Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities
Tom 88 / 2001
Streszczenie
Consider the nonlinear heat equation (E): $u_t-{\mit \Delta } u=|u|^{p-1}u+b|\nabla u|^q$. We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C_1 (T-t)^{-1/(p-1)} \leq \| u(t)\| _\infty \leq C_2 (T-t)^{-1/(p-1)}$. Also, as an application of our method, we obtain the same upper estimate if $u$ only satisfies the nonlinear parabolic inequality $u_t-u_{xx}\geq u^p$. More general inequalities of the form $u_t-u_{xx}\geq f(u)$ with, for instance, $f(u)=(1+u)\mathop {\rm log}\nolimits ^p(1+u)$ 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 $\dot v\geq f(v)$.
