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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

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