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Refined uniform estimates at blow-up and applications for nonlinear heat equations. (English) Zbl 0926.35024

Consider the nonlinear heat equation \(u_t=\Delta u+u^p\) on \(\mathbb R^N\times[0,T)\) with the initial condition \(u(0)=u_0\geq 0\), where \(1<p<(N+2)/(N-2)\) and \(u_0\in H^1(\mathbb R^n)\), and the solutions \(u\) of this equation which blow-up in a finite time \(T>0\), i.e. for which \(\lim_{t\to T}\| u(t)\|_{H^1}=+\infty\) and \(\lim_{t\to T}\| u(t)\|_{L^\infty}=+\infty\). The authors study the blow-up behaviour of \(u\) as \(t\to T\). The self-similar transformation \(y=(x-a)(T-t)^{-1/2}\), \(s=-\log(T-t)\), leads to the equation \(\partial w/\partial s=\Delta w-\frac 12y\nabla w-(p-1)^{-1}w+w^p\) which is satisfied by \(w(y,s)=(T-t)^{1/(p-1)}u(x,t)\). The behaviour of \(u(t)\) near \((x_0,T)\) where \(x_0\) is a blow-up point is thus transformed to the equivalent problem of the long-time behaviour of \(w_{x_0}\) as \(s\to\infty\). The authors first prove refined \(L^\infty\) estimates for \(w\), \(\nabla w\), \(\nabla^2w\) and \(\nabla^3w\) as \(s\to\infty\). They use this estimates and the classification of the behaviour of \(w(y,s)\) for bounded \(| y|\) obtained by S. Filippas and W. Liu [Ann. Inst. Henri Poincaré, Anal. Non Lineaire 10, 821-869 (1992)] and by J. J. L. Velázquez, [Commun. Partial Differ. Equations 17, No. 9/10, 1567-1596 (1992; Zbl 0813.35009), Trans. Am. Math. Soc. 338, No. 1, 441-464 (1993; Zbl 0803.35015)], to establish a blow-up profile classification theorem in the variable \(z=y/\sqrt s\). Finally, they show that in the case of a solution \(u(t)\) which blows up at some \(x_0\in\mathbb R^n\) in a non-degenerate way, three different notions of profile in the scales are equivalent: \(| y|\) bounded, \(z=| y|/\sqrt s\) bounded and \(| x-x_0|\) small.

MSC:

35B45 A priori estimates in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K15 Initial value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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