Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents (Q1868907)

Let \(\Omega\) be the unit ball in \(\mathbb{R}^N\), \(N\geq 3\) and \(p>1\). Consider positive radial solutions of the problem \(u_t-\Delta u=u^p\) in \(\Omega\times(0,\infty)\), \(u=0\) on \(\partial\Omega\times(0,\infty)\). If \(p=(N+2)/(N-2)\) then this problem admits global classical solutions which are not uniformly bounded (such phenomenon does not occur if \(p\neq(N+2)/(N-2)\)). The authors show that these solutions exhibit the following asymptotic behavior as \(t\to\infty\): \[ \begin{alignedat}{2} \ln\|u(\cdot,t)\|_\infty &= (\pi^2/4)t(1+o(1)) &\quad &\text{for } N=3,\\ \ln\|u(\cdot,t)\|_\infty &= 2\sqrt{t}(1+o(1)) &\quad &\text{for } N=4,\\ \|u(\cdot,t)\|_\infty &= \gamma_0(N)t^{(N-2)/2(N-4)}(1+o(1)) &\quad &\text{for } N\geq 5.\end{alignedat} \] An explicit value of \(\gamma_0(N)\) is calculated in a formal way. The authors also consider some quasilinear and fully nonlinear parabolic equations with scaling invariant operators. The proofs are based on matched asymptotic expansion techniques.