Blow-up of positive solutions of semilinear heat equations. (English) Zbl 0576.35068

The following problem is discussed:
\(u_ t=\Delta u+f(u)\) in \(\Omega\) \(\times (0,T)\), \(f\in C^ 1\), \(f(s)>0\) if \(s>0\), \(\Omega \subset R^ n\); \(u(x,0)=\phi (x)\) if \(x\in \Omega\), \(\phi \in C^ 1({\bar \Omega})\), \(\phi\geq 0\), \(\phi =0\) on \(\partial \Omega\), \(u(x,t)=0\), \(x\in \partial \Omega\), \(0<t<T.\)
As \(U(t)=\max_{x\in \Omega} u(x,t)\) grows with t, it is assumed that \(T<\infty\) is the supremum of all \(\sigma\) such that the solution to the problem above exists for \(t<\sigma\), and that \(U(T-)=+\infty:x\in \Omega\) is a blow-up point if there is \(\{(x_ mt_ m)\}\), \(t_ m\uparrow T\), \(x_ m\to X\), and \(u(x_ m,t_ m)\to \infty\), \(m\to \infty.\)
A partial outline of the results can be divided into:
Case (i): \(\Omega\) is a ball, u(.,t) are radial functions, \(\phi_ r\leq 0\). The authors prove that the only blow-up point is \(x=0\). For the case \(f(u)=(u+\lambda)^ p\), \(p>1\), \(\lambda\geq 0\), they obtain: \(| u(r,t)| \leq C/r^{2/(\lambda -1)}\), any \(\gamma <p\), \(\lim_{t\to T} \sup \| u(.,t)\|_{L^ q(\Omega)}<\infty\), \(q<n(p-1)/2\), \(\lim_{t\to T} \inf \| u(.,t)\|_{L^ q(\Omega)}=\infty\), \(q>n(p-1)/2\); if moreover \(\Delta \phi +f(\phi)\geq 0\) and \(n=1,2\); or \(n\geq 3\) and \(p\leq (n+2)/n-2\), \((T-t)^{1/(p-1)}u(r,t)\to (1/p- 1)^{1/p-1}\) as \(t\uparrow T\) provided \(r\leq C(T-t)^{1/2}\), some \(C>0.\)
Case (ii): Non symmetric, \(\Omega\) is a convex domain: the blow-up points lie in a compact subset of \(\Omega\). For the particular \(f(u)=(u+\lambda)^ p\), u(x,t)\(\leq^ C/(T-t)^{1/(p-1)}\), all \(x\in \Omega\), and \(\lim_{t\to T} \inf \| u(.,t)\|_{L^ q(\Omega)}=\infty\) if \(q>n(p-1)/2.\)
In the one-dimensional case \(n=1\), if \(\phi\) ’ changes sign just once, then the solution blow-up at a single point. The authors extend some of the results to the case of a boundary condition \(\partial u/\partial \nu +\beta u=0\), \(x\in \partial \Omega\), \(0<t<T\), \(\beta\geq 0\) and \(\nu\) the outward normal vector.
Reviewer: J.E.Bouillet


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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