Global unbounded solutions of the Fujita equation in the intermediate range

We consider the Fujita equation $$u_t={\varDelta } u+u^{p}$$ut=Δu+up on $${\mathbb {R}}^N$$RN with $$N\ge 3$$N≥3. We prove the existence of global, unbounded solutions for $$p_S10,\\ \,\infty &{} \text {if } N\le 10. \end{array}\right. \end{aligned}$$pJL:=(N-2)2-4N+8N-1(N-2)(N-10)ifN>10,∞ifN≤10.Previously, it was known that global, unbounded solutions exist for $$p\ge p_{JL}$$p≥pJL, whereas for $$p

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