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A description of self-similar blow-up for dimensions n\(\geq 3\). (English) Zbl 0726.35018

The authors consider the initial-boundary value problem for semilinear parabolic equations \(u_ t=\Delta u+f(u)\) in \(\Omega\times (0,T)\), \(u(z,0)=\phi (z)\) in \(\Omega\), \(u=0\) on \(\partial \Omega \times (0,T)\), where \(\Omega =\{z\in {\mathbb{R}}^ N:| z| <R\}\), \(R=const>0\), \(\phi =\phi (| z|)\) is nonnegative, nonincreasing (with respect to \(| z|)\) and \(\Delta \phi +f(\phi)\geq 0\) in \(\Omega\). The two nonlinearities considered are \(f(u)=e^ u\) or \(f(u)=u^ p\), \(p=const>1\). It is assumed that the solution \(u=u(z,t)\geq 0\) blows up in finite time \(T>0\). The authors obtain the asymptotic behaviour of u(z,t) near the point \(z=0\), \(t=T^-\). In the case \(f(u)=e^ u\) it is obtained that, for \(n\geq 3\), \(u(z,t)+\ln (T-t)\to 0\) uniformly on any set \(\{(z,t):\;| z| \leq C(T-t)^{1/2}\}\) for arbitrary \(C=const\geq 0\) as \(t\to T^-\) (the same result for \(N=1\) or 2 was obtained by the authors and A. Bressan [Indiana Univ. Math. J. 36, 295-305 (1987; Zbl 0655.35042)]). If \(f(u)=u^ p\), then for \(N\geq 3\) and \(p>N/(N-2)\) the solution satisfies \((T-t)^ mu(z,t)\to m^ m,\quad m=1/(p-1),\) uniformly on \(\{(z,t):\;| z| \leq C(T-t)^{1/2}\}\) for any \(C=const\geq 0\) as \(t\to T^-\) (the same result for \(p\leq (N+2)/(N- 2)_+\) was obtained by Y. Giga and R. V. Kohn [Commun. Pure Appl. Math. 38, 297-319 (1985; Zbl 0585.35051)] and for \(N=1\) by the reviewer and S. A. Posashkov [Akad. Nauk SSSR, Inst. Prikl. Mat., Preprint No.97 (1985)]). The condition \(\Delta \phi +f(\phi)\geq 0\) in \(\Omega\) is essential for these results. The stabilization as \(t\to T^- \), to a nonconstant self-similar solution is possible without the condition \(u_ t\geq 0\) in \(\Omega\times (0,T)\). The main new idea in the proofs is the intersection comparison with the singular stationary solutions of equations considered which are unbounded near \(z=0\). For example, for \(f(u)=u^ p\) the self-similar stationary solution \(u(z)=\{- 4m(m+(2-N)/2)/| z|^{-2}\}^ m\) exists for the case \(N\geq 3\) and \(p>N/(N-2)\).

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
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References:

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