Global unbounded solutions of the Fujita equation in the intermediate range (Q464141)

The authors consider the Cauchy problem for the Fujita equation \[ u_t = \Delta u +u^p, \quad x \in \mathbb{R}^N,\, t>0, \] with initial condition \[ u(x,0) = u_0 (x), \quad x \in \mathbb{R}^N, \] where \(p>1\) and \(N \geq 3\). Let \[ p_S =\frac{N+2}{N-2} \] be the Sobolev critical exponent and \(p_{\mathrm{JL}}\) be the Joseph-Lundgren exponent defined as \[ p_{\mathrm{JL}}=\frac{(N-2)^2-4N+8\sqrt{N-1}}{(N-2)(N-10)} \quad \text{if } N>10 \] and \(p_{\mathrm{JL}}=\infty\) if \(N \leq 10\). Then, the main result of the paper is that if \(p_s <p <p_{\mathrm{JL}}\), then there exists a radially symmetric function \(u_0 \in C(\mathbb{R}^N)\) such that the solution of the Fujita equation is global and unbounded.