner for document N > 10 + 8a document. This last result is in contrast to backward self-similar solutions previously obtained for document N, which blow up at document x = 0 document.