×

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
Full Text: DOI

References:

[1] Federer, Bull. Amer. Math. Soc. 76 pp 767– (1970)
[2] Partial Differential Equations of Parabolic Type, Prentice-Hall, New Jersey, 1964.
[3] Fujita, J. Fac. Sci. Univ. Tokyo, Sect. I 13 pp 109– (1966)
[4] Fujita, Proc. Symp. Pure Math. 18
[5] Amer. Math. Soc., 1968, pp. 131–161.
[6] Gidas, Comm. Pure Appl. Math. 34 pp 525– (1981)
[7] On elliptic equations related to self-Similar solutions of nonlinear heat equations, in preparation.
[8] Haraux, Indiana Univ. Math. J. 31 pp 167– (1982)
[9] , and , Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, AMS, 1968.
[10] Leray, Acta Math. 63 pp 193– (1934)
[11] , and , Focusing singularity for the nonlinear Schrödinger equation, in Nonlinear Partial Differential Equations, Proc. of the U. S.-Japan Seminar, Tokyo, 1982; , , eds; North Holland Math. Studies No. 81, 1984.
[12] Morse Theory, Princeton University Press, New Jersey, 1963. · Zbl 0108.10401 · doi:10.1515/9781400881802
[13] Weissler, Israel J. Math. 38 pp 29– (1981)
[14] Weissler, J. Differential Equations 55 pp 204– (1984)
[15] Weissler, Comm. Pure Appl. Math. 38 pp 291– (1985)
[16] Friedman, Indiana Univ. Math. J.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.