The blow-up rate for a non-scaling invariant semilinear heat equation

We consider the semilinear heat equation $${\mathrm{\partial }}_{t}u-\mathrm{\Delta }u=f(u),(x,t)\in {\mathbb{R}}^N\times [0,T),(1)$$ with $f(u)=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u{\log }^a(2+u^2)$, where $p>1$ is Sobolev subcritical and $a\in $\mathbb{R}$$. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely $u^{\mathrm{\prime }}=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u{\log }^a(2+u^2)$. In other terms, all blow-up solutions in the Sobolev subcritical range are Type I solutions. Up to our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous.