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Consider the nonlinear heat equation (E): ut − ∆u = |u| u + b|∇u| . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates C1(T − t) −1/(p−1) ≤ ‖u(t)‖∞ ≤ C2(T − t) . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality ut − uxx ≥ u . More general inequalities of the form ut − uxx ≥ f(u) with, for instance, f(u) = (1 + u) log (1 + u) are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions of the ordinary differential inequality v̇ ≥ f(v).

Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities