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Stability of the blow-up profile for equations of the type \(u_ t=\Delta u+| u| ^{p-1}u\). (English) Zbl 0872.35049

The authors deal with the following Cauchy problem for semilinear parabolic equations: \[ {(*)}\qquad\qquad u_{t}= \Delta u+|u|^{p-1}u,\quad u(\cdot,0)=u_{0}\in H, \] where \(u(t): x\in {\mathbb{R}}^{N}\to u(x,t)\in {\mathbb{R}}\) and \(H=W^{1,p+1}({\mathbb{R}}^{N})\cap L^{\infty}({\mathbb{R}}^{N}).\) The exponent \(p\) is assumed to be subcritical, i.e., if \(N\geq 3\) then \(1<p<(N+2)/(N-2),\) otherwise \(1<p<\infty.\) Since the local Cauchy problem \((*)\) can be solved in \(H,\) it is a well known fact that either the solution \(u(t)\) exists on \([0,+\infty),\) or on \([0,T)\) with \(T<\infty.\) In the last case, \(u\) blows up at the time \(T: |u|_{H} \to +\infty\quad \text{as}\quad t\to T.\) By now, there are a lot of results concerning the blow-up behavior of solutions to \((*):\) Y. Giga and R. V. Kohn [Commun. Pure Appl. Math. 42, No. 6, 845-884 (1989; Zbl 0703.35020)], S. Filippas and R. V. Kohn [Commun. Pure Appl. Math. 45, No. 7, 821-869 (1992; Zbl 0784.35010)], J. J. L. Velázquez [Trans. Am. Math. Soc. 338, No. 1, 441-464 (1993; Zbl 0803.35015)], M. Berger and R. V. Kohn [Commun. Pure Appl. Math. 41, No. 6, 841-863 (1988; Zbl 0652.65070)], M. A. Herrero and J. J. L. Velázquez [Differ. Integral Equ. 5, No. 5, 973-997 (1992; Zbl 0767.35036), Ann. Inst. Henri Poincaré, Anal. Non Linéaire 10, No. 2, 131-189 (1993; Zbl 0813.35007)], J. Bricmont and A. Kupiainen [Nonlinearity 7, No. 2, 539-575 (1994; Zbl 0857.35018)].
The general goal of the paper under review is to study the profile of the solution near blow-up and the stability of such behavior with respect to the initial data, providing more elementary proofs of the known results cited above. The authors’ approach is based on fine geometric considerations and techniques of a priori estimates. The main results proved can be summarized as follows.
Theorem 1. There exists \(T_{0}>0\) such that for each \(T\in (0,T_{0}],\) and \(g\in H\) with \(|g|_{L^{\infty}} \leq (\log T)^{-2},\) one can find \(d_{0}\in {\mathbb{R}}\) and \(d_{1}\in {\mathbb{R}}^{N}\) such that, for each \(a\in {\mathbb{R}}^{N},\) the problem \((*)\) with initial data \[ {u_{0}(x)=T^{-1/(p-1)}\left\{ f(z)\left( 1+ {{d_{0}+d_{1}z} \over {p-1+ {{(p-1)^{2}}\over {4p}}|z|^{2}}}\right) +g(z)\right\}, z= (x-a)(|\log T|T)^{-1/2},} \] has a unique classical solution \(u(x,t)\) defined on \({\mathbb{R}}^{N} \times [0,T)\) and
(i) \(u\) has one and only one blow-up \(|u(a,t)|\to\infty\) as \(t\to T;\)
(ii) \(\lim_{t\to T}(T-t)^{1/(p-1)}u(a+((T-t)|\log(T-t)|)^{1/2}z,t)=f(z)\) uniformly with respect to \(z\in {\mathbb{R}}^{N},\) with \[ f(z)=\left(p-1+ { {(p-1)^{2}}\over {4p} }|z|^{2}\right)^{-1/ (p-1)}. \] Theorem 2. Let \(\hat u_{0}\) be the initial data from Theorem 1. Denote by \(\hat u(t)\) the solution to \((*)\) with data \(\hat u_{0},\) and by \(\hat T\) and \(\hat a\) its blow-up time and blow-up point, respectively. Then there is a neighborhood \({\mathcal V}\) of \(\hat u_{0}\) in \(H\) with the property: For each \(u_{0}\) in \(\mathcal V,\) \(u(t)\) blows up in a finite time \(T=T(u_{0})\) at only one blow-up point \(a=a(u_{0}),\) where \(u(t)\) is the solution to \((*)\) with initial data \(u_{0}.\) Moreover, \(u(t)\) behaves near \(T(u_{0})\) and \(a(u_{0})\) in an analogous way as \(\hat u(t):\) \[ \lim_{t\to T} (T-t)^{1/(p-1)} u(a+((T-t)|\log(T-t)|)^{1/2}z,t)=f(z) \] uniformly with respect to \(z\in {\mathbb{R}}^{N}\).

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
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References:

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