Uniform boundedness and blow-up rate of solutions in non-scale-invariant superlinear heat equations

For superlinear heat equations with the Dirichlet boundary condition, the $L^{\mathrm{\infty }}$ estimates of radially symmetric solutions are studied. In particular, the uniform boundedness of global solutions and the non-existence of solutions with type II blow-up are proved. For the space dimension greater than $9$, our results are shown under the condition that an exponent representing the growth rate of a nonlinear term is between the Sobolev exponent and the Joseph-Lundgren exponent. In the case where the space dimension is greater than $2$ and smaller than $10$, our results are applicable for nonlinear terms growing extremely faster than the exponential function.