Asymptotically self-similar blow-up of semilinear heat equations.(English)Zbl 0585.35051

The authors’ goal is to characterize the asymptotic behaviour of a real solution u(x,t) to $(H)\quad u_ t-\Delta u-| u|^{p- 1}u=0,\quad p>1.$ The main theorem is:
Let u solve (H) on $$Q_ 1=B\times (-1,0)$$, $$B=\{| x| <1\}$$, and assume that $$| u(x,t)|.(-t)^{1/(p-1)}$$ is bounded in $$Q_ 1$$. If $$p\leq (n+2)/(n-2)$$ or if $$n\leq 2$$, then $\lim_{\lambda \to 0}(- \lambda^ 2t)^{1/(p-1)} u(\lambda x,\lambda^ 2t)=\pm (1/(p- 1)^{1/(p-\quad 1)})\quad or\quad zero.$ For each $$c>0$$, the limit above exists uniformly in $$\{(x,t)\in Q_ 1,\quad | x| <c(- t)^{1/2}\}.$$ The analysis is centered on the study of $(H^*)\quad w_ s-\Delta w+(1/2)y \cdot \nabla w+\beta w-| w|^{p-1} w=0,$ $$\beta =1/(p-1)$$, obtained from (H) by replacing $$w(y,s)=(- t)^{\beta} u(x,t),\quad x=(-t)^{1/2} y,\quad t=-\exp (-s).$$ The energy identity for bounded solutions of $$(H^*)$$, $\int^{b}_{a}\int | w_ s|^ 2 \rho dy ds=E[w](a)-E[w](b),$ with $$E[w](s):={1/2} \int | \nabla w|^ 2 \rho dy+{1/2}\beta \int | w|^ 2 \rho dy-((\quad 1/(p+1))\int | w|^{p+1} \rho dy,\quad \rho =\exp (-(1/4)y^ 2),$$ is employed in section 3 to prove that $$\lambda^{2\beta} u(\lambda x,\lambda^ 2t)$$ has a limit as $$\lambda\to 0$$. The characterization of the set of possible limits is done by discussing the self-similar solutions of (H), i.e. the stationary solutions of $$(H^*)$$. Integral inequalities are employed.
A Liouville type theorem for solutions of (H) is presented in Section 4: If $\sup_{x\in R^ n, t<0}| u(x,t)| (-t)^{\beta}<\infty \quad and\quad \limsup_{t\to 0}| u(0,t)| (-t)^{\beta}>0,$ then under the hypotheses of the main theorem $$u(x,t)=\pm \beta^{\beta}(-t)^{-\beta}.$$
The main theorem is presented in Section 5. Section 6 includes remarks and generalizations to systems, allowed by the fact that scaling and integral, energy type inequalities are the main tools for the proof.
The references include 15 items.
Reviewer: J.E.Bouillet

MSC:

 35K55 Nonlinear parabolic equations 35K05 Heat equation 35B40 Asymptotic behavior of solutions to PDEs 35B65 Smoothness and regularity of solutions to PDEs
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References:

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