Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities (Q2717709)

The authors establish the blowup rate for positive radial solutions of the equation \(u_t-\Delta u=u^p+b|\nabla u|^q\), where \(p>1\), \(q\geq 1\), \(b\in\mathbb R\), \(t\in(0,T)\) and the spatial domain is either \(\mathbb R^N\) or a ball in \(\mathbb R^N\). The equation is complemented by the homogeneous Dirichlet boundary conditions in the latter case. The solutions are supposed to be radially decreasing and increasing in time. If \(q<2p/(p+1)\) and \((N-2)p<N+2\) then these solutions satisfy the upper estimate NEWLINE\[NEWLINE\limsup_{t\to T}(T-t)^{1/(p-1)}\|u(t)\|_\infty<\infty.NEWLINE\]NEWLINE Such an estimate is obtained also in the case \(q=2p/(p+1)\) (under some assumptions on \(p\) and \(b\)), \(q>2p(p+1)\) (provided \(b>0\) and \(N=1\)) and for solutions of the parabolic inequality \(u_t-u_{xx}\geq u^p\). A generalization of this result is obtained for inequalities of the form \(u_t- u_{xx}\geq f(u)\). The corresponding lower estimate is obtained for the equation if \(q\leq 2\) and for inequalities of the form \(u_t-\Delta u\leq f(u)\). The proof of the main result is based on estimates for the rescaled function \(v(t,r)=u(t,r\alpha^{-1}(t))/u(t,0)\), where \(\alpha(t)=u(t,0)^{(p-1)/2}\). NEWLINENEWLINENEWLINEFurther results concerning the blowup rate for the equation mentioned above can be found in the survey paper by the first author [Electron. J. Differ. Equ. 2001, Paper No. 20 (2001; Zbl 0982.35054)].