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Positive solution to semilinear parabolic equation associated with critical Sobolev exponent. (English) Zbl 1277.35073

The authors consider the semilinear parabolic equation \(u_t-\Delta u=|u|^{p-1}u\) in \(\mathbb{R}^N\times (0,T)\) with initial condition \(u(x,0)=u_0(x)\) in \(\mathbb{R}^N\) and critical exponent \(p=(N+2)/(N-2)\), for \(N\geq 3\).
The paper is mainly concerned with the existence of a threshold modulus for the blowup of positive solutions. Take an appropriate profile function \(\psi\), \(0\not\equiv\psi(x)\geq 0\), and a modulus \(\lambda>0\). Let \(u_0=\lambda \psi\) and \(T_{\lambda}\in (0,+\infty]\) be the existence time of the solution \(u_{\lambda}\). Then the following results hold: There is a \(\bar \lambda>0\) such that:
If \(\lambda\in (0,\bar \lambda)\), then \(T_{\lambda}=+\infty\) and \(\|u_{\lambda}(t)\|\sim t^{-N/2}\) as \(t\uparrow +\infty\).
If \(\lambda=\bar \lambda\), then either \(T_{\lambda}<+\infty\) or \(T_{\lambda}=+\infty\) and \(\displaystyle \lim_{t\uparrow +\infty}t^{N-2\over 4}\|u_{\lambda}(t)\|_{\infty}=+\infty\).
If \(\lambda>\bar \lambda\), then \(T_{\lambda}<+\infty\).

MSC:

35B44 Blow-up in context of PDEs
35K58 Semilinear parabolic equations
35B33 Critical exponents in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35B09 Positive solutions to PDEs
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