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

Citations:

Zbl 0655.35042; Zbl 0585.35051
Full Text:

References:

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