## 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

### Citations:

Zbl 0813.35009; Zbl 0803.35015
Full Text: