Initial blow-up rates and universal bounds for nonlinear heat equations. (English) Zbl 1044.35027

Summary: We establish a universal upper bound on the initial blow-up rate for all positive classical solutions of the Dirichlet problem for the nonlinear heat equation \[ u_t=\Delta u+ u^p\quad\text{on }(0,T)\times\Omega, \] where \(p> 1\) and \(\Omega\) is a smoothly bounded domain in \(\mathbb{R}^N\). Namely, we show that \[ \| u(t)\|_\infty\leq C(p,\Omega, T) t^{-\alpha}\quad\text{on }(0,T/2] \] for some \(\alpha= \alpha(N,p)> 0\). This is proved for a supcritical \(p\) i.e., \(p< (N+ 2)/(N- 2))\) if \(N\leq 4\) (and under a stronger assumption on \(p\) if \(N\geq 5\)). As a consequence, we improve the known results on universal bounds for global solutions. Furthermore, if \(p< (N+ 3)/(N+ 1)\), then we may take \(\alpha= (N +1)/2\) and we show that this value of a is optimal. Interestingly, the rate can be faster than the previously known, maximal initial blow-up rate of the Cauchy problem. Applications to universal blow-up estimates at \(t= T\) are given. The Neumann problem is also considered and we obtain the estimate on the initial rate for all subcritical \(p\) up to dimension \(N= 6\).


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B45 A priori estimates in context of PDEs
35B33 Critical exponents in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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